Solution Opal Quartz.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy.stats import Normal, density, E, Variance\n",
    "from sympy.plotting import plot\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "rho_opal, rho_quartz, sigma1, sigma2, rho = sympy.symbols('rho_opal, rho_quartz, sigma1, sigma2, rho', real=True, positive=True)\n",
    "x1_obs, x2_obs = sympy.symbols('x1_obs, x2_obs', real=True)\n",
    "qvalue_opal = sympy.symbols('qvalue_opal', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "model1 = Normal('X1', rho, sigma1)\n",
    "model2 = Normal('X2', rho, sigma2)\n",
    "model = sympy.simplify(density(model1)(x1_obs) * density(model2)(x2_obs))\n",
    "\n",
    "model_opal = model.subs({rho: rho_opal})\n",
    "model_quartz = model.subs({rho: rho_quartz})\n",
    "\n",
    "nll_opal = sympy.simplify(-sympy.log(model_opal))\n",
    "nll_quartz = sympy.simplify(-sympy.log(model_quartz))\n",
    "\n",
    "qvalue_obs = sympy.factor(2 * (nll_opal - nll_quartz))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\frac{e^{- \\frac{\\left(\\rho_{opal} - x_{2 obs}\\right)^{2}}{2 \\sigma_{2}^{2}} - \\frac{\\left(\\rho_{opal} - x_{1 obs}\\right)^{2}}{2 \\sigma_{1}^{2}}}}{2 \\pi \\sigma_{1} \\sigma_{2}}$$"
      ],
      "text/plain": [
       "                     2                     2\n",
       "   (ρ_opal - x_2_obs)    (ρ_opal - x_1_obs) \n",
       " - ─────────────────── - ───────────────────\n",
       "              2                     2       \n",
       "          2⋅σ₂                  2⋅σ₁        \n",
       "ℯ                                           \n",
       "────────────────────────────────────────────\n",
       "                 2⋅π⋅σ₁⋅σ₂                  "
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_opal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left(\\rho_{opal} - \\rho_{quartz}\\right) \\left(\\frac{\\rho_{opal}}{\\sigma_{2}^{2}} + \\frac{\\rho_{opal}}{\\sigma_{1}^{2}} + \\frac{\\rho_{quartz}}{\\sigma_{2}^{2}} + \\frac{\\rho_{quartz}}{\\sigma_{1}^{2}} - \\frac{2 x_{2 obs}}{\\sigma_{2}^{2}} - \\frac{2 x_{1 obs}}{\\sigma_{1}^{2}}\\right)$$"
      ],
      "text/plain": [
       "                    ⎛ρ_opal   ρ_opal   ρ_quartz   ρ_quartz   2⋅x_2_obs   2⋅x_1\n",
       "(ρ_opal - ρ_quartz)⋅⎜────── + ────── + ──────── + ──────── - ───────── - ─────\n",
       "                    ⎜   2        2         2          2           2           \n",
       "                    ⎝ σ₂       σ₁        σ₂         σ₁          σ₂          σ₁\n",
       "\n",
       "_obs⎞\n",
       "────⎟\n",
       "2   ⎟\n",
       "    ⎠"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qvalue_obs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAmUAAAAyBAMAAADvtbFyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\left(\\rho_{opal} - \\rho_{quartz}\\right) \\left(\\frac{\\rho_{opal}}{\\sigma_{2}^{2}} + \\frac{\\rho_{opal}}{\\sigma_{1}^{2}} + \\frac{\\rho_{quartz}}{\\sigma_{2}^{2}} + \\frac{\\rho_{quartz}}{\\sigma_{1}^{2}} - \\frac{2 X_{2}}{\\sigma_{2}^{2}} - \\frac{2 X_{1}}{\\sigma_{1}^{2}}\\right)$$"
      ],
      "text/plain": [
       "                    ⎛ρ_opal   ρ_opal   ρ_quartz   ρ_quartz   2⋅X₂   2⋅X₁⎞\n",
       "(ρ_opal - ρ_quartz)⋅⎜────── + ────── + ──────── + ──────── - ──── - ────⎟\n",
       "                    ⎜   2        2         2          2        2      2 ⎟\n",
       "                    ⎝ σ₂       σ₁        σ₂         σ₁       σ₂     σ₁  ⎠"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X1_opal = model1.subs({rho: rho_opal})\n",
    "X2_opal = model2.subs({rho: rho_opal})\n",
    "X1_quartz = model1.subs({rho: rho_quartz})\n",
    "X2_quartz = model2.subs({rho: rho_quartz})\n",
    "\n",
    "q_random_variable_opal = qvalue_obs.subs({x1_obs: X1_opal, x2_obs:X2_opal})\n",
    "q_random_variable_quartz = qvalue_obs.subs({x1_obs: X1_quartz, x2_obs:X2_quartz})\n",
    "\n",
    "q_random_variable_opal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "density(q_random_variable_opal)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "qvalue_opal_pdf = sympy.simplify(density(- 2 * X2_opal / sigma2 ** 2 - 2 * X1_opal / sigma1 ** 2)(qvalue_opal))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAggAAABNBAMAAAAx2IGmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} \\sigma_{1} \\sigma_{2} e^{\\frac{qvalue_{opal}^{2} \\sigma_{2}^{4}}{8 \\left(\\sigma_{1}^{2} + \\sigma_{2}^{2}\\right)} - \\frac{qvalue_{opal}^{2} \\sigma_{2}^{2}}{8} - \\frac{qvalue_{opal} \\rho_{opal}}{2} - \\frac{\\rho_{opal}^{2}}{2 \\sigma_{2}^{2}} - \\frac{\\rho_{opal}^{2}}{2 \\sigma_{1}^{2}}}}{4 \\sqrt{\\pi} \\sqrt{\\sigma_{1}^{2} + \\sigma_{2}^{2}}}$$"
      ],
      "text/plain": [
       "                     2   4              2   2                              2  \n",
       "          qvalue_opal ⋅σ₂    qvalue_opal ⋅σ₂    qvalue_opal⋅ρ_opal   ρ_opal   \n",
       "          ──────────────── - ──────────────── - ────────────────── - ─────── -\n",
       "             ⎛  2     2⎞            8                   2                 2   \n",
       "           8⋅⎝σ₁  + σ₂ ⎠                                              2⋅σ₂    \n",
       "√2⋅σ₁⋅σ₂⋅ℯ                                                                    \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                         ___________                          \n",
       "                                        ╱   2     2                           \n",
       "                                 4⋅√π⋅╲╱  σ₁  + σ₂                            \n",
       "\n",
       "       2\n",
       " ρ_opal \n",
       " ───────\n",
       "      2 \n",
       "  2⋅σ₁  \n",
       "        \n",
       "────────\n",
       "        \n",
       "        \n",
       "        "
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qvalue_opal_pdf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgQAAAA4BAMAAABwC/1aAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\frac{1.89157464718046 \\cdot 10^{-63} \\sqrt{2} e^{- 0.001 qvalue_{opal}^{2} - 0.75 qvalue_{opal}}}{\\sqrt{\\pi}}$$"
      ],
      "text/plain": [
       "                                            2                   \n",
       "                         - 0.001⋅qvalue_opal  - 0.75⋅qvalue_opal\n",
       "1.89157464718046e-63⋅√2⋅ℯ                                       \n",
       "────────────────────────────────────────────────────────────────\n",
       "                               √π                               "
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qvalue_opal_pdf.subs({rho_opal:1.5, rho_quartz:1.7, sigma1:0.1, sigma2:0.2})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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D4XCwYsUKXn/9dZKTk8nKysJqtTJu3DhPmY0bNxITE0NJSQk2m43Vq1fz3HPP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\nghBCCOHV/w9zbmIjyMZwpQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fb06ffa3bd0>"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(qvalue_opal_pdf.subs({rho_opal:1.5, rho_quartz:1.7, sigma1:0.1, sigma2:0.2}), (qvalue_opal, -600, -200))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAArIAAABdBAMAAACr2pjNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHZ6YAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} \\sigma_{1} \\sigma_{2} e^{- \\frac{\\rho_{opal}^{2}}{2 \\sigma_{1}^{2}}} e^{- \\frac{\\rho_{opal}^{2}}{2 \\sigma_{2}^{2}}} \\int_{0}^{1} e^{- \\frac{qvalue_{opal} \\rho_{opal}}{2}} e^{- \\frac{qvalue_{opal}^{2} \\sigma_{2}^{2}}{8}} e^{\\frac{qvalue_{opal}^{2} \\sigma_{2}^{4}}{8 \\left(\\sigma_{1}^{2} + \\sigma_{2}^{2}\\right)}}\\, dqvalue_{opal}}{4 \\sqrt{\\pi} \\sqrt{\\sigma_{1}^{2} + \\sigma_{2}^{2}}}$$"
      ],
      "text/plain": [
       "                               1                                              \n",
       "                               ⌠                                              \n",
       "                 2          2  ⎮                                              \n",
       "          -ρ_opal    -ρ_opal   ⎮                                    2   2   qv\n",
       "          ─────────  ───────── ⎮  -qvalue_opal⋅ρ_opal   -qvalue_opal ⋅σ₂    ──\n",
       "                2          2   ⎮  ────────────────────  ──────────────────    \n",
       "            2⋅σ₁       2⋅σ₂    ⎮           2                    8            8\n",
       "√2⋅σ₁⋅σ₂⋅ℯ         ⋅ℯ         ⋅⎮ ℯ                    ⋅ℯ                  ⋅ℯ  \n",
       "                               ⌡                                              \n",
       "                               0                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                    ___________               \n",
       "                                                   ╱   2     2                \n",
       "                                            4⋅√π⋅╲╱  σ₁  + σ₂                 \n",
       "\n",
       "                             \n",
       "                             \n",
       "         2   4               \n",
       "alue_opal ⋅σ₂                \n",
       "──────────────               \n",
       " ⎛  2     2⎞                 \n",
       "⋅⎝σ₁  + σ₂ ⎠                 \n",
       "               d(qvalue_opal)\n",
       "                             \n",
       "                             \n",
       "─────────────────────────────\n",
       "                             \n",
       "                             \n",
       "                             "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(qvalue_opal_pdf, (qvalue_opal, 0, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMwAAABKBAMAAAAF5aYEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\int_{0}^{1} e^{- x_{1 obs}} e^{- x_{1 obs}^{2}}\\, dx_{1 obs}$$"
      ],
      "text/plain": [
       "1                                \n",
       "⌠                                \n",
       "⎮                    2           \n",
       "⎮  -x_1_obs  -x_1_obs            \n",
       "⎮ ℯ        ⋅ℯ          d(x_1_obs)\n",
       "⌡                                \n",
       "0                                "
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.exp(-x1_obs**2 - x1_obs), (x1_obs, 0, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-32-271038a15f28>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mE\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mq_random_variable_opal\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/stats/rv.pyc\u001b[0m in \u001b[0;36mexpectation\u001b[0;34m(expr, condition, numsamples, evaluate, **kwargs)\u001b[0m\n\u001b[1;32m    655\u001b[0m     \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mpspace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mintegrate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'doit'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdoit\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    658\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/integrals/integrals.pyc\u001b[0m in \u001b[0;36mdoit\u001b[0;34m(self, **hints)\u001b[0m\n\u001b[1;32m    519\u001b[0m                         \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mfunction\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Poly\u001b[0m \u001b[0;32mand\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    520\u001b[0m                         (xab[1].has(oo, -oo) or xab[2].has(oo, -oo))):\n\u001b[0;32m--> 521\u001b[0;31m                     \u001b[0mret\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtry_meijerg\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxab\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    522\u001b[0m                     \u001b[0;32mif\u001b[0m \u001b[0mret\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    523\u001b[0m                         \u001b[0mfunction\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mret\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/integrals/integrals.pyc\u001b[0m in \u001b[0;36mtry_meijerg\u001b[0;34m(function, xab)\u001b[0m\n\u001b[1;32m    491\u001b[0m                         \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mxab\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    492\u001b[0m                         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 493\u001b[0;31m                             \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmeijerint_definite\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    494\u001b[0m                         \u001b[0;32mexcept\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    495\u001b[0m                             \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mintegrals\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmeijerint\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0m_debug\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/integrals/meijerint.pyc\u001b[0m in \u001b[0;36mmeijerint_definite\u001b[0;34m(f, x, a, b)\u001b[0m\n\u001b[1;32m   1792\u001b[0m                 \u001b[0m_debug\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'  Non-real splitting point.'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1793\u001b[0m                 \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1794\u001b[0;31m             \u001b[0mres1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_meijerint_definite_2\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1795\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mres1\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1796\u001b[0m                 \u001b[0m_debug\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'  But could not compute first integral.'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/integrals/meijerint.pyc\u001b[0m in \u001b[0;36m_meijerint_definite_2\u001b[0;34m(f, x)\u001b[0m\n\u001b[1;32m   1924\u001b[0m     \u001b[0;32mfor\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexplanation\u001b[0m \u001b[0;32min\u001b[0m \u001b[0m_guess_expansion\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1925\u001b[0m         \u001b[0m_debug\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Trying'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexplanation\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1926\u001b[0;31m         \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_meijerint_definite_3\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1927\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mres\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1928\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mres\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/integrals/meijerint.pyc\u001b[0m in \u001b[0;36m_meijerint_definite_3\u001b[0;34m(f, x)\u001b[0m\n\u001b[1;32m   1941\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1942\u001b[0m         \u001b[0m_debug\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Expanding and evaluating all terms.'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1943\u001b[0;31m         \u001b[0mress\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0m_meijerint_definite_4\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1944\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mNone\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mr\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mress\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1945\u001b[0m             \u001b[0mconds\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/integrals/meijerint.pyc\u001b[0m in \u001b[0;36m_meijerint_definite_4\u001b[0;34m(f, x, only_double)\u001b[0m\n\u001b[1;32m   2009\u001b[0m                     \u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf1_\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf2_\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2010\u001b[0m                     \u001b[0m_debug\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Saxena subst for yielded:'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf1_\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf2_\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2011\u001b[0;31m                     \u001b[0mcond\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mAnd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcond\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_check_antecedents\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf1_\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf2_\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2012\u001b[0m                     \u001b[0;32mif\u001b[0m \u001b[0mcond\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mFalse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2013\u001b[0m                         \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/integrals/meijerint.pyc\u001b[0m in \u001b[0;36m_check_antecedents\u001b[0;34m(g1, g2, x)\u001b[0m\n\u001b[1;32m   1027\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1028\u001b[0m         c14 = And(Eq(phi, 0), bstar - 1 + cstar <= 0,\n\u001b[0;32m-> 1029\u001b[0;31m                   Or(And(Ne(zos, 1), _cond(zos)),\n\u001b[0m\u001b[1;32m   1030\u001b[0m                      And(re(mu + rho + v - u) < 1, Eq(zos, 1))))\n\u001b[1;32m   1031\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/integrals/meijerint.pyc\u001b[0m in \u001b[0;36m_cond\u001b[0;34m(z)\u001b[0m\n\u001b[1;32m   1024\u001b[0m             \u001b[0;34m`\u001b[0m\u001b[0mmeijerint_definite\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mI\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m`\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1025\u001b[0m             '''\n\u001b[0;32m-> 1026\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mz\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0;36m1\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mz\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0mpi\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1027\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1028\u001b[0m         c14 = And(Eq(phi, 0), bstar - 1 + cstar <= 0,\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/expr.pyc\u001b[0m in \u001b[0;36m__abs__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    113\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__abs__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    114\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mAbs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 115\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mAbs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    116\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0m_sympifyit\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'other'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mNotImplemented\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/cache.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/compatibility.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwds)\u001b[0m\n\u001b[1;32m    848\u001b[0m                         \u001b[0mstats\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mHITS\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    849\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 850\u001b[0;31m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0muser_function\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    851\u001b[0m                 \u001b[0;32mwith\u001b[0m \u001b[0mlock\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    852\u001b[0m                     \u001b[0mroot\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnonlocal_root\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/function.pyc\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    439\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    440\u001b[0m         \u001b[0mevaluate\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'evaluate'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mglobal_evaluate\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 441\u001b[0;31m         \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mFunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0moptions\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    442\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    443\u001b[0m             \u001b[0mpr2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_should_evalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/cache.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/compatibility.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwds)\u001b[0m\n\u001b[1;32m    848\u001b[0m                         \u001b[0mstats\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mHITS\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    849\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 850\u001b[0;31m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0muser_function\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    851\u001b[0m                 \u001b[0;32mwith\u001b[0m \u001b[0mlock\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    852\u001b[0m                     \u001b[0mroot\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnonlocal_root\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/function.pyc\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    248\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    249\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 250\u001b[0;31m             \u001b[0mevaluated\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meval\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    251\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mevaluated\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    252\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mevaluated\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/functions/elementary/complexes.pyc\u001b[0m in \u001b[0;36meval\u001b[0;34m(cls, arg)\u001b[0m\n\u001b[1;32m    515\u001b[0m             \u001b[0munk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0ma\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mabs_free_arg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfree_symbols\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_real\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    516\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0munk\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mconj\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mhas\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mconjugate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mu\u001b[0m \u001b[0;32min\u001b[0m \u001b[0munk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 517\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0msqrt\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpand_mul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mconj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    518\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    519\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_eval_is_integer\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/function.pyc\u001b[0m in \u001b[0;36mexpand_mul\u001b[0;34m(expr, deep)\u001b[0m\n\u001b[1;32m   2357\u001b[0m     \"\"\"\n\u001b[1;32m   2358\u001b[0m     return sympify(expr).expand(deep=deep, mul=True, power_exp=False,\n\u001b[0;32m-> 2359\u001b[0;31m     power_base=False, basic=False, multinomial=False, log=False)\n\u001b[0m\u001b[1;32m   2360\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2361\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/cache.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/compatibility.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwds)\u001b[0m\n\u001b[1;32m    848\u001b[0m                         \u001b[0mstats\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mHITS\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    849\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 850\u001b[0;31m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0muser_function\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    851\u001b[0m                 \u001b[0;32mwith\u001b[0m \u001b[0mlock\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    852\u001b[0m                     \u001b[0mroot\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnonlocal_root\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/expr.pyc\u001b[0m in \u001b[0;36mexpand\u001b[0;34m(self, deep, modulus, power_base, power_exp, mul, log, multinomial, basic, **hints)\u001b[0m\n\u001b[1;32m   3122\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0muse_hint\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3123\u001b[0m                 \u001b[0mhint\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'_eval_expand_'\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3124\u001b[0;31m                 \u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mdeep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3125\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3126\u001b[0m         \u001b[0;32mwhile\u001b[0m \u001b[0mTrue\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/expr.pyc\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/expr.pyc\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3060\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3061\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mhit\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3062\u001b[0;31m                 \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0msargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3063\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3064\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/cache.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/compatibility.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwds)\u001b[0m\n\u001b[1;32m    848\u001b[0m                         \u001b[0mstats\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mHITS\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    849\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 850\u001b[0;31m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0muser_function\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    851\u001b[0m                 \u001b[0;32mwith\u001b[0m \u001b[0mlock\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    852\u001b[0m                     \u001b[0mroot\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnonlocal_root\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/function.pyc\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    439\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    440\u001b[0m         \u001b[0mevaluate\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'evaluate'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mglobal_evaluate\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 441\u001b[0;31m         \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mFunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0moptions\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    442\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    443\u001b[0m             \u001b[0mpr2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_should_evalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/cache.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     91\u001b[0m             \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     92\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 93\u001b[0;31m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     94\u001b[0m                 \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     95\u001b[0m                     \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/compatibility.pyc\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwds)\u001b[0m\n\u001b[1;32m    848\u001b[0m                         \u001b[0mstats\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mHITS\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    849\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 850\u001b[0;31m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0muser_function\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    851\u001b[0m                 \u001b[0;32mwith\u001b[0m \u001b[0mlock\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    852\u001b[0m                     \u001b[0mroot\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnonlocal_root\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/function.pyc\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m    248\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    249\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 250\u001b[0;31m             \u001b[0mevaluated\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meval\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    251\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mevaluated\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    252\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mevaluated\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/functions/elementary/complexes.pyc\u001b[0m in \u001b[0;36meval\u001b[0;34m(cls, arg)\u001b[0m\n\u001b[1;32m    618\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mperiodic_argument\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moo\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    619\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Atom\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 620\u001b[0;31m             \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marg_\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfactor_terms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_coeff_Mul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    621\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0marg_\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    622\u001b[0m                 arg_ = Mul(*[a if (sign(a) not in (-1, 1)) else\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mfactor_terms\u001b[0;34m(expr, radical, clear, fraction, sign)\u001b[0m\n\u001b[1;32m   1199\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1200\u001b[0m     \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1201\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1202\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1203\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1175\u001b[0m         \u001b[0mcont\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1176\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1177\u001b[0;31m             \u001b[0mlist_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1178\u001b[0m             \u001b[0;31m# get a common negative (if there) which gcd_terms does not remove\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1179\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_coeff_Mul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m0\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mlist_args\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1195\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1196\u001b[0m             p = p.func(\n\u001b[0;32m-> 1197\u001b[0;31m                 *[do(a) for a in p.args])\n\u001b[0m\u001b[1;32m   1198\u001b[0m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_keep_coeff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcont\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msign\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1199\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1165\u001b[0m                 \u001b[0mis_iterable\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'args_cnc'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1166\u001b[0m             \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1167\u001b[0;31m             \u001b[0mnewargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1168\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mnewargs\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1169\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1175\u001b[0m         \u001b[0mcont\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1176\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1177\u001b[0;31m             \u001b[0mlist_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1178\u001b[0m             \u001b[0;31m# get a common negative (if there) which gcd_terms does not remove\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1179\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_coeff_Mul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m0\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mlist_args\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1195\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1196\u001b[0m             p = p.func(\n\u001b[0;32m-> 1197\u001b[0;31m                 *[do(a) for a in p.args])\n\u001b[0m\u001b[1;32m   1198\u001b[0m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_keep_coeff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcont\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msign\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1199\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1165\u001b[0m                 \u001b[0mis_iterable\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'args_cnc'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1166\u001b[0m             \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1167\u001b[0;31m             \u001b[0mnewargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1168\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mnewargs\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1169\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1175\u001b[0m         \u001b[0mcont\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1176\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1177\u001b[0;31m             \u001b[0mlist_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1178\u001b[0m             \u001b[0;31m# get a common negative (if there) which gcd_terms does not remove\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1179\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_coeff_Mul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m0\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mlist_args\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1195\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1196\u001b[0m             p = p.func(\n\u001b[0;32m-> 1197\u001b[0;31m                 *[do(a) for a in p.args])\n\u001b[0m\u001b[1;32m   1198\u001b[0m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_keep_coeff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcont\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msign\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1199\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1165\u001b[0m                 \u001b[0mis_iterable\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'args_cnc'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1166\u001b[0m             \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1167\u001b[0;31m             \u001b[0mnewargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1168\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mnewargs\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1169\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/exprtools.pyc\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1173\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mfactor_sum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfraction\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mfraction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msign\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1174\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1175\u001b[0;31m         \u001b[0mcont\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1176\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1177\u001b[0m             \u001b[0mlist_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/add.pyc\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m    977\u001b[0m         \"\"\"\n\u001b[1;32m    978\u001b[0m         con, prim = self.func(*[_keep_coeff(*a.as_content_primitive(\n\u001b[0;32m--> 979\u001b[0;31m             radical=radical, clear=clear)) for a in self.args]).primitive()\n\u001b[0m\u001b[1;32m    980\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mclear\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Integer\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mprim\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    981\u001b[0m             \u001b[0mcon\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_numer_denom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/mul.pyc\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m   1690\u001b[0m         \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1691\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1692\u001b[0;31m             \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1693\u001b[0m             \u001b[0mcoef\u001b[0m \u001b[0;34m*=\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1694\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mp\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mOne\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/power.pyc\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m   1598\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1599\u001b[0m         \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0me\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_base_exp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1600\u001b[0;31m         \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_keep_coeff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1601\u001b[0m         \u001b[0mce\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpe\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1602\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Rational\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/add.pyc\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m    977\u001b[0m         \"\"\"\n\u001b[1;32m    978\u001b[0m         con, prim = self.func(*[_keep_coeff(*a.as_content_primitive(\n\u001b[0;32m--> 979\u001b[0;31m             radical=radical, clear=clear)) for a in self.args]).primitive()\n\u001b[0m\u001b[1;32m    980\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mclear\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Integer\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mprim\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    981\u001b[0m             \u001b[0mcon\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_numer_denom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/mul.pyc\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m   1690\u001b[0m         \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1691\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1692\u001b[0;31m             \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1693\u001b[0m             \u001b[0mcoef\u001b[0m \u001b[0;34m*=\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1694\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mp\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mOne\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/power.pyc\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m   1598\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1599\u001b[0m         \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0me\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_base_exp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1600\u001b[0;31m         \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_keep_coeff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1601\u001b[0m         \u001b[0mce\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpe\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1602\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Rational\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/add.pyc\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m    977\u001b[0m         \"\"\"\n\u001b[1;32m    978\u001b[0m         con, prim = self.func(*[_keep_coeff(*a.as_content_primitive(\n\u001b[0;32m--> 979\u001b[0;31m             radical=radical, clear=clear)) for a in self.args]).primitive()\n\u001b[0m\u001b[1;32m    980\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mclear\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Integer\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mprim\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    981\u001b[0m             \u001b[0mcon\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_numer_denom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/add.pyc\u001b[0m in \u001b[0;36mprimitive\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    914\u001b[0m         \u001b[0minf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    915\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 916\u001b[0;31m             \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_coeff_Mul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    917\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Rational\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    918\u001b[0m                 \u001b[0mc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mOne\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/turra/venv/lib/python2.7/site-packages/sympy/core/mul.pyc\u001b[0m in \u001b[0;36mas_coeff_Mul\u001b[0;34m(self, rational)\u001b[0m\n\u001b[1;32m    765\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mOne\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    766\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 767\u001b[0;31m     \u001b[0;32mdef\u001b[0m \u001b[0mas_coeff_Mul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrational\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    768\u001b[0m         \u001b[0;34m\"\"\"Efficiently extract the coefficient of a product. \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    769\u001b[0m         \u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "E(q_random_variable_opal)"
   ]
  }
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