sympysymbolicintegraltest.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "sp.init_printing()\n",
    "from sympy.integrals.risch import NonElementaryIntegral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sp.Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [],
   "source": [
    "#y=1/sp.sqrt(x)\n",
    "#y=1/x\n",
    "#y=(x**2)*sp.sin(1/x)\n",
    "#y=1 / sp.cos(x**3)\n",
    "#y=1/sp.cos(x**2)\n",
    "#y=1/sp.exp(x**2)\n",
    "#y=2/sp.cos(x**2)\n",
    "#y=sp.exp(-(x-1)**2)\n",
    "#y=1/(x**3 + 1)\n",
    "#y=sp.sqrt(x+sp.sqrt(x))\n",
    "y=sp.log(x)*sp.exp(-x**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "test to see the correct answer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJgAAAAuBAMAAAAmf816AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\int e^{- x^{2}} \\log{\\left (x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠               \n",
       "⎮    2          \n",
       "⎮  -x           \n",
       "⎮ ℯ   ⋅log(x) dx\n",
       "⌡               "
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.integrate(y,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Method 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isinstance(sp.integrate(y, x, risch=True), NonElementaryIntegral)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Method 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "not sp.integrate(y, x).has(sp.Integral)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Method 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "not isinstance(sp.integrate(y, x), sp.Integral)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "|   formula              |analytical |1|2|3|\n",
    "|---------------|-------|---|---|---|\n",
    "|   $ \\frac{1}{\\sqrt{x}} $ | true |$\\times$|$\\checkmark$|$\\checkmark$|\n",
    "|   $ \\frac{1}{x} $ | true |$\\times$|$\\checkmark$|$\\checkmark$|\n",
    "|   $ x^2 \\sin\\left(\\frac{1}{x}\\right) $ | true |$\\times$|$\\checkmark$|$\\checkmark$|\n",
    "|   $ \\frac{1}{\\cos\\left(x^3\\right)} $ | false |$\\checkmark$|$\\checkmark$|$\\checkmark$|\n",
    "|   $ \\frac{1}{\\cos\\left(x^2\\right)} $ | false |$\\checkmark$|$\\checkmark$|$\\checkmark$|\n",
    "|   $ \\frac{1}{e^{x^2}} $ | true |$\\checkmark$|$\\checkmark$|$\\checkmark$|\n",
    "|   $ \\frac{2}{\\cos\\left(x^2\\right)} $ | false |$\\checkmark$|$\\checkmark$|$\\times$|\n",
    "|$e^{\\left(-\\left(x-1\\right)^2\\right)}$|true|$\\checkmark$|$\\checkmark$|$\\checkmark$|\n",
    "|$\\frac{1}{x^3+1}$|true|$\\times$|$\\checkmark$|$\\checkmark$|\n",
    "|$\\sqrt{x+\\sqrt{x}}$|false|$\\checkmark$|$\\checkmark$|$\\checkmark$|\n",
    "|$e^{- x^{2}} \\log{\\left (x \\right )}$|false|$\\times$|$\\checkmark$|$\\checkmark$|"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "codemirror_mode": {
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   "file_extension": ".py",
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The goal of this Jupyter/IPython Notebook is to test different methods suggested here in this StackOverflow post to see how we can check if sympy is able to analytically/symbolically solve a mathematical expression. It seems that the .has(Integral) is the most accurate.