thekvs/sympy-vs-demidovich.ipynb

## sympy-vs-demidovich.ipynb
{
 "metadata": {
  "name": "sympy-vs-demidovich"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "from __future__ import division\nfrom sympy import *\nfrom sympy import init_printing\ninit_printing()\n\nx, y, z, t = symbols('x y z t')\nk, m, n, p, q = symbols('k m n p q', integer=True)\nf, g, h = symbols('f g h', cls=Function)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "\u041f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "diff(((x**p)*(1-x)**q)/(1+x), x)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\frac{p x^{p} \\left(- x + 1\\right)^{q}}{x \\left(x + 1\\right)} - \\frac{q x^{p} \\left(- x + 1\\right)^{q}}{\\left(- x + 1\\right) \\left(x + 1\\right)} - \\frac{x^{p} \\left(- x + 1\\right)^{q}}{\\left(x + 1\\right)^{2}}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAREAAAAnCAYAAAAowHyKAAAABHNCSVQICAgIfAhkiAAABwVJREFU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\n",
       "prompt_number": 2,
       "text": "   p         q       p         q     p         q\np\u22c5x \u22c5(-x + 1)     q\u22c5x \u22c5(-x + 1)     x \u22c5(-x + 1) \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  x\u22c5(x + 1)      (-x + 1)\u22c5(x + 1)            2  \n                                      (x + 1)   "
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "\u0418\u043d\u0442\u0435\u0433\u0440\u0430\u043b\u044b"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate((exp(3*x)+1)/(exp(x) + 1), x) # 1646",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$x + \\frac{e^{2 x}}{2} - e^{x}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAcCAYAAAC9I+tXAAAABHNCSVQICAgIfAhkiAAAA79JREFU\naIHt2VuIHFUQgOEvl40bE7OCQRZjUDfr5UETdXUlWVEUgwiK4uIFH0JCBNGAeAlBUfGSPKxRH3xQ\n8EEyi6IRJSAiYkAQ9UUhRl1QVLwgClGCgSiiaIwP1cO2nct0z5zZWZb5oZk+Naerq/qcrlOnmi4z\nglmdNqAiF2EV+jCCzXivoxZ1qcxCjOXaN+IPLOmMOV2aZTn+xbKsvQgHxYCm5PrE+qaEuZ02oAIT\nWIlvs/bS7Pfr7HcEp2MFdmEBrsTG3DWNOAtrsCOBvalJ4d+05AU8lZ0vwvrs/Fp8mJ2Po7+Czsdw\nU0E2LAb2LTGZnjf14TyVf9OO9dhqMlnrxbzsfAz3Nan3Y8zPtc/HThyftReK5OoXnNrkPZohlX9N\nsVx7QvfVJmdnr0Mf6C5cmJ33VdA7LN7yPG9isCA7T6zN2yvoTkmz/jVNTfpZe6kYxP7suE6sm9fg\n7ux+f6Mn639nBd1P46qC7Hf8gBML8n3YW0F3qzT0r537yBoewfeJ9A3gUxHe8vRhFEP4UoTGA2Jr\nsgM/l9A9B7tFKP0nJ5/AGSIJ+i4n34PjRMIxFazTwL/iQA6JrO2AGP1bcZtYI5bgYXxT8uY11Qfy\nXNwhZnwPFmMDfqugoxlWi7d7Q0G+QAzYnpzsJPyEd3FZYjuS+D+AZzA7a9fwlaikjIg93L0V9NVU\nC61rxRuXzwg3iSyt3WwTPpZhTEz0VYltWCuR/8+K2VfnVZNp7lI8iRMq6KspP5ArRezPP8xhkWwU\n16fU9OIT5ZaZQbFubklsQ8v+540/zf/XgR/FPuWBJo2rKR9a3xbZ4Hhm0xx8ITLD/U3evyyjuAD3\nN+h3DN7HB7in8N8K4W/ZnGO3WPfqtM3/M0WKfUWJvuNiRhePX/H5Ef4byl3fIxKM51oxuAVew9kN\n+szCS6JgkJq2+n87/sKxOdlARR015UJrv5g0myrqT0EfPirRbwseKsjWJLIhif/1xGa+qJSck7VX\n4zOR4tb7bWzlRkdhrwgfhyseLBOTql2MalxXXScSvc0F+cWJbEjq/6iYFTeLPdOEWAvqPCi+BVah\npnyys1WUvvJrzCWZjnbu1XbilKP8f7l40C8Wju14OaEdLftfv3AxnhBGH8SjIov9U4TY1/FOReNq\nyic7c/E4ThaVlHkiImwTb0M76Mcrolp0JPaZrLMWOVy4bZZO+F+amqktLFflLrH5nhG083vkfvFG\nT1duMDXFhi5tZFBstmcMsxt3mZHcIm2y0qVDvOHQryhdunTp0qVLlxnIf3L8zDgR1/s8AAAAAElF\nTkSuQmCC\n",
       "prompt_number": 3,
       "text": "     2\u22c5x     \n    \u212f       x\nx + \u2500\u2500\u2500\u2500 - \u212f \n     2       "
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate(x / sqrt(1 - x**2), x) # 1674",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$- \\sqrt{- x^{2} + 1}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAAhCAYAAAAxvj44AAAABHNCSVQICAgIfAhkiAAAAuZJREFU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\nJJoCEu4AAAAASUVORK5CYII=\n",
       "prompt_number": 4,
       "text": "    __________\n   \u2571    2     \n-\u2572\u2571  - x  + 1 "
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate(1/(exp(x) + exp(-x)),x) # 1691",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\operatorname{RootSum} {\\left(4 z^{2} + 1, \\Lambda {\\left (i, i \\log{\\left (2 i + e^{x} \\right )} \\right )}\\right)}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATUAAAAcCAYAAAAAyyhOAAAABHNCSVQICAgIfAhkiAAADSBJREFU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\n9QAAAABJRU5ErkJggg==\n",
       "prompt_number": 5,
       "text": "       \u239b   2               \u239b       x\u239e\u239e\nRootSum\u239d4\u22c5z  + 1, i \u21a6 i\u22c5log\u239d2\u22c5i + \u212f \u23a0\u23a0"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate((sin(x)**5)*cos(x), x) # 1695",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\frac{1}{6} \\sin^{6}{\\left (x \\right )}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEsAAAAcCAYAAAApxUKlAAAABHNCSVQICAgIfAhkiAAAA/dJREFU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\nRK5CYII=\n",
       "prompt_number": 6,
       "text": "   6   \nsin (x)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   6   "
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate(exp(x) / (2 + exp(x)), x) # 1690",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\log{\\left (e^{x} + 2 \\right )}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAXCAYAAACLbliwAAAABHNCSVQICAgIfAhkiAAAA/hJREFU\naIHt2GuIVVUUB/CfjxwrLUlLe0mNpkVl2ZSWldGDQRDTiMAoJktKKAxFjKDXkBZCRVj0Lhp7oV+K\n6El9qajAIsgiyg/lUPawFKMX+Sj7sM7tntlz7p25zR2nD/cPh3PO2muvs/Y6e/332psG9irm4T18\ngWV76ZvTcHCNfc7EAtyLy7EIz6O5rp5VxrTse6/hUzyBw3PtTZjdk5Gj8Dva6+5ed7SIINWCA7Aw\ne56L9dnzGoyrk1/VcArewKjsfQTewY8idiUswCU9GevU/4FuwgsYVGO/4RiWPa/CjXXwZQqG9lL3\nFUxMZFOxB2sT+bM4spqxTv0f6OWY30cbH+G07PnAPtjp0HU2VsNv+BqHJPLt2JrIpuOxvGBw7b71\nGW2C52rFHCwVgZmCjzP5FfVxq0dswljsn8h3YN9Eth4z87q9SZvpWIwtmf4o3ImNid5VmIVvcATe\nxLE4VaTWw5iMn7GzwrdOxrViluyDMbgOv2bPE0TAb8L1+APrejGGeuB0jMQPOdlhIvhvFeh/gHPx\ncpGxTl2pY44IaL46OC7Tm5qTLRKpVUrjY/CXCPIsXJjJ50tSKocF2KDrKn6DWPj6Cx16Tx1FWCXG\nOaOgrR03V+rYqRzoEYJ7lhTorVZOXfhEuQooYQueTmRLcFeBvTOwS5RvJUwTC1DKifVEh/8e6Ili\ncq2s0L4YD5ReqlHHbIwWtXWKjSJ1W8TCtBUHJTpNBfabFNNGu6CLebgIQ/A5LsUvVXwcKDThOTyq\n8qzdIcaB6oEubQJ2F7Ttyu4TRaBX4nUcj88EXQzDPUm/tOYkuPh8Ufwvr+JPX7AGJxXIx4vMKfr5\nC8XYUgzCk2K8t1b55mj8VHqpFujvs3tR6o7O7luy+5/ZR68WP2GYKPDTbNikzNd5W0PwZRVf+opK\nlUmHyKbOGmytENm2Iidrw1OJ3lhBqage6JfEqj65oK0Fm/Fu9j4Dr+YNV8D7eDyRbRX0UOTLBLTi\noR7s7i1cib91DTKcpXugZ+Du0ktaRw9VHvA2XCNS6NCcztGibGlTppVvcYeggHPEbB6n++5vJz4U\nZVwJu/GIoJu8/kzcUjCAgcJ5YiFvxjO5a60o+/IYI8a1OTUyV/DRHnHe8bbY8sLZgvgfxP2CS08s\nMLwt65+/NuOyRHdSZiOPoYLP12WDWS1+cH9vqDr0vurYrvv4Slc6w1figrp4mMN4wcWtyqvscPHn\nl4nVd1LS5zZR0g00OvStji7CBFGN1B1L8WKV9g24OJENxu1iwRhI3Ke+J39NIjP3q6PNf3GC2HYX\n7Y5a8ZWBD+iAo9ajykpoFjN7pDiXGCz+6nfikD493WqggQYaaKCB/w3+AQ+GxaNhs814AAAAAElF\nTkSuQmCC\n",
       "prompt_number": 7,
       "text": "   \u239b x    \u239e\nlog\u239d\u212f  + 2\u23a0"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate(x * exp(-x), x) # 1795",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\left(- x - 1\\right) e^{- x}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAATCAYAAADyIwHfAAAABHNCSVQICAgIfAhkiAAAAq5JREFU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\nehNry/7fAAAAAElFTkSuQmCC\n",
       "prompt_number": 8,
       "text": "          -x\n(-x - 1)\u22c5\u212f  "
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate(cos(x)/sin(x), x) # 1698",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\frac{1}{2} \\log{\\left (\\cos^{2}{\\left (x \\right )} - 1 \\right )}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI0AAAAcCAYAAABcZDxDAAAABHNCSVQICAgIfAhkiAAABjFJREFU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\nRK5CYII=\n",
       "prompt_number": 9,
       "text": "   \u239b   2       \u239e\nlog\u239dcos (x) - 1\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       2        "
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate(log(x), (x, 0, 1)) # 2335",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$-1$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAASCAYAAACq26WdAAAABHNCSVQICAgIfAhkiAAAAIZJREFU\nOI3t1LENwkAMQNEHSoEyADtQRRRMQ8U+DMEaqTIARRTEFsyAQnEpUhApQXeRAvzGxcn+Z9kyCyfH\nbQ7RAVe07x6zSJIdznjgGanmKC4GOlvP+Yu/LAr9bSyE4a5G5tY4fSprsJ+SPJXvnVkK2aaLeYLa\nYIsSd+F6tMLpqnBMJf1RXtvTD+xjlzAcAAAAAElFTkSuQmCC\n",
       "prompt_number": 10,
       "text": "-1"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate(1 / (x**2+x+1)**2, (x, -oo, oo))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": "$$\\frac{4 \\pi}{9} \\sqrt{3}$$",
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEsAAAAgCAYAAABafYVzAAAABHNCSVQICAgIfAhkiAAAA9lJREFU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\njfLEtk2bNm32af4FlXbEMVqBt5EAAAAASUVORK5CYII=\n",
       "prompt_number": 11,
       "text": "    ___  \n4\u22c5\u2572\u2571 3 \u22c5\u03c0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    9    "
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    }
   ],
   "metadata": {}
  }
 ]
}