Integrand size = 108, antiderivative size = 28 \[ \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{-x-e^3 x+e^{\frac {e^3}{15}} x+4 x^2-x^3} \, dx=\left (16+\log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )\right )^2 \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.91 (sec) , antiderivative size = 902, normalized size of antiderivative = 32.21, number of steps used = 46, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6, 1608, 6860, 1642, 642, 2608, 2604, 2404, 2338, 2354, 2438, 2465, 2441, 2352, 2437, 2440} \[ \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{-x-e^3 x+e^{\frac {e^3}{15}} x+4 x^2-x^3} \, dx=-\log ^2\left (-2 \left (-x-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2\right )\right )-\log ^2\left (-2 \left (-x+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2\right )\right )-\log ^2(x)-32 \log (x)-2 \log (x) \log \left (x-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}\right )-2 \log \left (-\frac {i \left (-x+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2\right )}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (2 x-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )\right )+2 \log \left (\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (2 x-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )\right )+2 \log \left (x-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}\right ) \log \left (2 x-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )\right )-2 \log \left (\frac {i \left (-x-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2\right )}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (2 x-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )\right )+2 \log \left (\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (2 x-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )\right )+2 \log \left (x-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}\right ) \log \left (2 x-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )\right )+2 \log (x) \log \left (1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \log (x) \log \left (1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+32 \log \left (x^2-4 x-e^{\frac {e^3}{15}}+e^3+1\right )-2 \operatorname {PolyLog}\left (2,-\frac {-i x-\sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 i}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )-2 \operatorname {PolyLog}\left (2,\frac {-i x+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 i}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \]
[In]
[Out]
Rule 6
Rule 642
Rule 1608
Rule 1642
Rule 2338
Rule 2352
Rule 2354
Rule 2404
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2465
Rule 2604
Rule 2608
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{e^{\frac {e^3}{15}} x+\left (-1-e^3\right ) x+4 x^2-x^3} \, dx \\ & = \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{\left (-1-e^3+e^{\frac {e^3}{15}}\right ) x+4 x^2-x^3} \, dx \\ & = \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{x \left (-1-e^3+e^{\frac {e^3}{15}}+4 x-x^2\right )} \, dx \\ & = \int \left (\frac {32 \left (-1-e^3+e^{\frac {e^3}{15}}+x^2\right )}{x \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )}+\frac {2 \left (-1-e^3+e^{\frac {e^3}{15}}+x^2\right ) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{x \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )}\right ) \, dx \\ & = 2 \int \frac {\left (-1-e^3+e^{\frac {e^3}{15}}+x^2\right ) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{x \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )} \, dx+32 \int \frac {-1-e^3+e^{\frac {e^3}{15}}+x^2}{x \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )} \, dx \\ & = 2 \int \left (-\frac {\log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{x}+\frac {2 (-2+x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}\right ) \, dx+32 \int \left (-\frac {1}{x}+\frac {2 (-2+x)}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}\right ) \, dx \\ & = -32 \log (x)-2 \int \frac {\log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{x} \, dx+4 \int \frac {(-2+x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2} \, dx+64 \int \frac {-2+x}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2} \, dx \\ & = -32 \log (x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )+2 \int \frac {\left (1-\frac {1+e^3-e^{\frac {e^3}{15}}}{x^2}\right ) \log (x)}{-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x} \, dx+4 \int \left (\frac {\log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x}+\frac {\log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x}\right ) \, dx \\ & = -32 \log (x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )+2 \int \left (-\frac {\log (x)}{x}+\frac {2 (-2+x) \log (x)}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}\right ) \, dx+4 \int \frac {\log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx+4 \int \frac {\log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx \\ & = -32 \log (x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )-2 \int \frac {\log (x)}{x} \, dx-2 \int \frac {\left (1-\frac {1+e^3-e^{\frac {e^3}{15}}}{x^2}\right ) \log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x} \, dx-2 \int \frac {\left (1-\frac {1+e^3-e^{\frac {e^3}{15}}}{x^2}\right ) \log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x} \, dx+4 \int \frac {(-2+x) \log (x)}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2} \, dx \\ & = -32 \log (x)-\log ^2(x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )-2 \int \left (-\frac {\log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{x}+\frac {2 (-2+x) \log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}\right ) \, dx-2 \int \left (-\frac {\log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{x}+\frac {2 (-2+x) \log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}\right ) \, dx+4 \int \left (\frac {\log (x)}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x}+\frac {\log (x)}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x}\right ) \, dx \\ & = -32 \log (x)-\log ^2(x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )+2 \int \frac {\log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{x} \, dx+2 \int \frac {\log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{x} \, dx+4 \int \frac {\log (x)}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx+4 \int \frac {\log (x)}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx-4 \int \frac {(-2+x) \log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2} \, dx-4 \int \frac {(-2+x) \log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{1+e^3-e^{\frac {e^3}{15}}-4 x+x^2} \, dx \\ & = -32 \log (x)-\log ^2(x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )+2 \log \left (\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log (x) \log \left (1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \log (x) \log \left (1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )-2 \int \frac {\log \left (1+\frac {2 x}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )}{x} \, dx-2 \int \frac {\log \left (1+\frac {2 x}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )}{x} \, dx-4 \int \frac {\log \left (\frac {2 x}{4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx-4 \int \frac {\log \left (\frac {2 x}{4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx-4 \int \left (\frac {\log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x}+\frac {\log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x}\right ) \, dx-4 \int \left (\frac {\log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x}+\frac {\log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x}\right ) \, dx \\ & = -32 \log (x)-\log ^2(x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )+2 \log \left (\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log (x) \log \left (1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \log (x) \log \left (1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )-4 \int \frac {\log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx-4 \int \frac {\log \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx-4 \int \frac {\log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx-4 \int \frac {\log \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx \\ & = -32 \log (x)-\log ^2(x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )-2 \log \left (-\frac {i \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )-2 \log \left (\frac {i \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log (x) \log \left (1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \log (x) \log \left (1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )-2 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )-2 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )+4 \int \frac {\log \left (\frac {2 \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{2 \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )-2 \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )}\right )}{-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx+4 \int \frac {\log \left (\frac {2 \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )}{-2 \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )}\right )}{-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x} \, dx \\ & = -\log ^2\left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )\right )-\log ^2\left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )\right )-32 \log (x)-\log ^2(x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )-2 \log \left (-\frac {i \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )-2 \log \left (\frac {i \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log (x) \log \left (1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \log (x) \log \left (1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )-2 \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )}\right )}{x} \, dx,x,-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right )+2 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 \left (-4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )}\right )}{x} \, dx,x,-4-2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+2 x\right ) \\ & = -\log ^2\left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )\right )-\log ^2\left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )\right )-32 \log (x)-\log ^2(x)-2 \log (x) \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right )-2 \log \left (-\frac {i \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )-2 \log \left (\frac {i \left (2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}-x\right )}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log \left (-4+\frac {1+e^3-e^{\frac {e^3}{15}}}{x}+x\right ) \log \left (-2 \left (2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}\right )+2 x\right )+2 \log (x) \log \left (1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \log (x) \log \left (1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+32 \log \left (1+e^3-e^{\frac {e^3}{15}}-4 x+x^2\right )-2 \operatorname {PolyLog}\left (2,-\frac {2 i-\sqrt {-3+e^3-e^{\frac {e^3}{15}}}-i x}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )-2 \operatorname {PolyLog}\left (2,\frac {2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}-i x}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.43 (sec) , antiderivative size = 949, normalized size of antiderivative = 33.89 \[ \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{-x-e^3 x+e^{\frac {e^3}{15}} x+4 x^2-x^3} \, dx=2 \left (\log \left (\frac {2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}-i x}{2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log (x)+\log \left (\frac {-2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}+i x}{-2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log (x)-\frac {\log ^2(x)}{2}-\log \left (\frac {2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}-i x}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (2 \left (-2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+x\right )\right )+\log \left (\frac {2 x}{4+2 i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (2 \left (-2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+x\right )\right )-\frac {1}{2} \log ^2\left (2 \left (-2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+x\right )\right )-\log \left (\frac {-2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}+i x}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (2 \left (-2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+x\right )\right )+\log \left (\frac {i x}{2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right ) \log \left (2 \left (-2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+x\right )\right )-\frac {1}{2} \log ^2\left (2 \left (-2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+x\right )\right )-\log (x) \left (16+\log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )\right )+\log \left (2 \left (-2-i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+x\right )\right ) \left (16+\log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )\right )+\log \left (2 \left (-2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}+x\right )\right ) \left (16+\log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )\right )-\operatorname {PolyLog}\left (2,\frac {2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}-i x}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+\operatorname {PolyLog}\left (2,\frac {2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}-i x}{2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )-\operatorname {PolyLog}\left (2,\frac {-2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}+i x}{2 \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+\operatorname {PolyLog}\left (2,\frac {-2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}+i x}{-2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+\operatorname {PolyLog}\left (2,\frac {x}{2+i \sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )+\operatorname {PolyLog}\left (2,\frac {i x}{2 i+\sqrt {-3+e^3-e^{\frac {e^3}{15}}}}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(23)=46\).
Time = 0.49 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79
method | result | size |
default | \(-32 \ln \left (x \right )+32 \ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{3}}{15}}+{\mathrm e}^{3}+x^{2}-4 x +1\right )+\ln \left (\frac {-{\mathrm e}^{\frac {{\mathrm e}^{3}}{15}}+{\mathrm e}^{3}+x^{2}-4 x +1}{x}\right )^{2}\) | \(50\) |
norman | \(\ln \left (\frac {-{\mathrm e}^{\frac {{\mathrm e}^{3}}{15}}+{\mathrm e}^{3}+x^{2}-4 x +1}{x}\right )^{2}+32 \ln \left (\frac {-{\mathrm e}^{\frac {{\mathrm e}^{3}}{15}}+{\mathrm e}^{3}+x^{2}-4 x +1}{x}\right )\) | \(50\) |
risch | \(-32 \ln \left (x \right )+32 \ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{3}}{15}}+{\mathrm e}^{3}+x^{2}-4 x +1\right )+\ln \left (\frac {-{\mathrm e}^{\frac {{\mathrm e}^{3}}{15}}+{\mathrm e}^{3}+x^{2}-4 x +1}{x}\right )^{2}\) | \(50\) |
parts | \(-32 \ln \left (x \right )+32 \ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{3}}{15}}+{\mathrm e}^{3}+x^{2}-4 x +1\right )+\ln \left (\frac {-{\mathrm e}^{\frac {{\mathrm e}^{3}}{15}}+{\mathrm e}^{3}+x^{2}-4 x +1}{x}\right )^{2}\) | \(50\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{-x-e^3 x+e^{\frac {e^3}{15}} x+4 x^2-x^3} \, dx=\log \left (\frac {x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1}{x}\right )^{2} + 32 \, \log \left (\frac {x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1}{x}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 3.88 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{-x-e^3 x+e^{\frac {e^3}{15}} x+4 x^2-x^3} \, dx=- 32 \log {\left (x \right )} + \log {\left (\frac {x^{2} - 4 x - e^{\frac {e^{3}}{15}} + 1 + e^{3}}{x} \right )}^{2} + 32 \log {\left (x^{2} - 4 x - e^{\frac {e^{3}}{15}} + 1 + e^{3} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (23) = 46\).
Time = 2.91 (sec) , antiderivative size = 399, normalized size of antiderivative = 14.25 \[ \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{-x-e^3 x+e^{\frac {e^3}{15}} x+4 x^2-x^3} \, dx=16 \, {\left (\frac {\log \left (x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right )}{e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1} - \frac {2 \, \log \left (x\right )}{e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1} - \frac {4 \, \arctan \left (\frac {x - 2}{\sqrt {e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} - 3}}\right )}{{\left (e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right )} \sqrt {e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} - 3}}\right )} e^{3} - 16 \, {\left (\frac {\log \left (x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right )}{e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1} - \frac {2 \, \log \left (x\right )}{e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1} - \frac {4 \, \arctan \left (\frac {x - 2}{\sqrt {e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} - 3}}\right )}{{\left (e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right )} \sqrt {e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} - 3}}\right )} e^{\left (\frac {1}{15} \, e^{3}\right )} + \log \left (x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right )^{2} - 2 \, \log \left (x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right ) \log \left (x\right ) + \log \left (x\right )^{2} + \frac {64 \, \arctan \left (\frac {x - 2}{\sqrt {e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} - 3}}\right )}{\sqrt {e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} - 3}} + \frac {16 \, \log \left (x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right )}{e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1} - \frac {32 \, \log \left (x\right )}{e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1} - \frac {64 \, \arctan \left (\frac {x - 2}{\sqrt {e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} - 3}}\right )}{{\left (e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right )} \sqrt {e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} - 3}} + 16 \, \log \left (x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1\right ) \]
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\[ \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{-x-e^3 x+e^{\frac {e^3}{15}} x+4 x^2-x^3} \, dx=\int { \frac {2 \, {\left (16 \, x^{2} + {\left (x^{2} - e^{3} + e^{\left (\frac {1}{15} \, e^{3}\right )} - 1\right )} \log \left (\frac {x^{2} - 4 \, x + e^{3} - e^{\left (\frac {1}{15} \, e^{3}\right )} + 1}{x}\right ) - 16 \, e^{3} + 16 \, e^{\left (\frac {1}{15} \, e^{3}\right )} - 16\right )}}{x^{3} - 4 \, x^{2} + x e^{3} - x e^{\left (\frac {1}{15} \, e^{3}\right )} + x} \,d x } \]
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Time = 37.89 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {32+32 e^3-32 e^{\frac {e^3}{15}}-32 x^2+\left (2+2 e^3-2 e^{\frac {e^3}{15}}-2 x^2\right ) \log \left (\frac {1+e^3-e^{\frac {e^3}{15}}-4 x+x^2}{x}\right )}{-x-e^3 x+e^{\frac {e^3}{15}} x+4 x^2-x^3} \, dx={\ln \left (\frac {x^2-4\,x+{\mathrm {e}}^3-{\left ({\mathrm {e}}^{{\mathrm {e}}^3}\right )}^{1/15}+1}{x}\right )}^2+32\,\ln \left (x^2-4\,x-{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{15}}+{\mathrm {e}}^3+1\right )-32\,\ln \left (x\right ) \]
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