Integrand size = 126, antiderivative size = 35 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=e^{e^2-\left (x-\log \left (\frac {1}{x}-x\right )\right )^2}-4 (4-x)^2 x^4 \] Output:
exp(exp(2)-(x-ln(1/x-x))^2)-4*x^4*(4-x)^2
Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1}{x}-x\right )^{2 x}-4 (-4+x)^2 x^4 \] Input:
Integrate[(256*x^4 - 160*x^5 - 232*x^6 + 160*x^7 - 24*x^8 + E^(E^2 - x^2 + 2*x*Log[(1 - x^2)/x] - Log[(1 - x^2)/x]^2)*(2*x + 2*x^2 + 2*x^3 - 2*x^4 + (-2 - 2*x - 2*x^2 + 2*x^3)*Log[(1 - x^2)/x]))/(-x + x^3),x]
Output:
E^(E^2 - x^2 - Log[x^(-1) - x]^2)*(x^(-1) - x)^(2*x) - 4*(-4 + x)^2*x^4
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-2 x^4+2 x^3+2 x^2+\left (2 x^3-2 x^2-2 x-2\right ) \log \left (\frac {1-x^2}{x}\right )+2 x\right ) \exp \left (-x^2-\log ^2\left (\frac {1-x^2}{x}\right )+2 x \log \left (\frac {1-x^2}{x}\right )+e^2\right )-24 x^8+160 x^7-232 x^6-160 x^5+256 x^4}{x^3-x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-2 x^4+2 x^3+2 x^2+\left (2 x^3-2 x^2-2 x-2\right ) \log \left (\frac {1-x^2}{x}\right )+2 x\right ) \exp \left (-x^2-\log ^2\left (\frac {1-x^2}{x}\right )+2 x \log \left (\frac {1-x^2}{x}\right )+e^2\right )-24 x^8+160 x^7-232 x^6-160 x^5+256 x^4}{x \left (x^2-1\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {24 x^7}{x^2-1}+\frac {160 x^6}{x^2-1}-\frac {232 x^5}{x^2-1}-\frac {160 x^4}{x^2-1}+\frac {256 x^3}{x^2-1}+\frac {2 \left (-x^3+x^2+x+1\right ) \left (\frac {1-x^2}{x}\right )^{2 x} e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} \left (\log \left (\frac {1}{x}-x\right )-x\right )}{x \left (1-x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} \left (\frac {1-x^2}{x}\right )^{2 x}dx-2 \int \frac {e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} \left (\frac {1-x^2}{x}\right )^{2 x}}{1-x}dx-2 \int e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} x \left (\frac {1-x^2}{x}\right )^{2 x}dx-2 \int \frac {e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} \left (\frac {1-x^2}{x}\right )^{2 x}}{x+1}dx+2 \int e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )dx+2 \int \frac {e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-x-1}dx+2 \int \frac {e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x}dx+2 \int \frac {e^{-x^2-\log ^2\left (\frac {1}{x}-x\right )+e^2} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x}dx-4 x^6+32 x^5-64 x^4\) |
Input:
Int[(256*x^4 - 160*x^5 - 232*x^6 + 160*x^7 - 24*x^8 + E^(E^2 - x^2 + 2*x*L og[(1 - x^2)/x] - Log[(1 - x^2)/x]^2)*(2*x + 2*x^2 + 2*x^3 - 2*x^4 + (-2 - 2*x - 2*x^2 + 2*x^3)*Log[(1 - x^2)/x]))/(-x + x^3),x]
Output:
$Aborted
Time = 1.47 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57
| method | result | size |
| parallelrisch | \(-4 x^{6}+32 x^{5}-64 x^{4}+{\mathrm e}^{-\ln \left (-\frac {x^{2}-1}{x}\right )^{2}+2 x \ln \left (-\frac {x^{2}-1}{x}\right )+{\mathrm e}^{2}-x^{2}}\) | \(55\) |
| default | \({\mathrm e}^{-\ln \left (\frac {-x^{2}+1}{x}\right )^{2}+2 x \ln \left (\frac {-x^{2}+1}{x}\right )+{\mathrm e}^{2}-x^{2}}-4 x^{6}+32 x^{5}-64 x^{4}\) | \(57\) |
| parts | \({\mathrm e}^{-\ln \left (\frac {-x^{2}+1}{x}\right )^{2}+2 x \ln \left (\frac {-x^{2}+1}{x}\right )+{\mathrm e}^{2}-x^{2}}-4 x^{6}+32 x^{5}-64 x^{4}\) | \(57\) |
| risch | \(\left (\frac {-x^{2}+1}{x}\right )^{2 x} {\mathrm e}^{-\ln \left (\frac {-x^{2}+1}{x}\right )^{2}+{\mathrm e}^{2}-x^{2}}-4 x^{6}+32 x^{5}-64 x^{4}\) | \(58\) |
Input:
int((((2*x^3-2*x^2-2*x-2)*ln((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp(-ln((- x^2+1)/x)^2+2*x*ln((-x^2+1)/x)+exp(2)-x^2)-24*x^8+160*x^7-232*x^6-160*x^5+ 256*x^4)/(x^3-x),x,method=_RETURNVERBOSE)
Output:
-4*x^6+32*x^5-64*x^4+exp(-ln(-(x^2-1)/x)^2+2*x*ln(-(x^2-1)/x)+exp(2)-x^2)
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=-4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (-\frac {x^{2} - 1}{x}\right ) - \log \left (-\frac {x^{2} - 1}{x}\right )^{2} + e^{2}\right )} \] Input:
integrate((((2*x^3-2*x^2-2*x-2)*log((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp (-log((-x^2+1)/x)^2+2*x*log((-x^2+1)/x)+exp(2)-x^2)-24*x^8+160*x^7-232*x^6 -160*x^5+256*x^4)/(x^3-x),x, algorithm="fricas")
Output:
-4*x^6 + 32*x^5 - 64*x^4 + e^(-x^2 + 2*x*log(-(x^2 - 1)/x) - log(-(x^2 - 1 )/x)^2 + e^2)
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=- 4 x^{6} + 32 x^{5} - 64 x^{4} + e^{- x^{2} + 2 x \log {\left (\frac {1 - x^{2}}{x} \right )} - \log {\left (\frac {1 - x^{2}}{x} \right )}^{2} + e^{2}} \] Input:
integrate((((2*x**3-2*x**2-2*x-2)*ln((-x**2+1)/x)-2*x**4+2*x**3+2*x**2+2*x )*exp(-ln((-x**2+1)/x)**2+2*x*ln((-x**2+1)/x)+exp(2)-x**2)-24*x**8+160*x** 7-232*x**6-160*x**5+256*x**4)/(x**3-x),x)
Output:
-4*x**6 + 32*x**5 - 64*x**4 + exp(-x**2 + 2*x*log((1 - x**2)/x) - log((1 - x**2)/x)**2 + exp(2))
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (31) = 62\).
Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.86 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=-4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (x + 1\right ) - \log \left (x + 1\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, \log \left (x + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, x \log \left (-x + 1\right ) - 2 \, \log \left (x + 1\right ) \log \left (-x + 1\right ) + 2 \, \log \left (x\right ) \log \left (-x + 1\right ) - \log \left (-x + 1\right )^{2} + e^{2}\right )} \] Input:
integrate((((2*x^3-2*x^2-2*x-2)*log((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp (-log((-x^2+1)/x)^2+2*x*log((-x^2+1)/x)+exp(2)-x^2)-24*x^8+160*x^7-232*x^6 -160*x^5+256*x^4)/(x^3-x),x, algorithm="maxima")
Output:
-4*x^6 + 32*x^5 - 64*x^4 + e^(-x^2 + 2*x*log(x + 1) - log(x + 1)^2 - 2*x*l og(x) + 2*log(x + 1)*log(x) - log(x)^2 + 2*x*log(-x + 1) - 2*log(x + 1)*lo g(-x + 1) + 2*log(x)*log(-x + 1) - log(-x + 1)^2 + e^2)
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=-4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (-x + \frac {1}{x}\right ) - \log \left (-x + \frac {1}{x}\right )^{2} + e^{2}\right )} \] Input:
integrate((((2*x^3-2*x^2-2*x-2)*log((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp (-log((-x^2+1)/x)^2+2*x*log((-x^2+1)/x)+exp(2)-x^2)-24*x^8+160*x^7-232*x^6 -160*x^5+256*x^4)/(x^3-x),x, algorithm="giac")
Output:
-4*x^6 + 32*x^5 - 64*x^4 + e^(-x^2 + 2*x*log(-x + 1/x) - log(-x + 1/x)^2 + e^2)
Time = 4.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=32\,x^5-64\,x^4-4\,x^6+{\mathrm {e}}^{-x^2-{\ln \left (-\frac {x^2-1}{x}\right )}^2+{\mathrm {e}}^2}\,{\left (-\frac {x^2-1}{x}\right )}^{2\,x} \] Input:
int(-(exp(exp(2) - log(-(x^2 - 1)/x)^2 + 2*x*log(-(x^2 - 1)/x) - x^2)*(2*x + 2*x^2 + 2*x^3 - 2*x^4 - log(-(x^2 - 1)/x)*(2*x + 2*x^2 - 2*x^3 + 2)) + 256*x^4 - 160*x^5 - 232*x^6 + 160*x^7 - 24*x^8)/(x - x^3),x)
Output:
32*x^5 - 64*x^4 - 4*x^6 + exp(exp(2) - log(-(x^2 - 1)/x)^2 - x^2)*(-(x^2 - 1)/x)^(2*x)
\[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=2 e^{e^{2}} \left (\int \frac {\left (-x^{2}+1\right )^{2 x}}{x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x^{2}-x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}}}d x \right )-2 e^{e^{2}} \left (\int \frac {\left (-x^{2}+1\right )^{2 x} x^{3}}{x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x^{2}-x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}}}d x \right )+2 e^{e^{2}} \left (\int \frac {\left (-x^{2}+1\right )^{2 x} x^{2}}{x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x^{2}-x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}}}d x \right )+2 e^{e^{2}} \left (\int \frac {\left (-x^{2}+1\right )^{2 x} \mathrm {log}\left (\frac {-x^{2}+1}{x}\right ) x^{2}}{x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x^{2}-x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}}}d x \right )-2 e^{e^{2}} \left (\int \frac {\left (-x^{2}+1\right )^{2 x} \mathrm {log}\left (\frac {-x^{2}+1}{x}\right ) x}{x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x^{2}-x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}}}d x \right )-2 e^{e^{2}} \left (\int \frac {\left (-x^{2}+1\right )^{2 x} \mathrm {log}\left (\frac {-x^{2}+1}{x}\right )}{x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x^{3}-x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x}d x \right )-2 e^{e^{2}} \left (\int \frac {\left (-x^{2}+1\right )^{2 x} \mathrm {log}\left (\frac {-x^{2}+1}{x}\right )}{x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x^{2}-x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}}}d x \right )+2 e^{e^{2}} \left (\int \frac {\left (-x^{2}+1\right )^{2 x} x}{x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}} x^{2}-x^{2 x} e^{\mathrm {log}\left (\frac {-x^{2}+1}{x}\right )^{2}+x^{2}}}d x \right )-4 x^{6}+32 x^{5}-64 x^{4} \] Input:
int((((2*x^3-2*x^2-2*x-2)*log((-x^2+1)/x)-2*x^4+2*x^3+2*x^2+2*x)*exp(-log( (-x^2+1)/x)^2+2*x*log((-x^2+1)/x)+exp(2)-x^2)-24*x^8+160*x^7-232*x^6-160*x ^5+256*x^4)/(x^3-x),x)
Output:
2*(e**(e**2)*int(( - x**2 + 1)**(2*x)/(x**(2*x)*e**(log(( - x**2 + 1)/x)** 2 + x**2)*x**2 - x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)),x) - e**(e* *2)*int((( - x**2 + 1)**(2*x)*x**3)/(x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)*x**2 - x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)),x) + e**(e**2 )*int((( - x**2 + 1)**(2*x)*x**2)/(x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)*x**2 - x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)),x) + e**(e**2)* int((( - x**2 + 1)**(2*x)*log(( - x**2 + 1)/x)*x**2)/(x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)*x**2 - x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x** 2)),x) - e**(e**2)*int((( - x**2 + 1)**(2*x)*log(( - x**2 + 1)/x)*x)/(x**( 2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)*x**2 - x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)),x) - e**(e**2)*int((( - x**2 + 1)**(2*x)*log(( - x**2 + 1)/x))/(x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)*x**3 - x**(2*x)*e** (log(( - x**2 + 1)/x)**2 + x**2)*x),x) - e**(e**2)*int((( - x**2 + 1)**(2* x)*log(( - x**2 + 1)/x))/(x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)*x** 2 - x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)),x) + e**(e**2)*int((( - x**2 + 1)**(2*x)*x)/(x**(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)*x**2 - x **(2*x)*e**(log(( - x**2 + 1)/x)**2 + x**2)),x) - 2*x**6 + 16*x**5 - 32*x* *4)