Integrand size = 183, antiderivative size = 20 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{-\frac {1}{2}+x-\frac {x^4}{\left (-2+\log \left (x^2\right )\right )^4}} \] Output:
exp(x-1/2-x^4/(-2+ln(x^2))^4)
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(20)=40\).
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{-\frac {1}{2}+x+\frac {-32+64 x-x^4}{\left (-2+\log \left (x^2\right )\right )^4}+\frac {16 (-1+2 x)}{\left (-2+\log \left (x^2\right )\right )^3}} \left (x^2\right )^{-\frac {16 (-1+2 x)}{\left (-2+\log \left (x^2\right )\right )^4}} \] Input:
Integrate[(E^((-16 + 32*x - 2*x^4 + (32 - 64*x)*Log[x^2] + (-24 + 48*x)*Lo g[x^2]^2 + (8 - 16*x)*Log[x^2]^3 + (-1 + 2*x)*Log[x^2]^4)/(32 - 64*Log[x^2 ] + 48*Log[x^2]^2 - 16*Log[x^2]^3 + 2*Log[x^2]^4))*(-32 + 16*x^3 + (80 - 4 *x^3)*Log[x^2] - 80*Log[x^2]^2 + 40*Log[x^2]^3 - 10*Log[x^2]^4 + Log[x^2]^ 5))/(-32 + 80*Log[x^2] - 80*Log[x^2]^2 + 40*Log[x^2]^3 - 10*Log[x^2]^4 + L og[x^2]^5),x]
Output:
E^(-1/2 + x + (-32 + 64*x - x^4)/(-2 + Log[x^2])^4 + (16*(-1 + 2*x))/(-2 + Log[x^2])^3)/(x^2)^((16*(-1 + 2*x))/(-2 + Log[x^2])^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.
| \(\displaystyle \int \frac {\left (16 x^3+\log ^5\left (x^2\right )-10 \log ^4\left (x^2\right )+40 \log ^3\left (x^2\right )-80 \log ^2\left (x^2\right )+\left (80-4 x^3\right ) \log \left (x^2\right )-32\right ) \exp \left (\frac {-2 x^4+(2 x-1) \log ^4\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(48 x-24) \log ^2\left (x^2\right )+(32-64 x) \log \left (x^2\right )+32 x-16}{2 \log ^4\left (x^2\right )-16 \log ^3\left (x^2\right )+48 \log ^2\left (x^2\right )-64 \log \left (x^2\right )+32}\right )}{\log ^5\left (x^2\right )-10 \log ^4\left (x^2\right )+40 \log ^3\left (x^2\right )-80 \log ^2\left (x^2\right )+80 \log \left (x^2\right )-32} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-16 x^3-\log ^5\left (x^2\right )+10 \log ^4\left (x^2\right )-40 \log ^3\left (x^2\right )+80 \log ^2\left (x^2\right )-\left (80-4 x^3\right ) \log \left (x^2\right )+32\right ) \exp \left (\frac {-2 x^4+(2 x-1) \log ^4\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(48 x-24) \log ^2\left (x^2\right )+(32-64 x) \log \left (x^2\right )+32 x-16}{2 \left (\log \left (x^2\right )-2\right )^4}\right )}{\left (2-\log \left (x^2\right )\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\exp \left (\frac {-2 x^4+(2 x-1) \log ^4\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(48 x-24) \log ^2\left (x^2\right )+(32-64 x) \log \left (x^2\right )+32 x-16}{2 \left (\log \left (x^2\right )-2\right )^4}\right )-\frac {4 x^3 \exp \left (\frac {-2 x^4+(2 x-1) \log ^4\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(48 x-24) \log ^2\left (x^2\right )+(32-64 x) \log \left (x^2\right )+32 x-16}{2 \left (\log \left (x^2\right )-2\right )^4}\right )}{\left (\log \left (x^2\right )-2\right )^4}+\frac {8 x^3 \exp \left (\frac {-2 x^4+(2 x-1) \log ^4\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(48 x-24) \log ^2\left (x^2\right )+(32-64 x) \log \left (x^2\right )+32 x-16}{2 \left (\log \left (x^2\right )-2\right )^4}\right )}{\left (\log \left (x^2\right )-2\right )^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \exp \left (\frac {-2 x^4+32 x+(2 x-1) \log ^4\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(48 x-24) \log ^2\left (x^2\right )+(32-64 x) \log \left (x^2\right )-16}{2 \left (\log \left (x^2\right )-2\right )^4}\right )dx+8 \int \frac {\exp \left (\frac {-2 x^4+32 x+(2 x-1) \log ^4\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(48 x-24) \log ^2\left (x^2\right )+(32-64 x) \log \left (x^2\right )-16}{2 \left (\log \left (x^2\right )-2\right )^4}\right ) x^3}{\left (\log \left (x^2\right )-2\right )^5}dx-4 \int \frac {\exp \left (\frac {-2 x^4+32 x+(2 x-1) \log ^4\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(48 x-24) \log ^2\left (x^2\right )+(32-64 x) \log \left (x^2\right )-16}{2 \left (\log \left (x^2\right )-2\right )^4}\right ) x^3}{\left (\log \left (x^2\right )-2\right )^4}dx\) |
Input:
Int[(E^((-16 + 32*x - 2*x^4 + (32 - 64*x)*Log[x^2] + (-24 + 48*x)*Log[x^2] ^2 + (8 - 16*x)*Log[x^2]^3 + (-1 + 2*x)*Log[x^2]^4)/(32 - 64*Log[x^2] + 48 *Log[x^2]^2 - 16*Log[x^2]^3 + 2*Log[x^2]^4))*(-32 + 16*x^3 + (80 - 4*x^3)* Log[x^2] - 80*Log[x^2]^2 + 40*Log[x^2]^3 - 10*Log[x^2]^4 + Log[x^2]^5))/(- 32 + 80*Log[x^2] - 80*Log[x^2]^2 + 40*Log[x^2]^3 - 10*Log[x^2]^4 + Log[x^2 ]^5),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(17)=34\).
Time = 3.92 (sec) , antiderivative size = 86, normalized size of antiderivative = 4.30
| method | result | size |
| risch | \({\mathrm e}^{\frac {2 \ln \left (x^{2}\right )^{4} x -\ln \left (x^{2}\right )^{4}-16 \ln \left (x^{2}\right )^{3} x -2 x^{4}+8 \ln \left (x^{2}\right )^{3}+48 x \ln \left (x^{2}\right )^{2}-24 \ln \left (x^{2}\right )^{2}-64 x \ln \left (x^{2}\right )+32 \ln \left (x^{2}\right )+32 x -16}{2 {\left (-2+\ln \left (x^{2}\right )\right )}^{4}}}\) | \(86\) |
| parallelrisch | \({\mathrm e}^{\frac {\left (-1+2 x \right ) \ln \left (x^{2}\right )^{4}+\left (-16 x +8\right ) \ln \left (x^{2}\right )^{3}+\left (48 x -24\right ) \ln \left (x^{2}\right )^{2}+\left (-64 x +32\right ) \ln \left (x^{2}\right )-2 x^{4}+32 x -16}{2 \ln \left (x^{2}\right )^{4}-16 \ln \left (x^{2}\right )^{3}+48 \ln \left (x^{2}\right )^{2}-64 \ln \left (x^{2}\right )+32}}\) | \(92\) |
Input:
int((ln(x^2)^5-10*ln(x^2)^4+40*ln(x^2)^3-80*ln(x^2)^2+(-4*x^3+80)*ln(x^2)+ 16*x^3-32)*exp(((-1+2*x)*ln(x^2)^4+(-16*x+8)*ln(x^2)^3+(48*x-24)*ln(x^2)^2 +(-64*x+32)*ln(x^2)-2*x^4+32*x-16)/(2*ln(x^2)^4-16*ln(x^2)^3+48*ln(x^2)^2- 64*ln(x^2)+32))/(ln(x^2)^5-10*ln(x^2)^4+40*ln(x^2)^3-80*ln(x^2)^2+80*ln(x^ 2)-32),x,method=_RETURNVERBOSE)
Output:
exp(1/2*(2*ln(x^2)^4*x-ln(x^2)^4-16*ln(x^2)^3*x-2*x^4+8*ln(x^2)^3+48*x*ln( x^2)^2-24*ln(x^2)^2-64*x*ln(x^2)+32*ln(x^2)+32*x-16)/(-2+ln(x^2))^4)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (17) = 34\).
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.70 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{\left (\frac {{\left (2 \, x - 1\right )} \log \left (x^{2}\right )^{4} - 2 \, x^{4} - 8 \, {\left (2 \, x - 1\right )} \log \left (x^{2}\right )^{3} + 24 \, {\left (2 \, x - 1\right )} \log \left (x^{2}\right )^{2} - 32 \, {\left (2 \, x - 1\right )} \log \left (x^{2}\right ) + 32 \, x - 16}{2 \, {\left (\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16\right )}}\right )} \] Input:
integrate((log(x^2)^5-10*log(x^2)^4+40*log(x^2)^3-80*log(x^2)^2+(-4*x^3+80 )*log(x^2)+16*x^3-32)*exp(((-1+2*x)*log(x^2)^4+(-16*x+8)*log(x^2)^3+(48*x- 24)*log(x^2)^2+(-64*x+32)*log(x^2)-2*x^4+32*x-16)/(2*log(x^2)^4-16*log(x^2 )^3+48*log(x^2)^2-64*log(x^2)+32))/(log(x^2)^5-10*log(x^2)^4+40*log(x^2)^3 -80*log(x^2)^2+80*log(x^2)-32),x, algorithm="fricas")
Output:
e^(1/2*((2*x - 1)*log(x^2)^4 - 2*x^4 - 8*(2*x - 1)*log(x^2)^3 + 24*(2*x - 1)*log(x^2)^2 - 32*(2*x - 1)*log(x^2) + 32*x - 16)/(log(x^2)^4 - 8*log(x^2 )^3 + 24*log(x^2)^2 - 32*log(x^2) + 16))
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (17) = 34\).
Time = 0.52 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.50 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{\frac {- 2 x^{4} + 32 x + \left (8 - 16 x\right ) \log {\left (x^{2} \right )}^{3} + \left (32 - 64 x\right ) \log {\left (x^{2} \right )} + \left (2 x - 1\right ) \log {\left (x^{2} \right )}^{4} + \left (48 x - 24\right ) \log {\left (x^{2} \right )}^{2} - 16}{2 \log {\left (x^{2} \right )}^{4} - 16 \log {\left (x^{2} \right )}^{3} + 48 \log {\left (x^{2} \right )}^{2} - 64 \log {\left (x^{2} \right )} + 32}} \] Input:
integrate((ln(x**2)**5-10*ln(x**2)**4+40*ln(x**2)**3-80*ln(x**2)**2+(-4*x* *3+80)*ln(x**2)+16*x**3-32)*exp(((-1+2*x)*ln(x**2)**4+(-16*x+8)*ln(x**2)** 3+(48*x-24)*ln(x**2)**2+(-64*x+32)*ln(x**2)-2*x**4+32*x-16)/(2*ln(x**2)**4 -16*ln(x**2)**3+48*ln(x**2)**2-64*ln(x**2)+32))/(ln(x**2)**5-10*ln(x**2)** 4+40*ln(x**2)**3-80*ln(x**2)**2+80*ln(x**2)-32),x)
Output:
exp((-2*x**4 + 32*x + (8 - 16*x)*log(x**2)**3 + (32 - 64*x)*log(x**2) + (2 *x - 1)*log(x**2)**4 + (48*x - 24)*log(x**2)**2 - 16)/(2*log(x**2)**4 - 16 *log(x**2)**3 + 48*log(x**2)**2 - 64*log(x**2) + 32))
Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (17) = 34\).
Time = 0.40 (sec) , antiderivative size = 322, normalized size of antiderivative = 16.10 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{\left (\frac {x \log \left (x\right )^{4}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {x^{4}}{16 \, {\left (\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )}} - \frac {4 \, x \log \left (x\right )^{3}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {\log \left (x\right )^{4}}{2 \, {\left (\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )}} + \frac {6 \, x \log \left (x\right )^{2}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} + \frac {2 \, \log \left (x\right )^{3}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {4 \, x \log \left (x\right )}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {3 \, \log \left (x\right )^{2}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} + \frac {x}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} + \frac {2 \, \log \left (x\right )}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {1}{2 \, {\left (\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )}}\right )} \] Input:
integrate((log(x^2)^5-10*log(x^2)^4+40*log(x^2)^3-80*log(x^2)^2+(-4*x^3+80 )*log(x^2)+16*x^3-32)*exp(((-1+2*x)*log(x^2)^4+(-16*x+8)*log(x^2)^3+(48*x- 24)*log(x^2)^2+(-64*x+32)*log(x^2)-2*x^4+32*x-16)/(2*log(x^2)^4-16*log(x^2 )^3+48*log(x^2)^2-64*log(x^2)+32))/(log(x^2)^5-10*log(x^2)^4+40*log(x^2)^3 -80*log(x^2)^2+80*log(x^2)-32),x, algorithm="maxima")
Output:
e^(x*log(x)^4/(log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 4*log(x) + 1) - 1/16*x ^4/(log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 4*log(x) + 1) - 4*x*log(x)^3/(log (x)^4 - 4*log(x)^3 + 6*log(x)^2 - 4*log(x) + 1) - 1/2*log(x)^4/(log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 4*log(x) + 1) + 6*x*log(x)^2/(log(x)^4 - 4*log( x)^3 + 6*log(x)^2 - 4*log(x) + 1) + 2*log(x)^3/(log(x)^4 - 4*log(x)^3 + 6* log(x)^2 - 4*log(x) + 1) - 4*x*log(x)/(log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 4*log(x) + 1) - 3*log(x)^2/(log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 4*log(x ) + 1) + x/(log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 4*log(x) + 1) + 2*log(x)/ (log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 4*log(x) + 1) - 1/2/(log(x)^4 - 4*lo g(x)^3 + 6*log(x)^2 - 4*log(x) + 1))
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (17) = 34\).
Time = 1.89 (sec) , antiderivative size = 427, normalized size of antiderivative = 21.35 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{\left (\frac {x \log \left (x^{2}\right )^{4}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {x^{4}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {8 \, x \log \left (x^{2}\right )^{3}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {\log \left (x^{2}\right )^{4}}{2 \, {\left (\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16\right )}} + \frac {24 \, x \log \left (x^{2}\right )^{2}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} + \frac {4 \, \log \left (x^{2}\right )^{3}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {32 \, x \log \left (x^{2}\right )}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {12 \, \log \left (x^{2}\right )^{2}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} + \frac {16 \, x}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} + \frac {16 \, \log \left (x^{2}\right )}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {8}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16}\right )} \] Input:
integrate((log(x^2)^5-10*log(x^2)^4+40*log(x^2)^3-80*log(x^2)^2+(-4*x^3+80 )*log(x^2)+16*x^3-32)*exp(((-1+2*x)*log(x^2)^4+(-16*x+8)*log(x^2)^3+(48*x- 24)*log(x^2)^2+(-64*x+32)*log(x^2)-2*x^4+32*x-16)/(2*log(x^2)^4-16*log(x^2 )^3+48*log(x^2)^2-64*log(x^2)+32))/(log(x^2)^5-10*log(x^2)^4+40*log(x^2)^3 -80*log(x^2)^2+80*log(x^2)-32),x, algorithm="giac")
Output:
e^(x*log(x^2)^4/(log(x^2)^4 - 8*log(x^2)^3 + 24*log(x^2)^2 - 32*log(x^2) + 16) - x^4/(log(x^2)^4 - 8*log(x^2)^3 + 24*log(x^2)^2 - 32*log(x^2) + 16) - 8*x*log(x^2)^3/(log(x^2)^4 - 8*log(x^2)^3 + 24*log(x^2)^2 - 32*log(x^2) + 16) - 1/2*log(x^2)^4/(log(x^2)^4 - 8*log(x^2)^3 + 24*log(x^2)^2 - 32*log (x^2) + 16) + 24*x*log(x^2)^2/(log(x^2)^4 - 8*log(x^2)^3 + 24*log(x^2)^2 - 32*log(x^2) + 16) + 4*log(x^2)^3/(log(x^2)^4 - 8*log(x^2)^3 + 24*log(x^2) ^2 - 32*log(x^2) + 16) - 32*x*log(x^2)/(log(x^2)^4 - 8*log(x^2)^3 + 24*log (x^2)^2 - 32*log(x^2) + 16) - 12*log(x^2)^2/(log(x^2)^4 - 8*log(x^2)^3 + 2 4*log(x^2)^2 - 32*log(x^2) + 16) + 16*x/(log(x^2)^4 - 8*log(x^2)^3 + 24*lo g(x^2)^2 - 32*log(x^2) + 16) + 16*log(x^2)/(log(x^2)^4 - 8*log(x^2)^3 + 24 *log(x^2)^2 - 32*log(x^2) + 16) - 8/(log(x^2)^4 - 8*log(x^2)^3 + 24*log(x^ 2)^2 - 32*log(x^2) + 16))
Time = 0.81 (sec) , antiderivative size = 419, normalized size of antiderivative = 20.95 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx={\mathrm {e}}^{-\frac {2\,x^4}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{-\frac {16}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{\frac {2\,x\,{\ln \left (x^2\right )}^4}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{-\frac {16\,x\,{\ln \left (x^2\right )}^3}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{\frac {48\,x\,{\ln \left (x^2\right )}^2}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{\frac {32\,x}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{-\frac {{\ln \left (x^2\right )}^4}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{\frac {8\,{\ln \left (x^2\right )}^3}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{-\frac {24\,{\ln \left (x^2\right )}^2}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\left (\frac {1}{x^{32}}\right )}^{\frac {2\,x-1}{{\ln \left (x^2\right )}^4-8\,{\ln \left (x^2\right )}^3+24\,{\ln \left (x^2\right )}^2-32\,\ln \left (x^2\right )+16}} \] Input:
int(-(exp(-(log(x^2)^3*(16*x - 8) - log(x^2)^4*(2*x - 1) - 32*x - log(x^2) ^2*(48*x - 24) + 2*x^4 + log(x^2)*(64*x - 32) + 16)/(48*log(x^2)^2 - 64*lo g(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32))*(log(x^2)*(4*x^3 - 80) + 80*l og(x^2)^2 - 40*log(x^2)^3 + 10*log(x^2)^4 - log(x^2)^5 - 16*x^3 + 32))/(80 *log(x^2) - 80*log(x^2)^2 + 40*log(x^2)^3 - 10*log(x^2)^4 + log(x^2)^5 - 3 2),x)
Output:
exp(-(2*x^4)/(48*log(x^2)^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32))*exp(-16/(48*log(x^2)^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32))*exp((2*x*log(x^2)^4)/(48*log(x^2)^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32))*exp(-(16*x*log(x^2)^3)/(48*log(x^2)^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32))*exp((48*x*log(x^2)^2)/(48*log(x^2)^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32))*exp((32*x)/(48*log(x^2 )^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32))*exp(-log(x^2)^4/(4 8*log(x^2)^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32))*exp((8*lo g(x^2)^3)/(48*log(x^2)^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*log(x^2)^4 + 32 ))*exp(-(24*log(x^2)^2)/(48*log(x^2)^2 - 64*log(x^2) - 16*log(x^2)^3 + 2*l og(x^2)^4 + 32))*(1/x^32)^((2*x - 1)/(24*log(x^2)^2 - 32*log(x^2) - 8*log( x^2)^3 + log(x^2)^4 + 16))
Time = 0.39 (sec) , antiderivative size = 147, normalized size of antiderivative = 7.35 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=\frac {e^{\frac {\mathrm {log}\left (x^{2}\right )^{4} x -8 \mathrm {log}\left (x^{2}\right )^{3} x +4 \mathrm {log}\left (x^{2}\right )^{3}+24 \mathrm {log}\left (x^{2}\right )^{2} x -32 \,\mathrm {log}\left (x^{2}\right ) x +16 \,\mathrm {log}\left (x^{2}\right )+16 x}{\mathrm {log}\left (x^{2}\right )^{4}-8 \mathrm {log}\left (x^{2}\right )^{3}+24 \mathrm {log}\left (x^{2}\right )^{2}-32 \,\mathrm {log}\left (x^{2}\right )+16}}}{e^{\frac {\mathrm {log}\left (x^{2}\right )^{4}+24 \mathrm {log}\left (x^{2}\right )^{2}+2 x^{4}+16}{2 \mathrm {log}\left (x^{2}\right )^{4}-16 \mathrm {log}\left (x^{2}\right )^{3}+48 \mathrm {log}\left (x^{2}\right )^{2}-64 \,\mathrm {log}\left (x^{2}\right )+32}}} \] Input:
int((log(x^2)^5-10*log(x^2)^4+40*log(x^2)^3-80*log(x^2)^2+(-4*x^3+80)*log( x^2)+16*x^3-32)*exp(((-1+2*x)*log(x^2)^4+(-16*x+8)*log(x^2)^3+(48*x-24)*lo g(x^2)^2+(-64*x+32)*log(x^2)-2*x^4+32*x-16)/(2*log(x^2)^4-16*log(x^2)^3+48 *log(x^2)^2-64*log(x^2)+32))/(log(x^2)^5-10*log(x^2)^4+40*log(x^2)^3-80*lo g(x^2)^2+80*log(x^2)-32),x)
Output:
e**((log(x**2)**4*x - 8*log(x**2)**3*x + 4*log(x**2)**3 + 24*log(x**2)**2* x - 32*log(x**2)*x + 16*log(x**2) + 16*x)/(log(x**2)**4 - 8*log(x**2)**3 + 24*log(x**2)**2 - 32*log(x**2) + 16))/e**((log(x**2)**4 + 24*log(x**2)**2 + 2*x**4 + 16)/(2*log(x**2)**4 - 16*log(x**2)**3 + 48*log(x**2)**2 - 64*l og(x**2) + 32))