Integrand size = 147, antiderivative size = 32 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=3-e^{4+x+\frac {x}{4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-\log (x)}} \] Output:
3-exp(x+x/(4*ln(2/5)+4*I*Pi-ln(x))+4)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(32)=64\).
Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.66 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-e^{4+x+\frac {4 \pi (4+x)+i \left (16 \log \left (\frac {5}{2}\right )+x \left (-1+4 \log \left (\frac {5}{2}\right )\right )\right )}{4 \pi +4 i \log \left (\frac {5}{2}\right )+i \log (x)}} x^{(4+x) \left (-\frac {1}{\log (x)}+\frac {1}{-4 i \pi +4 \log \left (\frac {5}{2}\right )+\log (x)}\right )} \] Input:
Integrate[(E^((-x + (-16 - 4*x)*(I*Pi - Log[5/2]) + (4 + x)*Log[x])/(-4*(I *Pi - Log[5/2]) + Log[x]))*(-1 - 4*(I*Pi - Log[5/2]) - 16*(I*Pi - Log[5/2] )^2 + (1 + 8*(I*Pi - Log[5/2]))*Log[x] - Log[x]^2))/(16*(I*Pi - Log[5/2])^ 2 - 8*(I*Pi - Log[5/2])*Log[x] + Log[x]^2),x]
Output:
-(E^(4 + x + (4*Pi*(4 + x) + I*(16*Log[5/2] + x*(-1 + 4*Log[5/2])))/(4*Pi + (4*I)*Log[5/2] + I*Log[x]))*x^((4 + x)*(-Log[x]^(-1) + ((-4*I)*Pi + 4*Lo g[5/2] + Log[x])^(-1))))
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.
| \(\displaystyle \int \frac {\left (-\log ^2(x)+\left (1+8 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )\right ) \log (x)-1-16 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )^2-4 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )\right ) \exp \left (\frac {-x+(-4 x-16) \left (-\log \left (\frac {5}{2}\right )+i \pi \right )+(x+4) \log (x)}{\log (x)-4 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )}\right )}{\log ^2(x)-8 \left (-\log \left (\frac {5}{2}\right )+i \pi \right ) \log (x)+16 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )^2} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (\log ^2(x)-\left (1+8 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )\right ) \log (x)+1-4 \left (\pi +i \log \left (\frac {5}{2}\right )\right ) \left (-i+4 \pi +4 i \log \left (\frac {5}{2}\right )\right )\right ) \exp \left (\frac {-x+(-4 x-16) \left (-\log \left (\frac {5}{2}\right )+i \pi \right )+(x+4) \log (x)}{\log (x)-4 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )}\right )}{\left (i \log (x)+4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\exp \left (\frac {-x+(-4 x-16) \left (-\log \left (\frac {5}{2}\right )+i \pi \right )+(x+4) \log (x)}{\log (x)-4 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )}\right )+\frac {i \exp \left (\frac {-x+(-4 x-16) \left (-\log \left (\frac {5}{2}\right )+i \pi \right )+(x+4) \log (x)}{\log (x)-4 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )}\right )}{i \log (x)+4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )}+\frac {\exp \left (\frac {-x+(-4 x-16) \left (-\log \left (\frac {5}{2}\right )+i \pi \right )+(x+4) \log (x)}{\log (x)-4 \left (-\log \left (\frac {5}{2}\right )+i \pi \right )}\right )}{\left (i \log (x)+4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int \exp \left (\frac {\left (i \pi -\log \left (\frac {5}{2}\right )\right ) (-4 x-16)-x+(x+4) \log (x)}{\log (x)-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )}\right )dx+\int \frac {\exp \left (\frac {\left (i \pi -\log \left (\frac {5}{2}\right )\right ) (-4 x-16)-x+(x+4) \log (x)}{\log (x)-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )}\right )}{\left (i \log (x)+4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )\right )^2}dx+i \int \frac {\exp \left (\frac {\left (i \pi -\log \left (\frac {5}{2}\right )\right ) (-4 x-16)-x+(x+4) \log (x)}{\log (x)-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )}\right )}{i \log (x)+4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )}dx\) |
Input:
Int[(E^((-x + (-16 - 4*x)*(I*Pi - Log[5/2]) + (4 + x)*Log[x])/(-4*(I*Pi - Log[5/2]) + Log[x]))*(-1 - 4*(I*Pi - Log[5/2]) - 16*(I*Pi - Log[5/2])^2 + (1 + 8*(I*Pi - Log[5/2]))*Log[x] - Log[x]^2))/(16*(I*Pi - Log[5/2])^2 - 8* (I*Pi - Log[5/2])*Log[x] + Log[x]^2),x]
Output:
$Aborted
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (25 ) = 50\).
Time = 1.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94
| method | result | size |
| risch | \(-{\mathrm e}^{\frac {4 i \pi x +16 i \pi +4 x \ln \left (2\right )-x \ln \left (x \right )-4 x \ln \left (5\right )+16 \ln \left (2\right )-4 \ln \left (x \right )-16 \ln \left (5\right )+x}{4 \ln \left (2\right )-4 \ln \left (5\right )+4 i \pi -\ln \left (x \right )}}\) | \(62\) |
| norman | \(\frac {\left (-16 \pi ^{2}-16 \ln \left (2\right )^{2}+32 \ln \left (2\right ) \ln \left (5\right )-16 \ln \left (5\right )^{2}\right ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (x \right )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \left (x \right )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}+\left (8 \ln \left (2\right )-8 \ln \left (5\right )\right ) \ln \left (x \right ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (x \right )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \left (x \right )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}-\ln \left (x \right )^{2} {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (x \right )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \left (x \right )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}}{16 \pi ^{2}+16 \ln \left (2\right )^{2}-32 \ln \left (2\right ) \ln \left (5\right )-8 \ln \left (2\right ) \ln \left (x \right )+16 \ln \left (5\right )^{2}+8 \ln \left (5\right ) \ln \left (x \right )+\ln \left (x \right )^{2}}\) | \(202\) |
Input:
int((-ln(x)^2+(8*ln(2/5)+8*I*Pi+1)*ln(x)-16*(ln(2/5)+I*Pi)^2-4*ln(2/5)-4*I *Pi-1)*exp(((4+x)*ln(x)+(-16-4*x)*(ln(2/5)+I*Pi)-x)/(ln(x)-4*ln(2/5)-4*I*P i))/(ln(x)^2-8*(ln(2/5)+I*Pi)*ln(x)+16*(ln(2/5)+I*Pi)^2),x,method=_RETURNV ERBOSE)
Output:
-exp((4*I*Pi*x+16*I*Pi+4*x*ln(2)-x*ln(x)-4*x*ln(5)+16*ln(2)-4*ln(x)-16*ln( 5)+x)/(4*ln(2)-4*ln(5)+4*I*Pi-ln(x)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (50) = 100\).
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.59 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-e^{\left (-\frac {4 i \, \pi x}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} - \frac {4 \, x \log \left (\frac {2}{5}\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} + \frac {x \log \left (x\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} - \frac {16 i \, \pi }{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} - \frac {x}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} - \frac {16 \, \log \left (\frac {2}{5}\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} + \frac {4 \, \log \left (x\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )}\right )} \] Input:
integrate((-log(x)^2+(8*log(2/5)+8*I*pi+1)*log(x)-16*(log(2/5)+I*pi)^2-4*l og(2/5)-4*I*pi-1)*exp(((4+x)*log(x)+(-16-4*x)*(log(2/5)+I*pi)-x)/(log(x)-4 *log(2/5)-4*I*pi))/(log(x)^2-8*(log(2/5)+I*pi)*log(x)+16*(log(2/5)+I*pi)^2 ),x, algorithm="fricas")
Output:
-e^(-4*I*pi*x/(-4*I*pi - 4*log(2/5) + log(x)) - 4*x*log(2/5)/(-4*I*pi - 4* log(2/5) + log(x)) + x*log(x)/(-4*I*pi - 4*log(2/5) + log(x)) - 16*I*pi/(- 4*I*pi - 4*log(2/5) + log(x)) - x/(-4*I*pi - 4*log(2/5) + log(x)) - 16*log (2/5)/(-4*I*pi - 4*log(2/5) + log(x)) + 4*log(x)/(-4*I*pi - 4*log(2/5) + l og(x)))
Timed out. \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=\text {Timed out} \] Input:
integrate((-ln(x)**2+(8*ln(2/5)+8*I*pi+1)*ln(x)-16*(ln(2/5)+I*pi)**2-4*ln( 2/5)-4*I*pi-1)*exp(((4+x)*ln(x)+(-16-4*x)*(ln(2/5)+I*pi)-x)/(ln(x)-4*ln(2/ 5)-4*I*pi))/(ln(x)**2-8*(ln(2/5)+I*pi)*ln(x)+16*(ln(2/5)+I*pi)**2),x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (50) = 100\).
Time = 2.58 (sec) , antiderivative size = 184, normalized size of antiderivative = 5.75 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-e^{\left (-\frac {4 i \, \pi x}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} + \frac {4 \, x \log \left (5\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} - \frac {4 \, x \log \left (2\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} + \frac {x \log \left (x\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} - \frac {16 i \, \pi }{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} - \frac {x}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} + \frac {16 \, \log \left (5\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} - \frac {16 \, \log \left (2\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} + \frac {4 \, \log \left (x\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )}\right )} \] Input:
integrate((-log(x)^2+(8*log(2/5)+8*I*pi+1)*log(x)-16*(log(2/5)+I*pi)^2-4*l og(2/5)-4*I*pi-1)*exp(((4+x)*log(x)+(-16-4*x)*(log(2/5)+I*pi)-x)/(log(x)-4 *log(2/5)-4*I*pi))/(log(x)^2-8*(log(2/5)+I*pi)*log(x)+16*(log(2/5)+I*pi)^2 ),x, algorithm="maxima")
Output:
-e^(-4*I*pi*x/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) + 4*x*log(5)/(-4*I* pi + 4*log(5) - 4*log(2) + log(x)) - 4*x*log(2)/(-4*I*pi + 4*log(5) - 4*lo g(2) + log(x)) + x*log(x)/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) - 16*I* pi/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) - x/(-4*I*pi + 4*log(5) - 4*lo g(2) + log(x)) + 16*log(5)/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) - 16*l og(2)/(-4*I*pi + 4*log(5) - 4*log(2) + log(x)) + 4*log(x)/(-4*I*pi + 4*log (5) - 4*log(2) + log(x)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (50) = 100\).
Time = 0.54 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.66 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-e^{\left (\frac {4 \, \pi x}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} + \frac {4 i \, x \log \left (5\right )}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} - \frac {4 i \, x \log \left (2\right )}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} + \frac {i \, x \log \left (x\right )}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} - \frac {i \, x}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} + 4\right )} \] Input:
integrate((-log(x)^2+(8*log(2/5)+8*I*pi+1)*log(x)-16*(log(2/5)+I*pi)^2-4*l og(2/5)-4*I*pi-1)*exp(((4+x)*log(x)+(-16-4*x)*(log(2/5)+I*pi)-x)/(log(x)-4 *log(2/5)-4*I*pi))/(log(x)^2-8*(log(2/5)+I*pi)*log(x)+16*(log(2/5)+I*pi)^2 ),x, algorithm="giac")
Output:
-e^(4*pi*x/(4*pi + 4*I*log(5) - 4*I*log(2) + I*log(x)) + 4*I*x*log(5)/(4*p i + 4*I*log(5) - 4*I*log(2) + I*log(x)) - 4*I*x*log(2)/(4*pi + 4*I*log(5) - 4*I*log(2) + I*log(x)) + I*x*log(x)/(4*pi + 4*I*log(5) - 4*I*log(2) + I* log(x)) - I*x/(4*pi + 4*I*log(5) - 4*I*log(2) + I*log(x)) + 4)
Time = 6.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-{\mathrm {e}}^{\frac {\Pi \,16{}\mathrm {i}}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}+\frac {x}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}+\frac {\Pi \,x\,4{}\mathrm {i}}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}}\,{\left (\frac {625\,x}{16}\right )}^{\frac {x\,1{}\mathrm {i}+4{}\mathrm {i}}{4\,\Pi -\ln \left (\frac {2}{5}\right )\,4{}\mathrm {i}+\ln \left (x\right )\,1{}\mathrm {i}}} \] Input:
int(-(exp((x + (4*x + 16)*(Pi*1i + log(2/5)) - log(x)*(x + 4))/(Pi*4i + 4* log(2/5) - log(x)))*(Pi*4i + 4*log(2/5) - log(x)*(Pi*8i + 8*log(2/5) + 1) + log(x)^2 + 16*(Pi*1i + log(2/5))^2 + 1))/(log(x)^2 - 8*log(x)*(Pi*1i + l og(2/5)) + 16*(Pi*1i + log(2/5))^2),x)
Output:
-exp((Pi*16i)/(Pi*4i - log((625*x)/16)) + x/(Pi*4i - log((625*x)/16)) + (P i*x*4i)/(Pi*4i - log((625*x)/16)))*((625*x)/16)^((x*1i + 4i)/(4*Pi - log(2 /5)*4i + log(x)*1i))
\[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx =\text {Too large to display} \] Input:
int((-log(x)^2+(8*log(2/5)+8*I*Pi+1)*log(x)-16*(log(2/5)+I*Pi)^2-4*log(2/5 )-4*I*Pi-1)*exp(((4+x)*log(x)+(-16-4*x)*(log(2/5)+I*Pi)-x)/(log(x)-4*log(2 /5)-4*I*Pi))/(log(x)^2-8*(log(2/5)+I*Pi)*log(x)+16*(log(2/5)+I*Pi)^2),x)
Output:
e**4*( - 16*int(e**((4*log(2/5)*x - log(x)*x + 4*i*pi*x + x)/(4*log(2/5) - log(x) + 4*i*pi))/(16*log(2/5)**2 - 8*log(2/5)*log(x) + 32*log(2/5)*i*pi + log(x)**2 - 8*log(x)*i*pi - 16*pi**2),x)*log(2/5)**2 - 32*int(e**((4*log (2/5)*x - log(x)*x + 4*i*pi*x + x)/(4*log(2/5) - log(x) + 4*i*pi))/(16*log (2/5)**2 - 8*log(2/5)*log(x) + 32*log(2/5)*i*pi + log(x)**2 - 8*log(x)*i*p i - 16*pi**2),x)*log(2/5)*i*pi - 4*int(e**((4*log(2/5)*x - log(x)*x + 4*i* pi*x + x)/(4*log(2/5) - log(x) + 4*i*pi))/(16*log(2/5)**2 - 8*log(2/5)*log (x) + 32*log(2/5)*i*pi + log(x)**2 - 8*log(x)*i*pi - 16*pi**2),x)*log(2/5) - 4*int(e**((4*log(2/5)*x - log(x)*x + 4*i*pi*x + x)/(4*log(2/5) - log(x) + 4*i*pi))/(16*log(2/5)**2 - 8*log(2/5)*log(x) + 32*log(2/5)*i*pi + log(x )**2 - 8*log(x)*i*pi - 16*pi**2),x)*i*pi + 16*int(e**((4*log(2/5)*x - log( x)*x + 4*i*pi*x + x)/(4*log(2/5) - log(x) + 4*i*pi))/(16*log(2/5)**2 - 8*l og(2/5)*log(x) + 32*log(2/5)*i*pi + log(x)**2 - 8*log(x)*i*pi - 16*pi**2), x)*pi**2 - int(e**((4*log(2/5)*x - log(x)*x + 4*i*pi*x + x)/(4*log(2/5) - log(x) + 4*i*pi))/(16*log(2/5)**2 - 8*log(2/5)*log(x) + 32*log(2/5)*i*pi + log(x)**2 - 8*log(x)*i*pi - 16*pi**2),x) - int((e**((4*log(2/5)*x - log(x )*x + 4*i*pi*x + x)/(4*log(2/5) - log(x) + 4*i*pi))*log(x)**2)/(16*log(2/5 )**2 - 8*log(2/5)*log(x) + 32*log(2/5)*i*pi + log(x)**2 - 8*log(x)*i*pi - 16*pi**2),x) + 8*int((e**((4*log(2/5)*x - log(x)*x + 4*i*pi*x + x)/(4*log( 2/5) - log(x) + 4*i*pi))*log(x))/(16*log(2/5)**2 - 8*log(2/5)*log(x) + ...