Integrand size = 94, antiderivative size = 27 \[ \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx=x \left (-\frac {9}{256}+e^{-2+x}-x+\log \left (-x+\log \left (4 e^{2 x}\right )\right )\right ) \]
Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx=-\frac {1}{256} x \left (9-256 e^{-2+x}+256 x-256 \log \left (-x+\log \left (4 e^{2 x}\right )\right )\right ) \]
Integrate[(265*x + 512*x^2 + E^(-2 + x)*(-256*x - 256*x^2) + (-9 - 512*x + E^(-2 + x)*(256 + 256*x))*Log[4*E^(2*x)] + (-256*x + 256*Log[4*E^(2*x)])* Log[-x + Log[4*E^(2*x)]])/(-256*x + 256*Log[4*E^(2*x)]),x]
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.
| \(\displaystyle \int \frac {512 x^2+e^{x-2} \left (-256 x^2-256 x\right )+265 x+\left (-512 x+e^{x-2} (256 x+256)-9\right ) \log \left (4 e^{2 x}\right )+\left (256 \log \left (4 e^{2 x}\right )-256 x\right ) \log \left (\log \left (4 e^{2 x}\right )-x\right )}{256 \log \left (4 e^{2 x}\right )-256 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-512 x^2-e^{x-2} \left (-256 x^2-256 x\right )-265 x-\left (-512 x+e^{x-2} (256 x+256)-9\right ) \log \left (4 e^{2 x}\right )-\left (256 \log \left (4 e^{2 x}\right )-256 x\right ) \log \left (\log \left (4 e^{2 x}\right )-x\right )}{256 \left (x-\log \left (4 e^{2 x}\right )\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{256} \int -\frac {512 x^2+265 x-256 e^{x-2} \left (x^2+x\right )-\left (512 x-256 e^{x-2} (x+1)+9\right ) \log \left (4 e^{2 x}\right )-256 \left (x-\log \left (4 e^{2 x}\right )\right ) \log \left (\log \left (4 e^{2 x}\right )-x\right )}{x-\log \left (4 e^{2 x}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{256} \int \frac {512 x^2+265 x-256 e^{x-2} \left (x^2+x\right )-\left (512 x-256 e^{x-2} (x+1)+9\right ) \log \left (4 e^{2 x}\right )-256 \left (x-\log \left (4 e^{2 x}\right )\right ) \log \left (\log \left (4 e^{2 x}\right )-x\right )}{x-\log \left (4 e^{2 x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{256} \int \left (\frac {512 x^2-512 \log \left (4 e^{2 x}\right ) x-256 \log \left (\log \left (4 e^{2 x}\right )-x\right ) x+265 x-9 \log \left (4 e^{2 x}\right )+256 \log \left (4 e^{2 x}\right ) \log \left (\log \left (4 e^{2 x}\right )-x\right )}{x-\log \left (4 e^{2 x}\right )}-256 e^{x-2} (x+1)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{256} \left (-256 \int \frac {x}{x-\log \left (4 e^{2 x}\right )}dx+256 \int \log \left (\log \left (4 e^{2 x}\right )-x\right )dx-256 x^2-9 x-256 e^{x-2}+256 e^{x-2} (x+1)\right )\) |
Int[(265*x + 512*x^2 + E^(-2 + x)*(-256*x - 256*x^2) + (-9 - 512*x + E^(-2 + x)*(256 + 256*x))*Log[4*E^(2*x)] + (-256*x + 256*Log[4*E^(2*x)])*Log[-x + Log[4*E^(2*x)]])/(-256*x + 256*Log[4*E^(2*x)]),x]
3.3.49.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(23)=46\).
Time = 1.42 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22
| method | result | size |
| parallelrisch | \(-\frac {9 x^{2}}{13}+x \,{\mathrm e}^{-2+x}-\frac {4 x \ln \left (4 \,{\mathrm e}^{2 x}\right )}{13}+\ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) x +\frac {\ln \left (4 \,{\mathrm e}^{2 x}\right )^{2}}{13}-\frac {9 x}{512}-\frac {9 \ln \left (4 \,{\mathrm e}^{2 x}\right )}{1024}\) | \(60\) |
| risch | \(x \ln \left (2 \ln \left (2\right )+2 \ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) {\left (-\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}^{2}}{2}-x \right )-x^{2}-\frac {9 x}{256}+x \,{\mathrm e}^{-2+x}\) | \(63\) |
| default | \(-x^{2}+\frac {503 x}{256}+\frac {\left (512 x -256 \ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) \ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right )}{256}+\ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right )-\ln \left (4 \,{\mathrm e}^{2 x}\right )-6 \,{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+2 \,{\mathrm e}^{-2+x}-5 \left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+\left (-2+x \right ) {\mathrm e}^{-2+x}-\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-2+x}-2 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) {\mathrm e}^{-2+x}-\left ({\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )}^{2}+4 \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) \left (\ln \left ({\mathrm e}^{x}\right )-x \right )+4 {\left (\ln \left ({\mathrm e}^{x}\right )-x \right )}^{2}+4 \ln \left (4 \,{\mathrm e}^{2 x}\right )-8 x +4\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )-3 \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )-6 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) \left ({\mathrm e}^{-2+x}-\left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )\right )+2 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left ({\mathrm e}^{-2+x}-\left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )\right )\) | \(483\) |
| parts | \(-x^{2}+\frac {503 x}{256}+\frac {\left (512 x -256 \ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) \ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right )}{256}+\ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right )-\ln \left (4 \,{\mathrm e}^{2 x}\right )-6 \,{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+2 \,{\mathrm e}^{-2+x}-5 \left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+\left (-2+x \right ) {\mathrm e}^{-2+x}-\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-2+x}-2 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) {\mathrm e}^{-2+x}-\left ({\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )}^{2}+4 \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) \left (\ln \left ({\mathrm e}^{x}\right )-x \right )+4 {\left (\ln \left ({\mathrm e}^{x}\right )-x \right )}^{2}+4 \ln \left (4 \,{\mathrm e}^{2 x}\right )-8 x +4\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )-3 \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )-6 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) \left ({\mathrm e}^{-2+x}-\left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )\right )+2 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left ({\mathrm e}^{-2+x}-\left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \operatorname {Ei}_{1}\left (x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )\right )\) | \(483\) |
int(((256*ln(4*exp(x)^2)-256*x)*ln(ln(4*exp(x)^2)-x)+((256*x+256)*exp(-2+x )-512*x-9)*ln(4*exp(x)^2)+(-256*x^2-256*x)*exp(-2+x)+512*x^2+265*x)/(256*l n(4*exp(x)^2)-256*x),x,method=_RETURNVERBOSE)
-9/13*x^2+x*exp(-2+x)-4/13*x*ln(4*exp(x)^2)+ln(ln(4*exp(x)^2)-x)*x+1/13*ln (4*exp(x)^2)^2-9/512*x-9/1024*ln(4*exp(x)^2)
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx=-x^{2} + x e^{\left (x - 2\right )} + x \log \left (x + 2 \, \log \left (2\right )\right ) - \frac {9}{256} \, x \]
integrate(((256*log(4*exp(x)^2)-256*x)*log(log(4*exp(x)^2)-x)+((256*x+256) *exp(-2+x)-512*x-9)*log(4*exp(x)^2)+(-256*x^2-256*x)*exp(-2+x)+512*x^2+265 *x)/(256*log(4*exp(x)^2)-256*x),x, algorithm=\
Timed out. \[ \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx=\text {Timed out} \]
integrate(((256*ln(4*exp(x)**2)-256*x)*ln(ln(4*exp(x)**2)-x)+((256*x+256)* exp(-2+x)-512*x-9)*ln(4*exp(x)**2)+(-256*x**2-256*x)*exp(-2+x)+512*x**2+26 5*x)/(256*ln(4*exp(x)**2)-256*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (28) = 56\).
Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.41 \[ \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx=8 \, \log \left (2\right )^{2} \log \left (x + 2 \, \log \left (2\right )\right ) + x^{2} - \frac {1}{128} \, {\left (256 \, x^{2} e^{2} - x {\left (512 \, \log \left (2\right ) - 137\right )} e^{2} - 128 \, x e^{x} + {\left ({\left (1024 \, \log \left (2\right )^{2} - 265 \, \log \left (2\right )\right )} e^{2} - 128 \, x e^{2}\right )} \log \left (x + 2 \, \log \left (2\right )\right )\right )} e^{\left (-2\right )} - 4 \, x \log \left (2\right ) - \frac {265}{128} \, \log \left (2\right ) \log \left (x + 2 \, \log \left (2\right )\right ) + \frac {265}{256} \, x \]
integrate(((256*log(4*exp(x)^2)-256*x)*log(log(4*exp(x)^2)-x)+((256*x+256) *exp(-2+x)-512*x-9)*log(4*exp(x)^2)+(-256*x^2-256*x)*exp(-2+x)+512*x^2+265 *x)/(256*log(4*exp(x)^2)-256*x),x, algorithm=\
8*log(2)^2*log(x + 2*log(2)) + x^2 - 1/128*(256*x^2*e^2 - x*(512*log(2) - 137)*e^2 - 128*x*e^x + ((1024*log(2)^2 - 265*log(2))*e^2 - 128*x*e^2)*log( x + 2*log(2)))*e^(-2) - 4*x*log(2) - 265/128*log(2)*log(x + 2*log(2)) + 26 5/256*x
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx=-\frac {1}{256} \, {\left (256 \, x^{2} e^{2} - 256 \, x e^{2} \log \left (x + 2 \, \log \left (2\right )\right ) + 9 \, x e^{2} - 256 \, x e^{x}\right )} e^{\left (-2\right )} \]
integrate(((256*log(4*exp(x)^2)-256*x)*log(log(4*exp(x)^2)-x)+((256*x+256) *exp(-2+x)-512*x-9)*log(4*exp(x)^2)+(-256*x^2-256*x)*exp(-2+x)+512*x^2+265 *x)/(256*log(4*exp(x)^2)-256*x),x, algorithm=\
Time = 9.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx=x\,{\mathrm {e}}^{x-2}-\frac {9\,x}{256}-x^2+x\,\ln \left (x+2\,\ln \left (2\right )\right ) \]