\(\int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log (\log (\frac {4-2 x-e^x x}{x}))+\log ^2(\log (\frac {4-2 x-e^x x}{x}))} x (e^{4 e^x} (8+2 e^x x^2)+(-4+2 x+e^x x+e^{8 e^x} (8 e^{2 x} x^2+e^x (-32 x+16 x^2))) \log (\frac {4-2 x-e^x x}{x})+(8+2 e^x x^2+e^{4 e^x} (8 e^{2 x} x^2+e^x (-32 x+16 x^2)) \log (\frac {4-2 x-e^x x}{x})) \log (\log (\frac {4-2 x-e^x x}{x})))}{(-4 x+2 x^2+e^x x^2) \log (\frac {4-2 x-e^x x}{x})} \, dx\) [220]

Optimal result

Integrand size = 247, antiderivative size = 28 \[ \int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )} x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx=e^{\left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )^2} x \]

[Out]

exp(ln(x)+(ln(ln(4/x-exp(x)-2))+exp(4*exp(x)))^2)

Rubi [F]

\[ \int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )} x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx=\int \frac {\exp \left (e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right ) x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx \]

[In]

Int[(E^(E^(8*E^x) + 2*E^(4*E^x)*Log[Log[(4 - 2*x - E^x*x)/x]] + Log[Log[(4 - 2*x - E^x*x)/x]]^2)*x*(E^(4*E^x)*
(8 + 2*E^x*x^2) + (-4 + 2*x + E^x*x + E^(8*E^x)*(8*E^(2*x)*x^2 + E^x*(-32*x + 16*x^2)))*Log[(4 - 2*x - E^x*x)/
x] + (8 + 2*E^x*x^2 + E^(4*E^x)*(8*E^(2*x)*x^2 + E^x*(-32*x + 16*x^2))*Log[(4 - 2*x - E^x*x)/x])*Log[Log[(4 -
2*x - E^x*x)/x]]))/((-4*x + 2*x^2 + E^x*x^2)*Log[(4 - 2*x - E^x*x)/x]),x]

[Out]

Defer[Int][E^(E^(8*E^x) + Log[Log[-2 - E^x + 4/x]]^2)*Log[-2 - E^x + 4/x]^(2*E^(4*E^x)), x] + 8*Defer[Int][E^(
E^(8*E^x) + 8*E^x + x + Log[Log[-2 - E^x + 4/x]]^2)*x*Log[-2 - E^x + 4/x]^(2*E^(4*E^x)), x] + 2*Defer[Int][E^(
E^(8*E^x) + 4*E^x + Log[Log[-2 - E^x + 4/x]]^2)*x*Log[-2 - E^x + 4/x]^(-1 + 2*E^(4*E^x)), x] + 8*Defer[Int][(E
^(E^(8*E^x) + 4*E^x + Log[Log[-2 - E^x + 4/x]]^2)*Log[-2 - E^x + 4/x]^(-1 + 2*E^(4*E^x)))/(-4 + 2*x + E^x*x),
x] + 8*Defer[Int][(E^(E^(8*E^x) + 4*E^x + Log[Log[-2 - E^x + 4/x]]^2)*x*Log[-2 - E^x + 4/x]^(-1 + 2*E^(4*E^x))
)/(-4 + 2*x + E^x*x), x] - 4*Defer[Int][(E^(E^(8*E^x) + 4*E^x + Log[Log[-2 - E^x + 4/x]]^2)*x^2*Log[-2 - E^x +
4/x]^(-1 + 2*E^(4*E^x)))/(-4 + 2*x + E^x*x), x] + 8*Defer[Int][E^(E^(8*E^x) + 4*E^x + x + Log[Log[-2 - E^x +
4/x]]^2)*x*Log[-2 - E^x + 4/x]^(2*E^(4*E^x))*Log[Log[-2 - E^x + 4/x]], x] + 2*Defer[Int][E^(E^(8*E^x) + Log[Lo
g[-2 - E^x + 4/x]]^2)*x*Log[-2 - E^x + 4/x]^(-1 + 2*E^(4*E^x))*Log[Log[-2 - E^x + 4/x]], x] + 8*Defer[Int][(E^
(E^(8*E^x) + Log[Log[-2 - E^x + 4/x]]^2)*Log[-2 - E^x + 4/x]^(-1 + 2*E^(4*E^x))*Log[Log[-2 - E^x + 4/x]])/(-4
+ 2*x + E^x*x), x] + 8*Defer[Int][(E^(E^(8*E^x) + Log[Log[-2 - E^x + 4/x]]^2)*x*Log[-2 - E^x + 4/x]^(-1 + 2*E^
(4*E^x))*Log[Log[-2 - E^x + 4/x]])/(-4 + 2*x + E^x*x), x] - 4*Defer[Int][(E^(E^(8*E^x) + Log[Log[-2 - E^x + 4/
x]]^2)*x^2*Log[-2 - E^x + 4/x]^(-1 + 2*E^(4*E^x))*Log[Log[-2 - E^x + 4/x]])/(-4 + 2*x + E^x*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (-2 \left (4+e^x x^2\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )-\left (-4+\left (2+e^x\right ) x\right ) \log \left (-2-e^x+\frac {4}{x}\right ) \left (1+8 e^{8 e^x+x} x+8 e^{4 e^x+x} x \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )\right )}{4-\left (2+e^x\right ) x} \, dx \\ & = \int \left (e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )+2 e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )+2 e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )+8 e^{e^{8 e^x}+4 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )-\frac {4 e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \left (-2-2 x+x^2\right ) \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )}{-4+2 x+e^x x}\right ) \, dx \\ & = 2 \int e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx+2 \int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right ) \, dx-4 \int \frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \left (-2-2 x+x^2\right ) \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )}{-4+2 x+e^x x} \, dx+8 \int e^{e^{8 e^x}+4 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right ) \, dx+\int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx \\ & = 2 \int e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx+2 \int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right ) \, dx-4 \int \left (-\frac {2 e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )}{-4+2 x+e^x x}-\frac {2 e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )}{-4+2 x+e^x x}+\frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x^2 \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )}{-4+2 x+e^x x}\right ) \, dx+8 \int \left (e^{e^{8 e^x}+8 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )+e^{e^{8 e^x}+4 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right ) \, dx+\int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx \\ & = 2 \int e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx+2 \int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right ) \, dx-4 \int \frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x^2 \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )}{-4+2 x+e^x x} \, dx+8 \int e^{e^{8 e^x}+8 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx+8 \int e^{e^{8 e^x}+4 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right ) \, dx+8 \int \frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )}{-4+2 x+e^x x} \, dx+8 \int \frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \left (e^{4 e^x}+\log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )\right )}{-4+2 x+e^x x} \, dx+\int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx \\ & = 2 \int e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx+2 \int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right ) \, dx-4 \int \left (\frac {e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x^2 \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )}{-4+2 x+e^x x}+\frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x^2 \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )}{-4+2 x+e^x x}\right ) \, dx+8 \int e^{e^{8 e^x}+8 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx+8 \int e^{e^{8 e^x}+4 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right ) \, dx+8 \int \left (\frac {e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )}{-4+2 x+e^x x}+\frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )}{-4+2 x+e^x x}\right ) \, dx+8 \int \left (\frac {e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )}{-4+2 x+e^x x}+\frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )}{-4+2 x+e^x x}\right ) \, dx+\int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx \\ & = 2 \int e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx+2 \int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right ) \, dx-4 \int \frac {e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x^2 \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )}{-4+2 x+e^x x} \, dx-4 \int \frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x^2 \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )}{-4+2 x+e^x x} \, dx+8 \int e^{e^{8 e^x}+8 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx+8 \int \frac {e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )}{-4+2 x+e^x x} \, dx+8 \int \frac {e^{e^{8 e^x}+4 e^x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right )}{-4+2 x+e^x x} \, dx+8 \int e^{e^{8 e^x}+4 e^x+x+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right ) \, dx+8 \int \frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )}{-4+2 x+e^x x} \, dx+8 \int \frac {e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{-1+2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \log \left (\log \left (-2-e^x+\frac {4}{x}\right )\right )}{-4+2 x+e^x x} \, dx+\int e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )} x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx=e^{e^{8 e^x}+\log ^2\left (\log \left (-2-e^x+\frac {4}{x}\right )\right )} x \log ^{2 e^{4 e^x}}\left (-2-e^x+\frac {4}{x}\right ) \]

[In]

Integrate[(E^(E^(8*E^x) + 2*E^(4*E^x)*Log[Log[(4 - 2*x - E^x*x)/x]] + Log[Log[(4 - 2*x - E^x*x)/x]]^2)*x*(E^(4
*E^x)*(8 + 2*E^x*x^2) + (-4 + 2*x + E^x*x + E^(8*E^x)*(8*E^(2*x)*x^2 + E^x*(-32*x + 16*x^2)))*Log[(4 - 2*x - E
^x*x)/x] + (8 + 2*E^x*x^2 + E^(4*E^x)*(8*E^(2*x)*x^2 + E^x*(-32*x + 16*x^2))*Log[(4 - 2*x - E^x*x)/x])*Log[Log
[(4 - 2*x - E^x*x)/x]]))/((-4*x + 2*x^2 + E^x*x^2)*Log[(4 - 2*x - E^x*x)/x]),x]

[Out]

E^(E^(8*E^x) + Log[Log[-2 - E^x + 4/x]]^2)*x*Log[-2 - E^x + 4/x]^(2*E^(4*E^x))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.13 (sec) , antiderivative size = 281, normalized size of antiderivative = 10.04

\[{\left (i \pi -\ln \left (x \right )+\ln \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right )+\operatorname {csgn}\left (i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )\right )\right )}{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right )}^{2} \left (\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right )-1\right )\right )}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{x}}} x \,{\mathrm e}^{{\ln \left (i \pi -\ln \left (x \right )+\ln \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right )+\operatorname {csgn}\left (i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )\right )\right )}{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right )}^{2} \left (\operatorname {csgn}\left (\frac {i \left (-4+\left ({\mathrm e}^{x}+2\right ) x \right )}{x}\right )-1\right )\right )}^{2}+{\mathrm e}^{8 \,{\mathrm e}^{x}}}\]

[In]

int((((8*exp(x)^2*x^2+(16*x^2-32*x)*exp(x))*exp(4*exp(x))*ln((-exp(x)*x+4-2*x)/x)+2*exp(x)*x^2+8)*ln(ln((-exp(
x)*x+4-2*x)/x))+((8*exp(x)^2*x^2+(16*x^2-32*x)*exp(x))*exp(4*exp(x))^2+exp(x)*x+2*x-4)*ln((-exp(x)*x+4-2*x)/x)
+(2*exp(x)*x^2+8)*exp(4*exp(x)))*exp(ln(ln((-exp(x)*x+4-2*x)/x))^2+2*exp(4*exp(x))*ln(ln((-exp(x)*x+4-2*x)/x))
+exp(4*exp(x))^2+ln(x))/(exp(x)*x^2+2*x^2-4*x)/ln((-exp(x)*x+4-2*x)/x),x)

[Out]

(I*Pi-ln(x)+ln(-4+(exp(x)+2)*x)-1/2*I*Pi*csgn(I/x*(-4+(exp(x)+2)*x))*(-csgn(I/x*(-4+(exp(x)+2)*x))+csgn(I/x))*
(-csgn(I/x*(-4+(exp(x)+2)*x))+csgn(I*(-4+(exp(x)+2)*x)))+I*Pi*csgn(I/x*(-4+(exp(x)+2)*x))^2*(csgn(I/x*(-4+(exp
(x)+2)*x))-1))^(2*exp(4*exp(x)))*x*exp(ln(I*Pi-ln(x)+ln(-4+(exp(x)+2)*x)-1/2*I*Pi*csgn(I/x*(-4+(exp(x)+2)*x))*
(-csgn(I/x*(-4+(exp(x)+2)*x))+csgn(I/x))*(-csgn(I/x*(-4+(exp(x)+2)*x))+csgn(I*(-4+(exp(x)+2)*x)))+I*Pi*csgn(I/
x*(-4+(exp(x)+2)*x))^2*(csgn(I/x*(-4+(exp(x)+2)*x))-1))^2+exp(8*exp(x)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )} x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx=e^{\left (2 \, e^{\left (4 \, e^{x}\right )} \log \left (\log \left (-\frac {x e^{x} + 2 \, x - 4}{x}\right )\right ) + \log \left (\log \left (-\frac {x e^{x} + 2 \, x - 4}{x}\right )\right )^{2} + e^{\left (8 \, e^{x}\right )} + \log \left (x\right )\right )} \]

[In]

integrate((((8*exp(x)^2*x^2+(16*x^2-32*x)*exp(x))*exp(4*exp(x))*log((-exp(x)*x+4-2*x)/x)+2*exp(x)*x^2+8)*log(l
og((-exp(x)*x+4-2*x)/x))+((8*exp(x)^2*x^2+(16*x^2-32*x)*exp(x))*exp(4*exp(x))^2+exp(x)*x+2*x-4)*log((-exp(x)*x
+4-2*x)/x)+(2*exp(x)*x^2+8)*exp(4*exp(x)))*exp(log(log((-exp(x)*x+4-2*x)/x))^2+2*exp(4*exp(x))*log(log((-exp(x
)*x+4-2*x)/x))+exp(4*exp(x))^2+log(x))/(exp(x)*x^2+2*x^2-4*x)/log((-exp(x)*x+4-2*x)/x),x, algorithm="fricas")

[Out]

e^(2*e^(4*e^x)*log(log(-(x*e^x + 2*x - 4)/x)) + log(log(-(x*e^x + 2*x - 4)/x))^2 + e^(8*e^x) + log(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )} x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx=\text {Timed out} \]

[In]

integrate((((8*exp(x)**2*x**2+(16*x**2-32*x)*exp(x))*exp(4*exp(x))*ln((-exp(x)*x+4-2*x)/x)+2*exp(x)*x**2+8)*ln
(ln((-exp(x)*x+4-2*x)/x))+((8*exp(x)**2*x**2+(16*x**2-32*x)*exp(x))*exp(4*exp(x))**2+exp(x)*x+2*x-4)*ln((-exp(
x)*x+4-2*x)/x)+(2*exp(x)*x**2+8)*exp(4*exp(x)))*exp(ln(ln((-exp(x)*x+4-2*x)/x))**2+2*exp(4*exp(x))*ln(ln((-exp
(x)*x+4-2*x)/x))+exp(4*exp(x))**2+ln(x))/(exp(x)*x**2+2*x**2-4*x)/ln((-exp(x)*x+4-2*x)/x),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).

Time = 0.68 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )} x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx=x e^{\left (2 \, e^{\left (4 \, e^{x}\right )} \log \left (\log \left (-x e^{x} - 2 \, x + 4\right ) - \log \left (x\right )\right ) + \log \left (\log \left (-x e^{x} - 2 \, x + 4\right ) - \log \left (x\right )\right )^{2} + e^{\left (8 \, e^{x}\right )}\right )} \]

[In]

integrate((((8*exp(x)^2*x^2+(16*x^2-32*x)*exp(x))*exp(4*exp(x))*log((-exp(x)*x+4-2*x)/x)+2*exp(x)*x^2+8)*log(l
og((-exp(x)*x+4-2*x)/x))+((8*exp(x)^2*x^2+(16*x^2-32*x)*exp(x))*exp(4*exp(x))^2+exp(x)*x+2*x-4)*log((-exp(x)*x
+4-2*x)/x)+(2*exp(x)*x^2+8)*exp(4*exp(x)))*exp(log(log((-exp(x)*x+4-2*x)/x))^2+2*exp(4*exp(x))*log(log((-exp(x
)*x+4-2*x)/x))+exp(4*exp(x))^2+log(x))/(exp(x)*x^2+2*x^2-4*x)/log((-exp(x)*x+4-2*x)/x),x, algorithm="maxima")

[Out]

x*e^(2*e^(4*e^x)*log(log(-x*e^x - 2*x + 4) - log(x)) + log(log(-x*e^x - 2*x + 4) - log(x))^2 + e^(8*e^x))

Giac [F]

\[ \int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )} x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx=\int { \frac {{\left (2 \, {\left (x^{2} e^{x} + 4\right )} e^{\left (4 \, e^{x}\right )} + {\left (x e^{x} + 8 \, {\left (x^{2} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} - 2 \, x\right )} e^{x}\right )} e^{\left (8 \, e^{x}\right )} + 2 \, x - 4\right )} \log \left (-\frac {x e^{x} + 2 \, x - 4}{x}\right ) + 2 \, {\left (x^{2} e^{x} + 4 \, {\left (x^{2} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} - 2 \, x\right )} e^{x}\right )} e^{\left (4 \, e^{x}\right )} \log \left (-\frac {x e^{x} + 2 \, x - 4}{x}\right ) + 4\right )} \log \left (\log \left (-\frac {x e^{x} + 2 \, x - 4}{x}\right )\right )\right )} e^{\left (2 \, e^{\left (4 \, e^{x}\right )} \log \left (\log \left (-\frac {x e^{x} + 2 \, x - 4}{x}\right )\right ) + \log \left (\log \left (-\frac {x e^{x} + 2 \, x - 4}{x}\right )\right )^{2} + e^{\left (8 \, e^{x}\right )} + \log \left (x\right )\right )}}{{\left (x^{2} e^{x} + 2 \, x^{2} - 4 \, x\right )} \log \left (-\frac {x e^{x} + 2 \, x - 4}{x}\right )} \,d x } \]

[In]

integrate((((8*exp(x)^2*x^2+(16*x^2-32*x)*exp(x))*exp(4*exp(x))*log((-exp(x)*x+4-2*x)/x)+2*exp(x)*x^2+8)*log(l
og((-exp(x)*x+4-2*x)/x))+((8*exp(x)^2*x^2+(16*x^2-32*x)*exp(x))*exp(4*exp(x))^2+exp(x)*x+2*x-4)*log((-exp(x)*x
+4-2*x)/x)+(2*exp(x)*x^2+8)*exp(4*exp(x)))*exp(log(log((-exp(x)*x+4-2*x)/x))^2+2*exp(4*exp(x))*log(log((-exp(x
)*x+4-2*x)/x))+exp(4*exp(x))^2+log(x))/(exp(x)*x^2+2*x^2-4*x)/log((-exp(x)*x+4-2*x)/x),x, algorithm="giac")

[Out]

undef

Mupad [B] (verification not implemented)

Time = 8.43 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {e^{e^{8 e^x}+2 e^{4 e^x} \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )+\log ^2\left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )} x \left (e^{4 e^x} \left (8+2 e^x x^2\right )+\left (-4+2 x+e^x x+e^{8 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )+\left (8+2 e^x x^2+e^{4 e^x} \left (8 e^{2 x} x^2+e^x \left (-32 x+16 x^2\right )\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )\right ) \log \left (\log \left (\frac {4-2 x-e^x x}{x}\right )\right )\right )}{\left (-4 x+2 x^2+e^x x^2\right ) \log \left (\frac {4-2 x-e^x x}{x}\right )} \, dx=x\,{\mathrm {e}}^{{\mathrm {e}}^{8\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{{\ln \left (\ln \left (-\frac {2\,x+x\,{\mathrm {e}}^x-4}{x}\right )\right )}^2}\,{\ln \left (-\frac {2\,x+x\,{\mathrm {e}}^x-4}{x}\right )}^{2\,{\mathrm {e}}^{4\,{\mathrm {e}}^x}} \]

[In]

int((exp(exp(8*exp(x)) + log(x) + log(log(-(2*x + x*exp(x) - 4)/x))^2 + 2*exp(4*exp(x))*log(log(-(2*x + x*exp(
x) - 4)/x)))*(log(-(2*x + x*exp(x) - 4)/x)*(2*x + exp(8*exp(x))*(8*x^2*exp(2*x) - exp(x)*(32*x - 16*x^2)) + x*
exp(x) - 4) + exp(4*exp(x))*(2*x^2*exp(x) + 8) + log(log(-(2*x + x*exp(x) - 4)/x))*(2*x^2*exp(x) + exp(4*exp(x
))*log(-(2*x + x*exp(x) - 4)/x)*(8*x^2*exp(2*x) - exp(x)*(32*x - 16*x^2)) + 8)))/(log(-(2*x + x*exp(x) - 4)/x)
*(x^2*exp(x) - 4*x + 2*x^2)),x)

[Out]

x*exp(exp(8*exp(x)))*exp(log(log(-(2*x + x*exp(x) - 4)/x))^2)*log(-(2*x + x*exp(x) - 4)/x)^(2*exp(4*exp(x)))