\(\int \frac {(\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8})^x (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x (-32 x^4+8 x^5)+(e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)) \log (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}))}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx\) [7442]

Optimal result

Integrand size = 194, antiderivative size = 28 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\left (1-\left (4+\frac {e^x}{x^4}\right )^2-(1+x)^2-\log (5)\right )^x \]

[Out]

exp(x*ln(1-ln(5)-(1+x)^2-(exp(x)/x^4+4)^2))

Rubi [F]

\[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx \]

[In]

Int[(((-E^(2*x) - 8*E^x*x^4 - 16*x^8 - 2*x^9 - x^10 - x^8*Log[5])/x^8)^x*(2*x^9 + 2*x^10 + E^(2*x)*(-8 + 2*x)
+ E^x*(-32*x^4 + 8*x^5) + (E^(2*x) + 8*E^x*x^4 + 16*x^8 + 2*x^9 + x^10 + x^8*Log[5])*Log[(-E^(2*x) - 8*E^x*x^4
- 16*x^8 - 2*x^9 - x^10 - x^8*Log[5])/x^8]))/(E^(2*x) + 8*E^x*x^4 + 16*x^8 + 2*x^9 + x^10 + x^8*Log[5]),x]

[Out]

-((16 + Log[5])*Log[-16 - E^(2*x)/x^8 - (8*E^x)/x^4 - 2*x - x^2 - Log[5]]*Defer[Int][(-(E^(2*x)/x^8) - (8*E^x)
/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x]) + 8*Defer[Int][(E^(2*x)*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2
*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^8, x] - Log[-16 - E^(2*x)/x^8 - (8*E^x)/x^4 - 2*x - x^2 - Log[5]]*D
efer[Int][(E^(2*x)*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^8, x] - 2*Defer
[Int][(E^(2*x)*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^7, x] + 32*Defer[In
t][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^4, x] - 8*Log[-16 - E^(2*x
)/x^8 - (8*E^x)/x^4 - 2*x - x^2 - Log[5]]*Defer[Int][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 +
Log[5]/16))^(-1 + x))/x^4, x] - 8*Defer[Int][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/1
6))^(-1 + x))/x^3, x] - 2*Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x
), x] - 2*Log[-16 - E^(2*x)/x^8 - (8*E^x)/x^4 - 2*x - x^2 - Log[5]]*Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^4
- 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x] - 2*Defer[Int][x^2*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 -
16*(1 + Log[5]/16))^(-1 + x), x] - Log[-16 - E^(2*x)/x^8 - (8*E^x)/x^4 - 2*x - x^2 - Log[5]]*Defer[Int][x^2*(
-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x] + 2*(16 + Log[5])*Defer[Int][Defer
[Int][(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x], x] - 8*(16 + Log[5])*Defer
[Int][Defer[Int][(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x]/x, x] + 8*(16 +
Log[5])*Defer[Int][(E^x*x^4*Defer[Int][(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x
), x])/(-E^(2*x) - 8*E^x*x^4 - 2*x^9 - x^10 - 16*x^8*(1 + Log[5]/16)), x] + 2*(16 + Log[5])*Defer[Int][(x^10*D
efer[Int][(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(-E^(2*x) - 8*E^x*x^4
- 2*x^9 - x^10 - 16*x^8*(1 + Log[5]/16)), x] + 32*(16 + Log[5])*Defer[Int][(E^x*x^3*Defer[Int][(-(E^(2*x)/x^8)
- (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1
+ Log[5]/16)), x] + 8*(16 + Log[5])^2*Defer[Int][(x^7*Defer[Int][(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 -
16*(1 + Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] - 2*(7 + L
og[5])*(16 + Log[5])*Defer[Int][(x^8*Defer[Int][(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16)
)^(-1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] + 6*(16 + Log[5])*Defer[Int]
[(x^9*Defer[Int][(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^
x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] + 2*Defer[Int][Defer[Int][(E^(2*x)*(-(E^(2*x)/x^8) - (8*E^x
)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^8, x], x] - 8*Defer[Int][Defer[Int][(E^(2*x)*(-(E^(2*x)/x^
8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^8, x]/x, x] + 8*Defer[Int][(E^x*x^4*Defer[Int][
(E^(2*x)*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^8, x])/(-E^(2*x) - 8*E^x*
x^4 - 2*x^9 - x^10 - 16*x^8*(1 + Log[5]/16)), x] + 2*Defer[Int][(x^10*Defer[Int][(E^(2*x)*(-(E^(2*x)/x^8) - (8
*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^8, x])/(-E^(2*x) - 8*E^x*x^4 - 2*x^9 - x^10 - 16*x^8*(
1 + Log[5]/16)), x] + 32*Defer[Int][(E^x*x^3*Defer[Int][(E^(2*x)*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 1
6*(1 + Log[5]/16))^(-1 + x))/x^8, x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] + 8*(1
6 + Log[5])*Defer[Int][(x^7*Defer[Int][(E^(2*x)*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16)
)^(-1 + x))/x^8, x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] - 2*(7 + Log[5])*Defer[
Int][(x^8*Defer[Int][(E^(2*x)*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^8, x
])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] + 6*Defer[Int][(x^9*Defer[Int][(E^(2*x)*(
-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^8, x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9
+ x^10 + 16*x^8*(1 + Log[5]/16)), x] + 16*Defer[Int][Defer[Int][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^
2 - 16*(1 + Log[5]/16))^(-1 + x))/x^4, x], x] - 64*Defer[Int][Defer[Int][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 -
2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^4, x]/x, x] + 64*Defer[Int][(E^x*x^4*Defer[Int][(E^x*(-(E^(2*x)/x^
8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^4, x])/(-E^(2*x) - 8*E^x*x^4 - 2*x^9 - x^10 - 1
6*x^8*(1 + Log[5]/16)), x] + 16*Defer[Int][(x^10*Defer[Int][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 1
6*(1 + Log[5]/16))^(-1 + x))/x^4, x])/(-E^(2*x) - 8*E^x*x^4 - 2*x^9 - x^10 - 16*x^8*(1 + Log[5]/16)), x] + 256
*Defer[Int][(E^x*x^3*Defer[Int][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))
/x^4, x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] + 64*(16 + Log[5])*Defer[Int][(x^7
*Defer[Int][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^4, x])/(E^(2*x) +
8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] - 16*(7 + Log[5])*Defer[Int][(x^8*Defer[Int][(E^x*(-(E
^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^4, x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 +
x^10 + 16*x^8*(1 + Log[5]/16)), x] + 48*Defer[Int][(x^9*Defer[Int][(E^x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x -
x^2 - 16*(1 + Log[5]/16))^(-1 + x))/x^4, x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x]
+ 4*Defer[Int][Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x], x]
- 16*Defer[Int][Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x]/x, x
] + 16*Defer[Int][(E^x*x^4*Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 +
x), x])/(-E^(2*x) - 8*E^x*x^4 - 2*x^9 - x^10 - 16*x^8*(1 + Log[5]/16)), x] + 4*Defer[Int][(x^10*Defer[Int][x*(
-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(-E^(2*x) - 8*E^x*x^4 - 2*x^9 - x
^10 - 16*x^8*(1 + Log[5]/16)), x] + 64*Defer[Int][(E^x*x^3*Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x -
x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] + 16
*(16 + Log[5])*Defer[Int][(x^7*Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-
1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] - 4*(7 + Log[5])*Defer[Int][(x^8
*Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^x*x
^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] + 12*Defer[Int][(x^9*Defer[Int][x*(-(E^(2*x)/x^8) - (8*E^x)/x^
4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)
), x] + 2*Defer[Int][Defer[Int][x^2*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x),
x], x] - 8*Defer[Int][Defer[Int][x^2*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x),
x]/x, x] + 8*Defer[Int][(E^x*x^4*Defer[Int][x^2*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16
))^(-1 + x), x])/(-E^(2*x) - 8*E^x*x^4 - 2*x^9 - x^10 - 16*x^8*(1 + Log[5]/16)), x] + 2*Defer[Int][(x^10*Defer
[Int][x^2*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(-E^(2*x) - 8*E^x*x^4
- 2*x^9 - x^10 - 16*x^8*(1 + Log[5]/16)), x] + 32*Defer[Int][(E^x*x^3*Defer[Int][x^2*(-(E^(2*x)/x^8) - (8*E^x)
/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/
16)), x] + 8*(16 + Log[5])*Defer[Int][(x^7*Defer[Int][x^2*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 +
Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] - 2*(7 + Log[5])*D
efer[Int][(x^8*Defer[Int][x^2*(-(E^(2*x)/x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(E
^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^8*(1 + Log[5]/16)), x] + 6*Defer[Int][(x^9*Defer[Int][x^2*(-(E^(2*x)/
x^8) - (8*E^x)/x^4 - 2*x - x^2 - 16*(1 + Log[5]/16))^(-1 + x), x])/(E^(2*x) + 8*E^x*x^4 + 2*x^9 + x^10 + 16*x^
8*(1 + Log[5]/16)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+2 x^9+x^{10}+x^8 (16+\log (5))} \, dx \\ & = \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \left (-2 e^{2 x} (-4+x)-8 e^x (-4+x) x^4-2 x^9-2 x^{10}-\left (e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )\right ) \log \left (-\frac {e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )}{x^8}\right )\right )}{x^8} \, dx \\ & = \int \left (-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x}-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )-x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )-16 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )-\frac {8 e^x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \left (-4+x+\log \left (-\frac {e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )}{x^8}\right )\right )}{x^4}-\frac {e^{2 x} \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \left (-8+2 x+\log \left (-\frac {e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )}{x^8}\right )\right )}{x^8}\right ) \, dx \\ & = -\left (2 \int x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx\right )-2 \int x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx-2 \int x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right ) \, dx-8 \int \frac {e^x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \left (-4+x+\log \left (-\frac {e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )}{x^8}\right )\right )}{x^4} \, dx-(16+\log (5)) \int \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right ) \, dx-\int x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right ) \, dx-\int \frac {e^{2 x} \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \left (-8+2 x+\log \left (-\frac {e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )}{x^8}\right )\right )}{x^8} \, dx \\ & = -\left (2 \int x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx\right )-2 \int x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx+2 \int \frac {2 \left (e^{2 x} (-4+x)+4 e^x (-4+x) x^4+x^9 (1+x)\right ) \int x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx}{x \left (e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )\right )} \, dx-8 \int \left (\frac {e^x (-4+x) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x}}{x^4}+\frac {e^x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )}{x^4}\right ) \, dx-(-16-\log (5)) \int \frac {2 \left (e^{2 x} (-4+x)+4 e^x (-4+x) x^4+x^9 (1+x)\right ) \int \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx}{x \left (e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )\right )} \, dx-\log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right ) \int x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx-\left (2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )\right ) \int x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx-\left ((16+\log (5)) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )\right ) \int \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx-\int \left (\frac {2 e^{2 x} (-4+x) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x}}{x^8}+\frac {e^{2 x} \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )}{x^8}\right ) \, dx+\int \frac {2 \left (e^{2 x} (-4+x)+4 e^x (-4+x) x^4+x^9 (1+x)\right ) \int x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx}{x \left (e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )\right )} \, dx \\ & = -\left (2 \int \frac {e^{2 x} (-4+x) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x}}{x^8} \, dx\right )-2 \int x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx-2 \int x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx+2 \int \frac {\left (e^{2 x} (-4+x)+4 e^x (-4+x) x^4+x^9 (1+x)\right ) \int x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx}{x \left (e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )\right )} \, dx+4 \int \frac {\left (e^{2 x} (-4+x)+4 e^x (-4+x) x^4+x^9 (1+x)\right ) \int x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx}{x \left (e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )\right )} \, dx-8 \int \frac {e^x (-4+x) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x}}{x^4} \, dx-8 \int \frac {e^x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )}{x^4} \, dx+(2 (16+\log (5))) \int \frac {\left (e^{2 x} (-4+x)+4 e^x (-4+x) x^4+x^9 (1+x)\right ) \int \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx}{x \left (e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )\right )} \, dx-\log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right ) \int x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx-\left (2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )\right ) \int x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx-\left ((16+\log (5)) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )\right ) \int \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \, dx-\int \frac {e^{2 x} \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )^{-1+x} \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-2 x-x^2-16 \left (1+\frac {\log (5)}{16}\right )\right )}{x^8} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\left (-\frac {e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )}{x^8}\right )^x \]

[In]

Integrate[(((-E^(2*x) - 8*E^x*x^4 - 16*x^8 - 2*x^9 - x^10 - x^8*Log[5])/x^8)^x*(2*x^9 + 2*x^10 + E^(2*x)*(-8 +
2*x) + E^x*(-32*x^4 + 8*x^5) + (E^(2*x) + 8*E^x*x^4 + 16*x^8 + 2*x^9 + x^10 + x^8*Log[5])*Log[(-E^(2*x) - 8*E
^x*x^4 - 16*x^8 - 2*x^9 - x^10 - x^8*Log[5])/x^8]))/(E^(2*x) + 8*E^x*x^4 + 16*x^8 + 2*x^9 + x^10 + x^8*Log[5])
,x]

[Out]

(-((E^(2*x) + 8*E^x*x^4 + x^8*(16 + 2*x + x^2 + Log[5]))/x^8))^x

Maple [C] (warning: unable to verify)

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

Time = 0.14 (sec) , antiderivative size = 754, normalized size of antiderivative = 26.93

\[\text {Expression too large to display}\]

[In]

int(((exp(x)^2+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)*ln((-exp(x)^2-8*exp(x)*x^4-x^8*ln(5)-x^10-2*x^9-16*x^
8)/x^8)+(2*x-8)*exp(x)^2+(8*x^5-32*x^4)*exp(x)+2*x^10+2*x^9)*exp(x*ln((-exp(x)^2-8*exp(x)*x^4-x^8*ln(5)-x^10-2
*x^9-16*x^8)/x^8))/(exp(x)^2+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8),x)

[Out]

x^(-8*x)*(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)^x*exp(1/2*I*x*Pi*(2-csgn(I*(exp(2*x)+8*exp(x)*x^4
+x^8*ln(5)+x^10+2*x^9+16*x^8)/x^8)*csgn(I/x^8)*csgn(I*(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8))+csg
n(I*x^2)*csgn(I*x)*csgn(I*x^3)+csgn(I*x)*csgn(I*x^3)*csgn(I*x^4)+csgn(I*x)*csgn(I*x^4)*csgn(I*x^5)+csgn(I*x)*c
sgn(I*x^5)*csgn(I*x^6)+csgn(I*x)*csgn(I*x^6)*csgn(I*x^7)+csgn(I*x)*csgn(I*x^7)*csgn(I*x^8)-2*csgn(I*(exp(2*x)+
8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)/x^8)^2+csgn(I*(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)/x^
8)^3+csgn(I*x^2)^3+csgn(I*x^3)^3+csgn(I*x^4)^3+csgn(I*x^5)^3+csgn(I*x^6)^3+csgn(I*x^7)^3+csgn(I*x^8)^3+csgn(I*
(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)/x^8)^2*csgn(I/x^8)+csgn(I*(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)
+x^10+2*x^9+16*x^8)/x^8)^2*csgn(I*(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8))-2*csgn(I*x^2)^2*csgn(I*
x)+csgn(I*x^2)*csgn(I*x)^2-csgn(I*x^2)*csgn(I*x^3)^2-csgn(I*x)*csgn(I*x^3)^2-csgn(I*x)*csgn(I*x^4)^2-csgn(I*x)
*csgn(I*x^5)^2-csgn(I*x)*csgn(I*x^6)^2-csgn(I*x)*csgn(I*x^7)^2-csgn(I*x)*csgn(I*x^8)^2-csgn(I*x^3)*csgn(I*x^4)
^2-csgn(I*x^4)*csgn(I*x^5)^2-csgn(I*x^5)*csgn(I*x^6)^2-csgn(I*x^6)*csgn(I*x^7)^2-csgn(I*x^7)*csgn(I*x^8)^2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\left (-\frac {x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}}{x^{8}}\right )^{x} \]

[In]

integrate(((exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8)*log((-exp(x)^2-8*exp(x)*x^4-x^8*log(5)-x^10-2*
x^9-16*x^8)/x^8)+(2*x-8)*exp(x)^2+(8*x^5-32*x^4)*exp(x)+2*x^10+2*x^9)*exp(x*log((-exp(x)^2-8*exp(x)*x^4-x^8*lo
g(5)-x^10-2*x^9-16*x^8)/x^8))/(exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8),x, algorithm="fricas")

[Out]

(-(x^10 + 2*x^9 + x^8*log(5) + 16*x^8 + 8*x^4*e^x + e^(2*x))/x^8)^x

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\text {Timed out} \]

[In]

integrate(((exp(x)**2+8*exp(x)*x**4+x**8*ln(5)+x**10+2*x**9+16*x**8)*ln((-exp(x)**2-8*exp(x)*x**4-x**8*ln(5)-x
**10-2*x**9-16*x**8)/x**8)+(2*x-8)*exp(x)**2+(8*x**5-32*x**4)*exp(x)+2*x**10+2*x**9)*exp(x*ln((-exp(x)**2-8*ex
p(x)*x**4-x**8*ln(5)-x**10-2*x**9-16*x**8)/x**8))/(exp(x)**2+8*exp(x)*x**4+x**8*ln(5)+x**10+2*x**9+16*x**8),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=e^{\left (x \log \left (-x^{10} - 2 \, x^{9} - x^{8} {\left (\log \left (5\right ) + 16\right )} - 8 \, x^{4} e^{x} - e^{\left (2 \, x\right )}\right ) - 8 \, x \log \left (x\right )\right )} \]

[In]

integrate(((exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8)*log((-exp(x)^2-8*exp(x)*x^4-x^8*log(5)-x^10-2*
x^9-16*x^8)/x^8)+(2*x-8)*exp(x)^2+(8*x^5-32*x^4)*exp(x)+2*x^10+2*x^9)*exp(x*log((-exp(x)^2-8*exp(x)*x^4-x^8*lo
g(5)-x^10-2*x^9-16*x^8)/x^8))/(exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8),x, algorithm="maxima")

[Out]

e^(x*log(-x^10 - 2*x^9 - x^8*(log(5) + 16) - 8*x^4*e^x - e^(2*x)) - 8*x*log(x))

Giac [F]

\[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\int { \frac {{\left (2 \, x^{10} + 2 \, x^{9} + 2 \, {\left (x - 4\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{5} - 4 \, x^{4}\right )} e^{x} + {\left (x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (-\frac {x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}}{x^{8}}\right )\right )} \left (-\frac {x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}}{x^{8}}\right )^{x}}{x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}} \,d x } \]

[In]

integrate(((exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8)*log((-exp(x)^2-8*exp(x)*x^4-x^8*log(5)-x^10-2*
x^9-16*x^8)/x^8)+(2*x-8)*exp(x)^2+(8*x^5-32*x^4)*exp(x)+2*x^10+2*x^9)*exp(x*log((-exp(x)^2-8*exp(x)*x^4-x^8*lo
g(5)-x^10-2*x^9-16*x^8)/x^8))/(exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8),x, algorithm="giac")

[Out]

integrate((2*x^10 + 2*x^9 + 2*(x - 4)*e^(2*x) + 8*(x^5 - 4*x^4)*e^x + (x^10 + 2*x^9 + x^8*log(5) + 16*x^8 + 8*
x^4*e^x + e^(2*x))*log(-(x^10 + 2*x^9 + x^8*log(5) + 16*x^8 + 8*x^4*e^x + e^(2*x))/x^8))*(-(x^10 + 2*x^9 + x^8
*log(5) + 16*x^8 + 8*x^4*e^x + e^(2*x))/x^8)^x/(x^10 + 2*x^9 + x^8*log(5) + 16*x^8 + 8*x^4*e^x + e^(2*x)), x)

Mupad [B] (verification not implemented)

Time = 12.75 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx={\left (\frac {1}{x^8}\right )}^x\,{\left (-{\mathrm {e}}^{2\,x}-8\,x^4\,{\mathrm {e}}^x-x^8\,\ln \left (5\right )-16\,x^8-2\,x^9-x^{10}\right )}^x \]

[In]

int((exp(x*log(-(exp(2*x) + 8*x^4*exp(x) + x^8*log(5) + 16*x^8 + 2*x^9 + x^10)/x^8))*(log(-(exp(2*x) + 8*x^4*e
xp(x) + x^8*log(5) + 16*x^8 + 2*x^9 + x^10)/x^8)*(exp(2*x) + 8*x^4*exp(x) + x^8*log(5) + 16*x^8 + 2*x^9 + x^10
) - exp(x)*(32*x^4 - 8*x^5) + exp(2*x)*(2*x - 8) + 2*x^9 + 2*x^10))/(exp(2*x) + 8*x^4*exp(x) + x^8*log(5) + 16
*x^8 + 2*x^9 + x^10),x)

[Out]

(1/x^8)^x*(- exp(2*x) - 8*x^4*exp(x) - x^8*log(5) - 16*x^8 - 2*x^9 - x^10)^x