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

Optimal. Leaf size=28 \[ \left (1-\left (4+\frac {e^x}{x^4}\right )^2-(1+x)^2-\log (5)\right )^x \]

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Rubi [F]  time = 59.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 35, normalized size = 1.25 \begin {gather*} \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 \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.98, size = 38, normalized size = 1.36 \begin {gather*} \left (-\frac {x^{10} + 2 \, x^{9} + x^{8} \log \relax (5) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}}{x^{8}}\right )^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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maple [C]  time = 0.30, size = 843, normalized size = 30.11

method	result	size
risch	\({\mathrm e}^{-\frac {x \left (16 \ln \relax (x )-2 i \pi -i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{8}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{5}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{6}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{7}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{4}+x^{8} \ln \relax (5)+x^{10}+2 x^{9}+16 x^{8}\right )}{x^{8}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{8}}\right )-i \pi \mathrm {csgn}\left (i x^{4}\right )^{3}-i \pi \mathrm {csgn}\left (i x^{5}\right )^{3}-i \pi \mathrm {csgn}\left (i x^{6}\right )^{3}-i \pi \mathrm {csgn}\left (i x^{7}\right )^{3}-i \pi \mathrm {csgn}\left (i x^{8}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{4}+x^{8} \ln \relax (5)+x^{10}+2 x^{9}+16 x^{8}\right )}{x^{8}}\right )^{3}-i \pi \mathrm {csgn}\left (i x^{3}\right )^{3}+2 i \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{4}+x^{8} \ln \relax (5)+x^{10}+2 x^{9}+16 x^{8}\right )}{x^{8}}\right )^{2} \pi -i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 \ln \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{4}+x^{8} \ln \relax (5)+x^{10}+2 x^{9}+16 x^{8}\right )-i \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{4}+x^{8} \ln \relax (5)+x^{10}+2 x^{9}+16 x^{8}\right )}{x^{8}}\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{4}+x^{8} \ln \relax (5)+x^{10}+2 x^{9}+16 x^{8}\right )\right )+i \pi \,\mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (i x^{5}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{5}\right ) \mathrm {csgn}\left (i x^{6}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{6}\right ) \mathrm {csgn}\left (i x^{7}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{7}\right ) \mathrm {csgn}\left (i x^{8}\right )^{2}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{5}\right ) \mathrm {csgn}\left (i x^{6}\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{6}\right ) \mathrm {csgn}\left (i x^{7}\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{7}\right ) \mathrm {csgn}\left (i x^{8}\right )+i \pi \,\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{4}+x^{8} \ln \relax (5)+x^{10}+2 x^{9}+16 x^{8}\right )}{x^{8}}\right ) \mathrm {csgn}\left (\frac {i}{x^{8}}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{4}+x^{8} \ln \relax (5)+x^{10}+2 x^{9}+16 x^{8}\right )\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (i x^{5}\right )\right )}{2}}\)	\(843\)

Verification of antiderivative is not currently implemented for this CAS.

[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,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.54, size = 43, normalized size = 1.54 \begin {gather*} e^{\left (x \log \left (-x^{10} - 2 \, x^{9} - x^{8} {\left (\log \relax (5) + 16\right )} - 8 \, x^{4} e^{x} - e^{\left (2 \, x\right )}\right ) - 8 \, x \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 5.98, size = 44, normalized size = 1.57 \begin {gather*} {\left (\frac {1}{x^8}\right )}^x\,{\left (-{\mathrm {e}}^{2\,x}-8\,x^4\,{\mathrm {e}}^x-x^8\,\ln \relax (5)-16\,x^8-2\,x^9-x^{10}\right )}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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