\(\int \frac {(2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)) \log (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)})+(e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)) \log ^2(\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)})}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx\) [5958]

Optimal result

Integrand size = 137, antiderivative size = 27 \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\frac {1}{2} x \log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \]

[Out]

1/2*ln(exp(5/exp(x)/x)+x/ln(18))^2*x

Rubi [F]

\[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx \]

[In]

Int[((2*E^x*x^2 + E^(5/(E^x*x))*(-10 - 10*x)*Log[18])*Log[(x + E^(5/(E^x*x))*Log[18])/Log[18]] + (E^x*x^2 + E^
(5/(E^x*x) + x)*x*Log[18])*Log[(x + E^(5/(E^x*x))*Log[18])/Log[18]]^2)/(2*E^x*x^2 + 2*E^(5/(E^x*x) + x)*x*Log[
18]),x]

[Out]

-5*Log[18]*Log[E^(5/(E^x*x)) + x/Log[18]]*Defer[Int][E^(5/(E^x*x) - x)/(x + E^(5/(E^x*x))*Log[18]), x] - 5*Log
[18]*Log[E^(5/(E^x*x)) + x/Log[18]]*Defer[Int][E^(5/(E^x*x) - x)/(x*(x + E^(5/(E^x*x))*Log[18])), x] + Log[E^(
5/(E^x*x)) + x/Log[18]]*Defer[Int][x/(x + E^(5/(E^x*x))*Log[18]), x] + Defer[Int][Log[E^(5/(E^x*x)) + x/Log[18
]]^2, x]/2 + 5*Log[18]*Defer[Int][Defer[Int][E^(5/(E^x*x) - x)/(x + E^(5/(E^x*x))*Log[18]), x]/(x + E^(5/(E^x*
x))*Log[18]), x] - 25*Log[18]^2*Defer[Int][(E^(5/(E^x*x) - x)*Defer[Int][E^(5/(E^x*x) - x)/(x + E^(5/(E^x*x))*
Log[18]), x])/(x^2*(x + E^(5/(E^x*x))*Log[18])), x] - 25*Log[18]^2*Defer[Int][(E^(5/(E^x*x) - x)*Defer[Int][E^
(5/(E^x*x) - x)/(x + E^(5/(E^x*x))*Log[18]), x])/(x*(x + E^(5/(E^x*x))*Log[18])), x] - Defer[Int][Defer[Int][x
/(x + E^(5/(E^x*x))*Log[18]), x]/(x + E^(5/(E^x*x))*Log[18]), x] + 5*Log[18]*Defer[Int][(E^(5/(E^x*x) - x)*Def
er[Int][x/(x + E^(5/(E^x*x))*Log[18]), x])/(x^2*(x + E^(5/(E^x*x))*Log[18])), x] + 5*Log[18]*Defer[Int][(E^(5/
(E^x*x) - x)*Defer[Int][x/(x + E^(5/(E^x*x))*Log[18]), x])/(x*(x + E^(5/(E^x*x))*Log[18])), x] + 5*Log[18]*Def
er[Int][Defer[Int][E^(5/(E^x*x) - x)/(x^2 + E^(5/(E^x*x))*x*Log[18]), x]/(x + E^(5/(E^x*x))*Log[18]), x] - 25*
Log[18]^2*Defer[Int][(E^(5/(E^x*x) - x)*Defer[Int][E^(5/(E^x*x) - x)/(x^2 + E^(5/(E^x*x))*x*Log[18]), x])/(x^2
*(x + E^(5/(E^x*x))*Log[18])), x] - 25*Log[18]^2*Defer[Int][(E^(5/(E^x*x) - x)*Defer[Int][E^(5/(E^x*x) - x)/(x
^2 + E^(5/(E^x*x))*x*Log[18]), x])/(x*(x + E^(5/(E^x*x))*Log[18])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )\right )}{2 x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx \\ & = \frac {1}{2} \int \frac {e^{-x} \left (\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )\right )}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx \\ & = \frac {1}{2} \int \left (-\frac {10 e^{\frac {5 e^{-x}}{x}-x} (1+x) \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}+\frac {\log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \left (2 x+x \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )+e^{\frac {5 e^{-x}}{x}} \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right )}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}\right ) \, dx \\ & = \frac {1}{2} \int \frac {\log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \left (2 x+x \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )+e^{\frac {5 e^{-x}}{x}} \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right )}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-(5 \log (18)) \int \frac {e^{\frac {5 e^{-x}}{x}-x} (1+x) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx \\ & = \frac {1}{2} \int \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \left (\frac {2 x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}+\log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \, dx+(5 \log (18)) \int \frac {e^{-x} \left (e^x x^2-5 e^{\frac {5 e^{-x}}{x}} (1+x) \log (18)\right ) \left (\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx\right )}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx \\ & = \frac {1}{2} \int \left (\frac {2 x \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}+\log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \, dx+(5 \log (18)) \int \left (\frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}-\frac {5 e^{\frac {5 e^{-x}}{x}-x} (1+x) \log (18) \left (\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx\right )}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}\right ) \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx \\ & = \frac {1}{2} \int \log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} (1+x) \left (\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx\right )}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx+\int \frac {x \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx \\ & = \frac {1}{2} \int \log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \, dx+(5 \log (18)) \int \left (\frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}+\frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}\right ) \, dx-\left (25 \log ^2(18)\right ) \int \left (\frac {e^{\frac {5 e^{-x}}{x}-x} \left (\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx\right )}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}+\frac {e^{\frac {5 e^{-x}}{x}-x} \left (\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx\right )}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}\right ) \, dx+\log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\int \frac {e^{-x} \left (e^x x^2-5 e^{\frac {5 e^{-x}}{x}} (1+x) \log (18)\right ) \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx \\ & = \frac {1}{2} \int \log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \left (\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx\right )}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \left (\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx\right )}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx+\log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\int \left (\frac {\int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)}-\frac {5 e^{\frac {5 e^{-x}}{x}-x} (1+x) \log (18) \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}\right ) \, dx \\ & = \frac {1}{2} \int \log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+(5 \log (18)) \int \frac {e^{\frac {5 e^{-x}}{x}-x} (1+x) \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (25 \log ^2(18)\right ) \int \left (\frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}+\frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}\right ) \, dx-\left (25 \log ^2(18)\right ) \int \left (\frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}+\frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}\right ) \, dx+\log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\int \frac {\int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx \\ & = \frac {1}{2} \int \log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+(5 \log (18)) \int \left (\frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}+\frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )}\right ) \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx+\log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\int \frac {\int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx \\ & = \frac {1}{2} \int \log ^2\left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx+(5 \log (18)) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx+(5 \log (18)) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx+(5 \log (18)) \int \frac {\int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x^2 \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\left (25 \log ^2(18)\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x^2+e^{\frac {5 e^{-x}}{x}} x \log (18)} \, dx}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx+\log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right ) \int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx-\left (5 \log (18) \log \left (e^{\frac {5 e^{-x}}{x}}+\frac {x}{\log (18)}\right )\right ) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x \left (x+e^{\frac {5 e^{-x}}{x}} \log (18)\right )} \, dx-\int \frac {\int \frac {x}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx}{x+e^{\frac {5 e^{-x}}{x}} \log (18)} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx \]

[In]

Integrate[((2*E^x*x^2 + E^(5/(E^x*x))*(-10 - 10*x)*Log[18])*Log[(x + E^(5/(E^x*x))*Log[18])/Log[18]] + (E^x*x^
2 + E^(5/(E^x*x) + x)*x*Log[18])*Log[(x + E^(5/(E^x*x))*Log[18])/Log[18]]^2)/(2*E^x*x^2 + 2*E^(5/(E^x*x) + x)*
x*Log[18]),x]

[Out]

Integrate[((2*E^x*x^2 + E^(5/(E^x*x))*(-10 - 10*x)*Log[18])*Log[(x + E^(5/(E^x*x))*Log[18])/Log[18]] + (E^x*x^
2 + E^(5/(E^x*x) + x)*x*Log[18])*Log[(x + E^(5/(E^x*x))*Log[18])/Log[18]]^2)/(2*E^x*x^2 + 2*E^(5/(E^x*x) + x)*
x*Log[18]), x]

Maple [A] (verified)

Time = 27.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {\ln \left (18\right ) {\mathrm e}^{\frac {5 \,{\mathrm e}^{-x}}{x}}+x}{\ln \left (18\right )}\right )^{2} x}{2}\)	\(27\)
risch	\(\frac {x \ln \left (\frac {\left (\ln \left (2\right )+2 \ln \left (3\right )\right ) {\mathrm e}^{\frac {5 \,{\mathrm e}^{-x}}{x}}+x}{\ln \left (2\right )+2 \ln \left (3\right )}\right )^{2}}{2}\)	\(37\)

[In]

int(((x*ln(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x^2)*ln((ln(18)*exp(5/exp(x)/x)+x)/ln(18))^2+((-10*x-10)*ln(18)*e
xp(5/exp(x)/x)+2*exp(x)*x^2)*ln((ln(18)*exp(5/exp(x)/x)+x)/ln(18)))/(2*x*ln(18)*exp(x)*exp(5/exp(x)/x)+2*exp(x
)*x^2),x,method=_RETURNVERBOSE)

[Out]

1/2*ln((ln(18)*exp(5/exp(x)/x)+x)/ln(18))^2*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\frac {1}{2} \, x \log \left (\frac {{\left (x e^{x} + e^{\left (\frac {{\left (x^{2} e^{x} + 5\right )} e^{\left (-x\right )}}{x}\right )} \log \left (18\right )\right )} e^{\left (-x\right )}}{\log \left (18\right )}\right )^{2} \]

[In]

integrate(((x*log(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x^2)*log((log(18)*exp(5/exp(x)/x)+x)/log(18))^2+((-10*x-10
)*log(18)*exp(5/exp(x)/x)+2*exp(x)*x^2)*log((log(18)*exp(5/exp(x)/x)+x)/log(18)))/(2*x*log(18)*exp(x)*exp(5/ex
p(x)/x)+2*exp(x)*x^2),x, algorithm="fricas")

[Out]

1/2*x*log((x*e^x + e^((x^2*e^x + 5)*e^(-x)/x)*log(18))*e^(-x)/log(18))^2

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\frac {x \log {\left (\frac {x + e^{\frac {5 e^{- x}}{x}} \log {\left (18 \right )}}{\log {\left (18 \right )}} \right )}^{2}}{2} \]

[In]

integrate(((x*ln(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x**2)*ln((ln(18)*exp(5/exp(x)/x)+x)/ln(18))**2+((-10*x-10)*
ln(18)*exp(5/exp(x)/x)+2*exp(x)*x**2)*ln((ln(18)*exp(5/exp(x)/x)+x)/ln(18)))/(2*x*ln(18)*exp(x)*exp(5/exp(x)/x
)+2*exp(x)*x**2),x)

[Out]

x*log((x + exp(5*exp(-x)/x)*log(18))/log(18))**2/2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).

Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67 \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\frac {1}{2} \, x \log \left ({\left (2 \, \log \left (3\right ) + \log \left (2\right )\right )} e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} + x\right )^{2} - x \log \left ({\left (2 \, \log \left (3\right ) + \log \left (2\right )\right )} e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} + x\right ) \log \left (2 \, \log \left (3\right ) + \log \left (2\right )\right ) + \frac {1}{2} \, x \log \left (2 \, \log \left (3\right ) + \log \left (2\right )\right )^{2} \]

[In]

integrate(((x*log(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x^2)*log((log(18)*exp(5/exp(x)/x)+x)/log(18))^2+((-10*x-10
)*log(18)*exp(5/exp(x)/x)+2*exp(x)*x^2)*log((log(18)*exp(5/exp(x)/x)+x)/log(18)))/(2*x*log(18)*exp(x)*exp(5/ex
p(x)/x)+2*exp(x)*x^2),x, algorithm="maxima")

[Out]

1/2*x*log((2*log(3) + log(2))*e^(5*e^(-x)/x) + x)^2 - x*log((2*log(3) + log(2))*e^(5*e^(-x)/x) + x)*log(2*log(
3) + log(2)) + 1/2*x*log(2*log(3) + log(2))^2

Giac [F]

\[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\int { \frac {{\left (x^{2} e^{x} + x e^{\left (x + \frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right )\right )} \log \left (\frac {e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right ) + x}{\log \left (18\right )}\right )^{2} + 2 \, {\left (x^{2} e^{x} - 5 \, {\left (x + 1\right )} e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right )\right )} \log \left (\frac {e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right ) + x}{\log \left (18\right )}\right )}{2 \, {\left (x^{2} e^{x} + x e^{\left (x + \frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (18\right )\right )}} \,d x } \]

[In]

integrate(((x*log(18)*exp(x)*exp(5/exp(x)/x)+exp(x)*x^2)*log((log(18)*exp(5/exp(x)/x)+x)/log(18))^2+((-10*x-10
)*log(18)*exp(5/exp(x)/x)+2*exp(x)*x^2)*log((log(18)*exp(5/exp(x)/x)+x)/log(18)))/(2*x*log(18)*exp(x)*exp(5/ex
p(x)/x)+2*exp(x)*x^2),x, algorithm="giac")

[Out]

integrate(1/2*((x^2*e^x + x*e^(x + 5*e^(-x)/x)*log(18))*log((e^(5*e^(-x)/x)*log(18) + x)/log(18))^2 + 2*(x^2*e
^x - 5*(x + 1)*e^(5*e^(-x)/x)*log(18))*log((e^(5*e^(-x)/x)*log(18) + x)/log(18)))/(x^2*e^x + x*e^(x + 5*e^(-x)
/x)*log(18)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (2 e^x x^2+e^{\frac {5 e^{-x}}{x}} (-10-10 x) \log (18)\right ) \log \left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )+\left (e^x x^2+e^{\frac {5 e^{-x}}{x}+x} x \log (18)\right ) \log ^2\left (\frac {x+e^{\frac {5 e^{-x}}{x}} \log (18)}{\log (18)}\right )}{2 e^x x^2+2 e^{\frac {5 e^{-x}}{x}+x} x \log (18)} \, dx=\int \frac {\left (x^2\,{\mathrm {e}}^x+x\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,{\mathrm {e}}^x\,\ln \left (18\right )\right )\,{\ln \left (\frac {x+{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,\ln \left (18\right )}{\ln \left (18\right )}\right )}^2+\left (2\,x^2\,{\mathrm {e}}^x-{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,\ln \left (18\right )\,\left (10\,x+10\right )\right )\,\ln \left (\frac {x+{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,\ln \left (18\right )}{\ln \left (18\right )}\right )}{2\,x^2\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,{\mathrm {e}}^x\,\ln \left (18\right )} \,d x \]

[In]

int((log((x + exp((5*exp(-x))/x)*log(18))/log(18))^2*(x^2*exp(x) + x*exp((5*exp(-x))/x)*exp(x)*log(18)) + log(
(x + exp((5*exp(-x))/x)*log(18))/log(18))*(2*x^2*exp(x) - exp((5*exp(-x))/x)*log(18)*(10*x + 10)))/(2*x^2*exp(
x) + 2*x*exp((5*exp(-x))/x)*exp(x)*log(18)),x)

[Out]

int((log((x + exp((5*exp(-x))/x)*log(18))/log(18))^2*(x^2*exp(x) + x*exp((5*exp(-x))/x)*exp(x)*log(18)) + log(
(x + exp((5*exp(-x))/x)*log(18))/log(18))*(2*x^2*exp(x) - exp((5*exp(-x))/x)*log(18)*(10*x + 10)))/(2*x^2*exp(
x) + 2*x*exp((5*exp(-x))/x)*exp(x)*log(18)), x)