\(\int \frac {e^{-e^x} (e^{e^x} (-4 x+10 x^3)+(2-5 x^2) \log ^2(2-5 x^2)+\log (x) (-20 x^2 \log (2-5 x^2)+e^x (-2 x+5 x^3) \log ^2(2-5 x^2)))}{-2 x+5 x^3} \, dx\) [9650]

Optimal result

Integrand size = 93, antiderivative size = 26 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=-4+2 x-e^{-e^x} \log (x) \log ^2\left (2-5 x^2\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=\int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx \]

[In]

Int[(E^E^x*(-4*x + 10*x^3) + (2 - 5*x^2)*Log[2 - 5*x^2]^2 + Log[x]*(-20*x^2*Log[2 - 5*x^2] + E^x*(-2*x + 5*x^3
)*Log[2 - 5*x^2]^2))/(E^E^x*(-2*x + 5*x^3)),x]

[Out]

2*x + 2*Sqrt[5]*Log[x]*Log[2 - 5*x^2]*Defer[Int][1/(E^E^x*(Sqrt[2] - Sqrt[5]*x)), x] - 2*Sqrt[5]*Log[x]*Log[2
- 5*x^2]*Defer[Int][1/(E^E^x*(Sqrt[2] + Sqrt[5]*x)), x] - Defer[Int][Log[2 - 5*x^2]^2/(E^E^x*x), x] + Defer[In
t][E^(-E^x + x)*Log[x]*Log[2 - 5*x^2]^2, x] - 2*Sqrt[5]*Log[2 - 5*x^2]*Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2]
- Sqrt[5]*x)), x]/x, x] + 10*Log[x]*Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] - Sqrt[5]*x)), x]/(Sqrt[2] - Sqrt
[5]*x), x] - 10*Log[x]*Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] - Sqrt[5]*x)), x]/(Sqrt[2] + Sqrt[5]*x), x] + 2
*Sqrt[5]*Log[2 - 5*x^2]*Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] + Sqrt[5]*x)), x]/x, x] - 10*Log[x]*Defer[Int]
[Defer[Int][1/(E^E^x*(Sqrt[2] + Sqrt[5]*x)), x]/(Sqrt[2] - Sqrt[5]*x), x] + 10*Log[x]*Defer[Int][Defer[Int][1/
(E^E^x*(Sqrt[2] + Sqrt[5]*x)), x]/(Sqrt[2] + Sqrt[5]*x), x] - 10*Defer[Int][Defer[Int][Defer[Int][1/(E^E^x*(Sq
rt[2] - Sqrt[5]*x)), x]/x, x]/(Sqrt[2] - Sqrt[5]*x), x] + 10*Defer[Int][Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2
] - Sqrt[5]*x)), x]/x, x]/(Sqrt[2] + Sqrt[5]*x), x] - 10*Defer[Int][Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] -
Sqrt[5]*x)), x]/(Sqrt[2] - Sqrt[5]*x), x]/x, x] + 10*Defer[Int][Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] - Sqrt
[5]*x)), x]/(Sqrt[2] + Sqrt[5]*x), x]/x, x] + 10*Defer[Int][Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] + Sqrt[5]*
x)), x]/x, x]/(Sqrt[2] - Sqrt[5]*x), x] - 10*Defer[Int][Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] + Sqrt[5]*x)),
x]/x, x]/(Sqrt[2] + Sqrt[5]*x), x] + 10*Defer[Int][Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] + Sqrt[5]*x)), x]/
(Sqrt[2] - Sqrt[5]*x), x]/x, x] - 10*Defer[Int][Defer[Int][Defer[Int][1/(E^E^x*(Sqrt[2] + Sqrt[5]*x)), x]/(Sqr
t[2] + Sqrt[5]*x), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{x \left (-2+5 x^2\right )} \, dx \\ & = \int e^{-e^x} \left (2 e^{e^x}-\frac {\log ^2\left (2-5 x^2\right )}{x}+\log (x) \log \left (2-5 x^2\right ) \left (\frac {20 x}{2-5 x^2}+e^x \log \left (2-5 x^2\right )\right )\right ) \, dx \\ & = \int \left (2-\frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x}+\frac {e^{-e^x} \log (x) \log \left (2-5 x^2\right ) \left (-20 x-2 e^x \log \left (2-5 x^2\right )+5 e^x x^2 \log \left (2-5 x^2\right )\right )}{-2+5 x^2}\right ) \, dx \\ & = 2 x-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int \frac {e^{-e^x} \log (x) \log \left (2-5 x^2\right ) \left (-20 x-2 e^x \log \left (2-5 x^2\right )+5 e^x x^2 \log \left (2-5 x^2\right )\right )}{-2+5 x^2} \, dx \\ & = 2 x-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int \left (-\frac {20 e^{-e^x} x \log (x) \log \left (2-5 x^2\right )}{-2+5 x^2}+e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right )\right ) \, dx \\ & = 2 x-20 \int \frac {e^{-e^x} x \log (x) \log \left (2-5 x^2\right )}{-2+5 x^2} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x+20 \int \frac {\sqrt {5} x \log (x) \left (\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx\right )}{2-5 x^2} \, dx+20 \int \frac {\log \left (2-5 x^2\right ) \left (-\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx+\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx\right )}{2 \sqrt {5} x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x+\left (2 \sqrt {5}\right ) \int \frac {\log \left (2-5 x^2\right ) \left (-\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx+\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx\right )}{x} \, dx+\left (20 \sqrt {5}\right ) \int \frac {x \log (x) \left (\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx\right )}{2-5 x^2} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x-\left (2 \sqrt {5}\right ) \int -\frac {10 x \left (-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx\right )}{2-5 x^2} \, dx-\left (20 \sqrt {5}\right ) \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{2 \sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (20 \sqrt {5}\right ) \int \frac {x \left (-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx\right )}{2-5 x^2} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x-10 \int \left (\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx}{x}+\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x}\right ) \, dx+\left (20 \sqrt {5}\right ) \int \left (\frac {x \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{-2+5 x^2}-\frac {x \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{-2+5 x^2}\right ) \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (20 \sqrt {5}\right ) \int \frac {x \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{-2+5 x^2} \, dx-\left (20 \sqrt {5}\right ) \int \frac {x \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{-2+5 x^2} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x-10 \int \left (\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x}-\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx}{x}\right ) \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (20 \sqrt {5}\right ) \int \left (-\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{2 \sqrt {5} \left (\sqrt {2}-\sqrt {5} x\right )}+\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{2 \sqrt {5} \left (\sqrt {2}+\sqrt {5} x\right )}\right ) \, dx-\left (20 \sqrt {5}\right ) \int \left (-\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{2 \sqrt {5} \left (\sqrt {2}-\sqrt {5} x\right )}+\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{2 \sqrt {5} \left (\sqrt {2}+\sqrt {5} x\right )}\right ) \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-10 \int \left (\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx}{x}-\frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x}\right ) \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ & = 2 x-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx+10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx-10 \int \frac {\int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}-\sqrt {5} x} \, dx+(10 \log (x)) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{\sqrt {2}+\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log \left (2-5 x^2\right )\right ) \int \frac {\int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx}{x} \, dx+\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}-\sqrt {5} x} \, dx-\left (2 \sqrt {5} \log (x) \log \left (2-5 x^2\right )\right ) \int \frac {e^{-e^x}}{\sqrt {2}+\sqrt {5} x} \, dx-\int \frac {e^{-e^x} \log ^2\left (2-5 x^2\right )}{x} \, dx+\int e^{-e^x+x} \log (x) \log ^2\left (2-5 x^2\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=2 x-e^{-e^x} \log (x) \log ^2\left (2-5 x^2\right ) \]

[In]

Integrate[(E^E^x*(-4*x + 10*x^3) + (2 - 5*x^2)*Log[2 - 5*x^2]^2 + Log[x]*(-20*x^2*Log[2 - 5*x^2] + E^x*(-2*x +
5*x^3)*Log[2 - 5*x^2]^2))/(E^E^x*(-2*x + 5*x^3)),x]

[Out]

2*x - (Log[x]*Log[2 - 5*x^2]^2)/E^E^x

Maple [A] (verified)

Time = 17.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
risch	\(2 x -\ln \left (x \right ) \ln \left (-5 x^{2}+2\right )^{2} {\mathrm e}^{-{\mathrm e}^{x}}\)	\(24\)
parallelrisch	\(\frac {\left (-100 \ln \left (-5 x^{2}+2\right )^{2} \ln \left (x \right )+200 x \,{\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{-{\mathrm e}^{x}}}{100}\)	\(29\)

[In]

int(((10*x^3-4*x)*exp(exp(x))+((5*x^3-2*x)*exp(x)*ln(-5*x^2+2)^2-20*x^2*ln(-5*x^2+2))*ln(x)+(-5*x^2+2)*ln(-5*x
^2+2)^2)/(5*x^3-2*x)/exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

2*x-ln(x)*ln(-5*x^2+2)^2*exp(-exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=-{\left (\log \left (-5 \, x^{2} + 2\right )^{2} \log \left (x\right ) - 2 \, x e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )} \]

[In]

integrate(((10*x^3-4*x)*exp(exp(x))+((5*x^3-2*x)*exp(x)*log(-5*x^2+2)^2-20*x^2*log(-5*x^2+2))*log(x)+(-5*x^2+2
)*log(-5*x^2+2)^2)/(5*x^3-2*x)/exp(exp(x)),x, algorithm="fricas")

[Out]

-(log(-5*x^2 + 2)^2*log(x) - 2*x*e^(e^x))*e^(-e^x)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=\text {Timed out} \]

[In]

integrate(((10*x**3-4*x)*exp(exp(x))+((5*x**3-2*x)*exp(x)*ln(-5*x**2+2)**2-20*x**2*ln(-5*x**2+2))*ln(x)+(-5*x*
*2+2)*ln(-5*x**2+2)**2)/(5*x**3-2*x)/exp(exp(x)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=-e^{\left (-e^{x}\right )} \log \left (-5 \, x^{2} + 2\right )^{2} \log \left (x\right ) + 2 \, x \]

[In]

integrate(((10*x^3-4*x)*exp(exp(x))+((5*x^3-2*x)*exp(x)*log(-5*x^2+2)^2-20*x^2*log(-5*x^2+2))*log(x)+(-5*x^2+2
)*log(-5*x^2+2)^2)/(5*x^3-2*x)/exp(exp(x)),x, algorithm="maxima")

[Out]

-e^(-e^x)*log(-5*x^2 + 2)^2*log(x) + 2*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=-{\left (e^{\left (x - e^{x}\right )} \log \left (-5 \, x^{2} + 2\right )^{2} \log \left (x\right ) - 2 \, x e^{x}\right )} e^{\left (-x\right )} \]

[In]

integrate(((10*x^3-4*x)*exp(exp(x))+((5*x^3-2*x)*exp(x)*log(-5*x^2+2)^2-20*x^2*log(-5*x^2+2))*log(x)+(-5*x^2+2
)*log(-5*x^2+2)^2)/(5*x^3-2*x)/exp(exp(x)),x, algorithm="giac")

[Out]

-(e^(x - e^x)*log(-5*x^2 + 2)^2*log(x) - 2*x*e^x)*e^(-x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=\int \frac {{\mathrm {e}}^{-{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{{\mathrm {e}}^x}\,\left (4\,x-10\,x^3\right )+{\ln \left (2-5\,x^2\right )}^2\,\left (5\,x^2-2\right )+\ln \left (x\right )\,\left (20\,x^2\,\ln \left (2-5\,x^2\right )+{\ln \left (2-5\,x^2\right )}^2\,{\mathrm {e}}^x\,\left (2\,x-5\,x^3\right )\right )\right )}{2\,x-5\,x^3} \,d x \]

[In]

int((exp(-exp(x))*(exp(exp(x))*(4*x - 10*x^3) + log(2 - 5*x^2)^2*(5*x^2 - 2) + log(x)*(20*x^2*log(2 - 5*x^2) +
log(2 - 5*x^2)^2*exp(x)*(2*x - 5*x^3))))/(2*x - 5*x^3),x)

[Out]

int((exp(-exp(x))*(exp(exp(x))*(4*x - 10*x^3) + log(2 - 5*x^2)^2*(5*x^2 - 2) + log(x)*(20*x^2*log(2 - 5*x^2) +
log(2 - 5*x^2)^2*exp(x)*(2*x - 5*x^3))))/(2*x - 5*x^3), x)