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]
\[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=\int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx \]
[In]
[Out]
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*}
Time = 0.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=2 x-e^{-e^x} \log (x) \log ^2\left (2-5 x^2\right ) \]
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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 | \(\frac {\left (-100 \ln \left (-5 x^{2}+2\right )^{2} \ln \left (x \right )+200 x \,{\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{-{\mathrm e}^{x}}}{100}\) | \(29\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=-{\left (\log \left (-5 \, x^{2} + 2\right )^{2} \log \left (x\right ) - 2 \, x e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )} \]
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Timed out. \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=\text {Timed out} \]
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Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=-e^{\left (-e^{x}\right )} \log \left (-5 \, x^{2} + 2\right )^{2} \log \left (x\right ) + 2 \, x \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=-{\left (e^{\left (x - e^{x}\right )} \log \left (-5 \, x^{2} + 2\right )^{2} \log \left (x\right ) - 2 \, x e^{x}\right )} e^{\left (-x\right )} \]
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Timed out. \[ \int \frac {e^{-e^x} \left (e^{e^x} \left (-4 x+10 x^3\right )+\left (2-5 x^2\right ) \log ^2\left (2-5 x^2\right )+\log (x) \left (-20 x^2 \log \left (2-5 x^2\right )+e^x \left (-2 x+5 x^3\right ) \log ^2\left (2-5 x^2\right )\right )\right )}{-2 x+5 x^3} \, dx=\int \frac {{\mathrm {e}}^{-{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{{\mathrm {e}}^x}\,\left (4\,x-10\,x^3\right )+{\ln \left (2-5\,x^2\right )}^2\,\left (5\,x^2-2\right )+\ln \left (x\right )\,\left (20\,x^2\,\ln \left (2-5\,x^2\right )+{\ln \left (2-5\,x^2\right )}^2\,{\mathrm {e}}^x\,\left (2\,x-5\,x^3\right )\right )\right )}{2\,x-5\,x^3} \,d x \]
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