Integrand size = 139, antiderivative size = 29 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2}{\log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \]
[Out]
\[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{e \left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \log ^2\left (e \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )} \, dx \\ & = \frac {\int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \log ^2\left (e \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )} \, dx}{e} \\ & = \frac {\int \left (\frac {2 e^{e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )-x^2} x^3}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2}+\frac {e x \left (2 e^{1+e^2+e^{8-x^2}}+2 e^x-e^x x+2 e^{1+e^2+e^{8-x^2}} \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+2 e^x \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2}\right ) \, dx}{e} \\ & = \frac {2 \int \frac {e^{e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )-x^2} x^3}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx}{e}+\int \frac {x \left (2 e^{1+e^2+e^{8-x^2}}+2 e^x-e^x x+2 e^{1+e^2+e^{8-x^2}} \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )+2 e^x \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx \\ & = \frac {2 \int \frac {e^{e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )-x^2} x^3}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx}{e}+\int \frac {x \left (2 e^{1+e^2+e^{8-x^2}}-e^x (-2+x)+2 \left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx \\ & = \frac {2 \int \frac {e^{e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )-x^2} x^3}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx}{e}+\int \left (\frac {e^{1+e^2+e^{8-x^2}} x^2}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2}-\frac {x \left (-2+x-2 \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}{\left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2}\right ) \, dx \\ & = \frac {2 \int \frac {e^{e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )-x^2} x^3}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx}{e}+\int \frac {e^{1+e^2+e^{8-x^2}} x^2}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx-\int \frac {x \left (-2+x-2 \log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )}{\left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx \\ & = \frac {2 \int \frac {e^{e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )-x^2} x^3}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx}{e}+\int \frac {e^{1+e^2+e^{8-x^2}} x^2}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx-\int \left (\frac {x^2}{\left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2}-\frac {2 x}{1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )}\right ) \, dx \\ & = 2 \int \frac {x}{1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )} \, dx+\frac {2 \int \frac {e^{e^{8-x^2}+10 \left (1+\frac {e^2}{10}\right )-x^2} x^3}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx}{e}-\int \frac {x^2}{\left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx+\int \frac {e^{1+e^2+e^{8-x^2}} x^2}{\left (e^{1+e^2+e^{8-x^2}}+e^x\right ) \left (1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )\right )^2} \, dx \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2}{1+\log \left (e^{1+e^2+e^{8-x^2}}+e^x\right )} \]
[In]
[Out]
Time = 0.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {x^{2}}{\ln \left ({\mathrm e}^{{\mathrm e}^{-x^{2}+8}+{\mathrm e}^{2}+2}+{\mathrm e}^{1+x}\right )}\) | \(26\) |
parallelrisch | \(\frac {x^{2}}{\ln \left ({\mathrm e}^{{\mathrm e}^{-x^{2}+8}+{\mathrm e}^{2}+2}+{\mathrm e}^{1+x}\right )}\) | \(26\) |
[In]
[Out]
none
Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log \left (e^{\left (x + 1\right )} + e^{\left (e^{2} + e^{\left (-x^{2} + 8\right )} + 2\right )}\right )} \]
[In]
[Out]
Time = 1.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log {\left (e^{x + 1} + e^{e^{8 - x^{2}} + 2 + e^{2}} \right )}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^{2}}{\log \left (e^{x} + e^{\left (e^{2} + e^{\left (-x^{2} + 8\right )} + 1\right )}\right ) + 1} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (25) = 50\).
Time = 14.59 (sec) , antiderivative size = 814, normalized size of antiderivative = 28.07 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 12.21 (sec) , antiderivative size = 286, normalized size of antiderivative = 9.86 \[ \int \frac {-e^{1+x} x^2+2 e^{10+e^2+e^{8-x^2}-x^2} x^3+\left (2 e^{2+e^2+e^{8-x^2}} x+2 e^{1+x} x\right ) \log \left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )}{\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right ) \log ^2\left (e^{2+e^2+e^{8-x^2}}+e^{1+x}\right )} \, dx=\frac {x^2-\frac {2\,x\,\ln \left (\mathrm {e}\,{\mathrm {e}}^x+{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^2}\right )\,\left ({\mathrm {e}}^{x+1}+{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}+2}\right )}{{\mathrm {e}}^{x+1}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}-x^2+10}}}{\ln \left (\mathrm {e}\,{\mathrm {e}}^x+{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^2}\right )}-{\mathrm {e}}^{x^2-8}+\frac {4\,x^2\,{\mathrm {e}}^{-x^2+2\,x+10}-{\mathrm {e}}^{2\,x+2}+4\,x^3\,{\mathrm {e}}^{-x^2+2\,x+10}+4\,x^3\,{\mathrm {e}}^{-2\,x^2+2\,x+18}+x\,{\mathrm {e}}^{2\,x+2}-2\,x\,{\mathrm {e}}^{-x^2+2\,x+10}+2\,x^2\,{\mathrm {e}}^{2\,x+2}}{\left ({\mathrm {e}}^{x+1}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^2+{\mathrm {e}}^8\,{\mathrm {e}}^{-x^2}-x^2+10}\right )\,\left (x\,{\mathrm {e}}^{-x^2+x+9}-{\mathrm {e}}^{-x^2+x+9}+2\,x^2\,{\mathrm {e}}^{-x^2+x+9}+2\,x^2\,{\mathrm {e}}^{-2\,x^2+x+17}\right )} \]
[In]
[Out]