Integrand size = 113, antiderivative size = 31 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {e^{6-2 x-\frac {2 \left (x+\log \left (\frac {x}{2}\right )\right )^2}{x^2}} \left (e^4+x\right )}{x} \]
[Out]
\[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \left (-4 e^4-\left (4+e^4\right ) x-2 e^4 x^2-2 x^3\right )}{x^3}+\frac {4 \exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) (-1+x) \left (e^4+x\right ) \log \left (\frac {x}{2}\right )}{x^4}+\frac {4 \exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \left (e^4+x\right ) \log ^2\left (\frac {x}{2}\right )}{x^4}\right ) \, dx \\ & = 4 \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) (-1+x) \left (e^4+x\right ) \log \left (\frac {x}{2}\right )}{x^4} \, dx+4 \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \left (e^4+x\right ) \log ^2\left (\frac {x}{2}\right )}{x^4} \, dx+\int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \left (-4 e^4-\left (4+e^4\right ) x-2 e^4 x^2-2 x^3\right )}{x^3} \, dx \\ & = 4 \int \left (-\frac {\exp \left (4+\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log \left (\frac {x}{2}\right )}{x^4}+\frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \left (-1+e^4\right ) \log \left (\frac {x}{2}\right )}{x^3}+\frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log \left (\frac {x}{2}\right )}{x^2}\right ) \, dx+4 \int \left (\frac {\exp \left (4+\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log ^2\left (\frac {x}{2}\right )}{x^4}+\frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log ^2\left (\frac {x}{2}\right )}{x^3}\right ) \, dx+\int \left (-2 \exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )-\frac {4 \exp \left (4+\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )}{x^3}+\frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \left (-4-e^4\right )}{x^2}-\frac {2 \exp \left (4+\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )}{x}\right ) \, dx \\ & = -\left (2 \int \exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \, dx\right )-2 \int \frac {\exp \left (4+\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )}{x} \, dx-4 \int \frac {\exp \left (4+\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )}{x^3} \, dx-4 \int \frac {\exp \left (4+\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log \left (\frac {x}{2}\right )}{x^4} \, dx+4 \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log \left (\frac {x}{2}\right )}{x^2} \, dx+4 \int \frac {\exp \left (4+\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log ^2\left (\frac {x}{2}\right )}{x^4} \, dx+4 \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log ^2\left (\frac {x}{2}\right )}{x^3} \, dx+\left (-4-e^4\right ) \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )}{x^2} \, dx-\left (4 \left (1-e^4\right )\right ) \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log \left (\frac {x}{2}\right )}{x^3} \, dx \\ & = -\left (2 \int \exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \, dx\right )-2 \int \frac {\exp \left (-\frac {2 \left (-4 x^2+x^3+2 x \log \left (\frac {x}{2}\right )+\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )}{x} \, dx-4 \int \frac {\exp \left (-\frac {2 \left (-4 x^2+x^3+2 x \log \left (\frac {x}{2}\right )+\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )}{x^3} \, dx-4 \int \frac {\exp \left (-\frac {2 \left (-4 x^2+x^3+2 x \log \left (\frac {x}{2}\right )+\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log \left (\frac {x}{2}\right )}{x^4} \, dx+4 \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log \left (\frac {x}{2}\right )}{x^2} \, dx+4 \int \frac {\exp \left (-\frac {2 \left (-4 x^2+x^3+2 x \log \left (\frac {x}{2}\right )+\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log ^2\left (\frac {x}{2}\right )}{x^4} \, dx+4 \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log ^2\left (\frac {x}{2}\right )}{x^3} \, dx+\left (-4-e^4\right ) \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right )}{x^2} \, dx-\left (4 \left (1-e^4\right )\right ) \int \frac {\exp \left (\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}\right ) \log \left (\frac {x}{2}\right )}{x^3} \, dx \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=16^{\frac {1}{x}} e^{4-2 x-\frac {2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^{-\frac {4+x}{x}} \left (e^4+x\right ) \]
[In]
[Out]
Time = 1.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32
method | result | size |
risch | \(\frac {\left (x +{\mathrm e}^{4}\right ) \left (\frac {x}{2}\right )^{-\frac {4}{x}} {\mathrm e}^{-\frac {2 \left (x^{3}+\ln \left (\frac {x}{2}\right )^{2}-2 x^{2}\right )}{x^{2}}}}{x}\) | \(41\) |
parallelrisch | \(\frac {{\mathrm e}^{4} {\mathrm e}^{-\frac {2 \left (x^{3}+\ln \left (\frac {x}{2}\right )^{2}+2 x \ln \left (\frac {x}{2}\right )-2 x^{2}\right )}{x^{2}}}+{\mathrm e}^{-\frac {2 \left (x^{3}+\ln \left (\frac {x}{2}\right )^{2}+2 x \ln \left (\frac {x}{2}\right )-2 x^{2}\right )}{x^{2}}} x}{x}\) | \(71\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {{\left (x + e^{4}\right )} e^{\left (-\frac {2 \, {\left (x^{3} - 2 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )}}{x} \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {\left (x + e^{4}\right ) e^{\frac {2 \left (- x^{3} + 2 x^{2} - 2 x \log {\left (\frac {x}{2} \right )} - \log {\left (\frac {x}{2} \right )}^{2}\right )}{x^{2}}}}{x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).
Time = 0.40 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {{\left (x e^{4} + e^{8}\right )} e^{\left (-2 \, x + \frac {4 \, \log \left (2\right )}{x} - \frac {2 \, \log \left (2\right )^{2}}{x^{2}} - \frac {4 \, \log \left (x\right )}{x} + \frac {4 \, \log \left (2\right ) \log \left (x\right )}{x^{2}} - \frac {2 \, \log \left (x\right )^{2}}{x^{2}}\right )}}{x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.49 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {x e^{\left (-\frac {2 \, {\left (x^{3} - 2 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )} + e^{\left (-\frac {2 \, {\left (x^{3} - 4 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )}}{x} \]
[In]
[Out]
Time = 13.80 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {2^{4/x}\,x^{\frac {4\,\ln \left (2\right )}{x^2}}\,{\mathrm {e}}^{4-\frac {2\,{\ln \left (2\right )}^2}{x^2}-\frac {2\,{\ln \left (x\right )}^2}{x^2}-2\,x}\,\left (x+{\mathrm {e}}^4\right )}{x^{4/x}\,x} \]
[In]
[Out]