Integrand size = 51, antiderivative size = 22 \[ \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx=\frac {1}{2} x \left (x+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}\right ) \]
[Out]
\[ \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx=\int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{x^2} \, dx \\ & = \frac {1}{2} \int \left (2 x+\frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2 (1+x)} x^4\right )\right )}{x^2}\right ) \, dx \\ & = \frac {x^2}{2}+\frac {1}{2} \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2 (1+x)} x^4\right )\right )}{x^2} \, dx \\ & = \frac {x^2}{2}+\frac {1}{2} \int \left (\frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2\right )}{x^2}-\frac {2 \left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \log \left (e^{-2-2 x} x^4\right )}{x^2}\right ) \, dx \\ & = \frac {x^2}{2}+\frac {1}{2} \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2\right )}{x^2} \, dx-\int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \log \left (e^{-2-2 x} x^4\right )}{x^2} \, dx \\ & = \frac {x^2}{2}+\frac {1}{2} \int \left (\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}+\frac {4 \left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2}-\frac {2 \left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x}\right ) \, dx-\log \left (e^{-2-2 x} x^4\right ) \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx+\int \frac {2 (2-x) \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx}{x} \, dx \\ & = \frac {x^2}{2}+\frac {1}{2} \int \left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \, dx+2 \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx+2 \int \frac {(2-x) \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx}{x} \, dx-\log \left (e^{-2-2 x} x^4\right ) \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx-\int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x} \, dx \\ & = \frac {x^2}{2}+\frac {1}{2} \int \left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \, dx+2 \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx+2 \int \left (-\int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx+\frac {2 \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx}{x}\right ) \, dx-\log \left (e^{-2-2 x} x^4\right ) \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx-\int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x} \, dx \\ & = \frac {x^2}{2}+\frac {1}{2} \int \left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \, dx+2 \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx-2 \int \left (\int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx\right ) \, dx+4 \int \frac {\int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx}{x} \, dx-\log \left (e^{-2-2 x} x^4\right ) \int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x^2} \, dx-\int \frac {\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}}}{x} \, dx \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx=\frac {1}{2} x \left (x+\left (e^{-2 (1+x)} x^4\right )^{\frac {1}{x^2}}\right ) \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {{\mathrm e}^{\frac {\ln \left (x^{4} {\mathrm e}^{-2-2 x}\right )}{x^{2}}} x}{2}+\frac {x^{2}}{2}\) | \(26\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {\ln \left (x^{4} {\mathrm e}^{-2-2 x}\right )}{x^{2}}} x}{2}+\frac {x^{2}}{2}\) | \(26\) |
parts | \(\frac {{\mathrm e}^{\frac {\ln \left (x^{4} {\mathrm e}^{-2-2 x}\right )}{x^{2}}} x}{2}+\frac {x^{2}}{2}\) | \(26\) |
risch | \(\frac {x^{2}}{2}+\frac {x \,x^{\frac {4}{x^{2}}} \left ({\mathrm e}^{1+x}\right )^{-\frac {2}{x^{2}}} {\mathrm e}^{-\frac {i \pi \left (-\operatorname {csgn}\left (i {\mathrm e}^{-2-2 x}\right ) \operatorname {csgn}\left (i x^{4} {\mathrm e}^{-2-2 x}\right )^{2}+\operatorname {csgn}\left (i {\mathrm e}^{-2-2 x}\right ) \operatorname {csgn}\left (i x^{4} {\mathrm e}^{-2-2 x}\right ) \operatorname {csgn}\left (i x^{4}\right )+\operatorname {csgn}\left (i x^{2}\right )^{3}-2 \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )-\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{3}\right )^{3}-\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{4} {\mathrm e}^{-2-2 x}\right )^{3}-\operatorname {csgn}\left (i x^{4} {\mathrm e}^{-2-2 x}\right )^{2} \operatorname {csgn}\left (i x^{4}\right )+\operatorname {csgn}\left (i x^{4}\right )^{3}-\operatorname {csgn}\left (i {\mathrm e}^{2+2 x}\right )^{3}+2 \operatorname {csgn}\left (i {\mathrm e}^{2+2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{1+x}\right )-\operatorname {csgn}\left (i {\mathrm e}^{2+2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{1+x}\right )^{2}\right )}{2 x^{2}}}}{2}\) | \(357\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx=\frac {1}{2} \, \left (x^{4} e^{\left (-2 \, x - 2\right )}\right )^{\left (\frac {1}{x^{2}}\right )} x + \frac {1}{2} \, x^{2} \]
[In]
[Out]
\[ \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx=\frac {\int 2 x\, dx + \int e^{- \frac {2}{x^{2}}} e^{\frac {\log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}}\, dx + \int \frac {8 e^{- \frac {2}{x^{2}}} e^{\frac {\log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}}}{x^{2}}\, dx + \int \left (- \frac {2 e^{- \frac {2}{x^{2}}} e^{\frac {\log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}}}{x}\right )\, dx + \int \left (- \frac {2 e^{- \frac {2}{x^{2}}} e^{\frac {\log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}} \log {\left (x^{4} e^{- 2 x} \right )}}{x^{2}}\right )\, dx}{2} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{\left (-\frac {2}{x} + \frac {4 \, \log \left (x\right )}{x^{2}} - \frac {2}{x^{2}}\right )} \]
[In]
[Out]
\[ \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx=\int { \frac {2 \, x^{3} + {\left (x^{2} - 2 \, x - 2 \, \log \left (x^{4} e^{\left (-2 \, x - 2\right )}\right ) + 4\right )} \left (x^{4} e^{\left (-2 \, x - 2\right )}\right )^{\left (\frac {1}{x^{2}}\right )}}{2 \, x^{2}} \,d x } \]
[In]
[Out]
Time = 7.92 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {2 x^3+\left (e^{-2-2 x} x^4\right )^{\frac {1}{x^2}} \left (4-2 x+x^2-2 \log \left (e^{-2-2 x} x^4\right )\right )}{2 x^2} \, dx=\frac {x^2}{2}+\frac {x\,{\mathrm {e}}^{-\frac {2}{x}}\,{\mathrm {e}}^{-\frac {2}{x^2}}\,{\left (x^4\right )}^{\frac {1}{x^2}}}{2} \]
[In]
[Out]