Integrand size = 51, antiderivative size = 25 \[ \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{5 x} \, dx=9 e^{\frac {1}{5} (-4+2 x-\log (5))} \left (4-\log \left (x^4\right )\right ) \]
[Out]
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {12, 14, 2225, 2209, 2634} \[ \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{5 x} \, dx=\frac {36 e^{\frac {2 x}{5}-\frac {4}{5}}}{\sqrt [5]{5}}-\frac {9 e^{\frac {2 x}{5}-\frac {4}{5}} \log \left (x^4\right )}{\sqrt [5]{5}} \]
[In]
[Out]
Rule 12
Rule 14
Rule 2209
Rule 2225
Rule 2634
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{x} \, dx \\ & = \frac {1}{5} \int \left (\frac {72 e^{-\frac {4}{5}+\frac {2 x}{5}}}{\sqrt [5]{5}}-\frac {36\ 5^{4/5} e^{-\frac {4}{5}+\frac {2 x}{5}}}{x}-\frac {18 e^{-\frac {4}{5}+\frac {2 x}{5}} \log \left (x^4\right )}{\sqrt [5]{5}}\right ) \, dx \\ & = -\frac {18 \int e^{-\frac {4}{5}+\frac {2 x}{5}} \log \left (x^4\right ) \, dx}{5 \sqrt [5]{5}}+\frac {72 \int e^{-\frac {4}{5}+\frac {2 x}{5}} \, dx}{5 \sqrt [5]{5}}-\frac {36 \int \frac {e^{-\frac {4}{5}+\frac {2 x}{5}}}{x} \, dx}{\sqrt [5]{5}} \\ & = \frac {36 e^{-\frac {4}{5}+\frac {2 x}{5}}}{\sqrt [5]{5}}-\frac {36 \text {Ei}\left (\frac {2 x}{5}\right )}{\sqrt [5]{5} e^{4/5}}-\frac {9 e^{-\frac {4}{5}+\frac {2 x}{5}} \log \left (x^4\right )}{\sqrt [5]{5}}+\frac {18 \int \frac {10 e^{-\frac {4}{5}+\frac {2 x}{5}}}{x} \, dx}{5 \sqrt [5]{5}} \\ & = \frac {36 e^{-\frac {4}{5}+\frac {2 x}{5}}}{\sqrt [5]{5}}-\frac {36 \text {Ei}\left (\frac {2 x}{5}\right )}{\sqrt [5]{5} e^{4/5}}-\frac {9 e^{-\frac {4}{5}+\frac {2 x}{5}} \log \left (x^4\right )}{\sqrt [5]{5}}+\frac {36 \int \frac {e^{-\frac {4}{5}+\frac {2 x}{5}}}{x} \, dx}{\sqrt [5]{5}} \\ & = \frac {36 e^{-\frac {4}{5}+\frac {2 x}{5}}}{\sqrt [5]{5}}-\frac {9 e^{-\frac {4}{5}+\frac {2 x}{5}} \log \left (x^4\right )}{\sqrt [5]{5}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{5 x} \, dx=-\frac {9 e^{\frac {2}{5} (-2+x)} \left (-4+\log \left (x^4\right )\right )}{\sqrt [5]{5}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20
method | result | size |
norman | \(-9 \,{\mathrm e}^{-\frac {\ln \left (5\right )}{5}+\frac {2 x}{5}-\frac {4}{5}} \ln \left (x^{4}\right )+36 \,{\mathrm e}^{-\frac {\ln \left (5\right )}{5}+\frac {2 x}{5}-\frac {4}{5}}\) | \(30\) |
parallelrisch | \(-9 \,{\mathrm e}^{-\frac {\ln \left (5\right )}{5}+\frac {2 x}{5}-\frac {4}{5}} \ln \left (x^{4}\right )+36 \,{\mathrm e}^{-\frac {\ln \left (5\right )}{5}+\frac {2 x}{5}-\frac {4}{5}}\) | \(30\) |
default | \(\frac {\left (180-45 \ln \left (x^{4}\right )+180 \ln \left (x \right )\right ) {\mathrm e}^{-\frac {\ln \left (5\right )}{5}+\frac {2 x}{5}-\frac {4}{5}}}{5}-36 \ln \left (x \right ) {\mathrm e}^{-\frac {\ln \left (5\right )}{5}+\frac {2 x}{5}-\frac {4}{5}}\) | \(40\) |
risch | \(-\frac {36 \ln \left (x \right ) 5^{\frac {4}{5}} {\mathrm e}^{-\frac {4}{5}+\frac {2 x}{5}}}{5}+\frac {9 i 5^{\frac {4}{5}} {\mathrm e}^{-\frac {4}{5}+\frac {2 x}{5}} \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+\pi \operatorname {csgn}\left (i x^{3}\right )^{3}-\pi \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\pi \operatorname {csgn}\left (i x^{4}\right )^{3}-8 i\right )}{10}\) | \(210\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{5 x} \, dx=-9 \, e^{\left (\frac {2}{5} \, x - \frac {1}{5} \, \log \left (5\right ) - \frac {4}{5}\right )} \log \left (x^{4}\right ) + 36 \, e^{\left (\frac {2}{5} \, x - \frac {1}{5} \, \log \left (5\right ) - \frac {4}{5}\right )} \]
[In]
[Out]
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{5 x} \, dx=\frac {\left (- 9 \cdot 5^{\frac {4}{5}} \log {\left (x^{4} \right )} + 36 \cdot 5^{\frac {4}{5}}\right ) e^{\frac {2 x}{5} - \frac {4}{5}}}{5} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{5 x} \, dx=-9 \, e^{\left (\frac {2}{5} \, x - \frac {1}{5} \, \log \left (5\right ) - \frac {4}{5}\right )} \log \left (x^{4}\right ) + 36 \, e^{\left (\frac {2}{5} \, x - \frac {1}{5} \, \log \left (5\right ) - \frac {4}{5}\right )} \]
[In]
[Out]
\[ \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{5 x} \, dx=\int { -\frac {18 \, {\left (x e^{\left (\frac {2}{5} \, x - \frac {1}{5} \, \log \left (5\right ) - \frac {4}{5}\right )} \log \left (x^{4}\right ) - 2 \, {\left (2 \, x - 5\right )} e^{\left (\frac {2}{5} \, x - \frac {1}{5} \, \log \left (5\right ) - \frac {4}{5}\right )}\right )}}{5 \, x} \,d x } \]
[In]
[Out]
Time = 10.64 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {1}{5} (-4+2 x-\log (5))} (-180+72 x)-18 e^{\frac {1}{5} (-4+2 x-\log (5))} x \log \left (x^4\right )}{5 x} \, dx=-\frac {9\,5^{4/5}\,{\mathrm {e}}^{\frac {2\,x}{5}-\frac {4}{5}}\,\left (\ln \left (x^4\right )-4\right )}{5} \]
[In]
[Out]