Integrand size = 43, antiderivative size = 24 \[ \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{76 \log ^2(2 x)} \, dx=\frac {x \left (5-e^{e^5}+\frac {3 x^2}{76}\right )}{\log (2 x)} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {12, 6873, 6874, 2367, 2334, 2335, 2343, 2346, 2209} \[ \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{76 \log ^2(2 x)} \, dx=\frac {3 x^3}{76 \log (2 x)}+\frac {\left (5-e^{e^5}\right ) x}{\log (2 x)} \]
[In]
[Out]
Rule 12
Rule 2209
Rule 2334
Rule 2335
Rule 2343
Rule 2346
Rule 2367
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{76} \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{\log ^2(2 x)} \, dx \\ & = \frac {1}{76} \int \frac {-380 \left (1-\frac {e^{e^5}}{5}\right )-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{\log ^2(2 x)} \, dx \\ & = \frac {1}{76} \int \left (\frac {-380+76 e^{e^5}-3 x^2}{\log ^2(2 x)}-\frac {-380+76 e^{e^5}-9 x^2}{\log (2 x)}\right ) \, dx \\ & = \frac {1}{76} \int \frac {-380+76 e^{e^5}-3 x^2}{\log ^2(2 x)} \, dx-\frac {1}{76} \int \frac {-380+76 e^{e^5}-9 x^2}{\log (2 x)} \, dx \\ & = \frac {1}{76} \int \left (-\frac {380 \left (1-\frac {e^{e^5}}{5}\right )}{\log ^2(2 x)}-\frac {3 x^2}{\log ^2(2 x)}\right ) \, dx-\frac {1}{76} \int \left (-\frac {380 \left (1-\frac {e^{e^5}}{5}\right )}{\log (2 x)}-\frac {9 x^2}{\log (2 x)}\right ) \, dx \\ & = -\left (\frac {3}{76} \int \frac {x^2}{\log ^2(2 x)} \, dx\right )+\frac {9}{76} \int \frac {x^2}{\log (2 x)} \, dx+\left (-5+e^{e^5}\right ) \int \frac {1}{\log ^2(2 x)} \, dx-\left (-5+e^{e^5}\right ) \int \frac {1}{\log (2 x)} \, dx \\ & = \frac {\left (5-e^{e^5}\right ) x}{\log (2 x)}+\frac {3 x^3}{76 \log (2 x)}+\frac {1}{2} \left (5-e^{e^5}\right ) \text {li}(2 x)+\frac {9}{608} \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (2 x)\right )-\frac {9}{76} \int \frac {x^2}{\log (2 x)} \, dx+\left (-5+e^{e^5}\right ) \int \frac {1}{\log (2 x)} \, dx \\ & = \frac {9}{608} \text {Ei}(3 \log (2 x))+\frac {\left (5-e^{e^5}\right ) x}{\log (2 x)}+\frac {3 x^3}{76 \log (2 x)}-\frac {9}{608} \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (2 x)\right ) \\ & = \frac {\left (5-e^{e^5}\right ) x}{\log (2 x)}+\frac {3 x^3}{76 \log (2 x)} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{76 \log ^2(2 x)} \, dx=\frac {x \left (380-76 e^{e^5}+3 x^2\right )}{76 \log (2 x)} \]
[In]
[Out]
Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {x \left (-3 x^{2}+76 \,{\mathrm e}^{{\mathrm e}^{5}}-380\right )}{76 \ln \left (2 x \right )}\) | \(22\) |
norman | \(\frac {\left (-{\mathrm e}^{{\mathrm e}^{5}}+5\right ) x +\frac {3 x^{3}}{76}}{\ln \left (2 x \right )}\) | \(23\) |
parallelrisch | \(\frac {3 x^{3}-76 x \,{\mathrm e}^{{\mathrm e}^{5}}+380 x}{76 \ln \left (2 x \right )}\) | \(24\) |
derivativedivides | \(\frac {{\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (-\ln \left (2 x \right )\right )}{2}+\frac {3 x^{3}}{76 \ln \left (2 x \right )}+\frac {{\mathrm e}^{{\mathrm e}^{5}} \left (-\frac {2 x}{\ln \left (2 x \right )}-\operatorname {Ei}_{1}\left (-\ln \left (2 x \right )\right )\right )}{2}+\frac {5 x}{\ln \left (2 x \right )}\) | \(60\) |
default | \(\frac {{\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (-\ln \left (2 x \right )\right )}{2}+\frac {3 x^{3}}{76 \ln \left (2 x \right )}+\frac {{\mathrm e}^{{\mathrm e}^{5}} \left (-\frac {2 x}{\ln \left (2 x \right )}-\operatorname {Ei}_{1}\left (-\ln \left (2 x \right )\right )\right )}{2}+\frac {5 x}{\ln \left (2 x \right )}\) | \(60\) |
parts | \(\frac {{\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (-\ln \left (2 x \right )\right )}{2}+\frac {3 x^{3}}{76 \ln \left (2 x \right )}+\frac {{\mathrm e}^{{\mathrm e}^{5}} \left (-\frac {2 x}{\ln \left (2 x \right )}-\operatorname {Ei}_{1}\left (-\ln \left (2 x \right )\right )\right )}{2}+\frac {5 x}{\ln \left (2 x \right )}\) | \(60\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{76 \log ^2(2 x)} \, dx=\frac {3 \, x^{3} - 76 \, x e^{\left (e^{5}\right )} + 380 \, x}{76 \, \log \left (2 \, x\right )} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{76 \log ^2(2 x)} \, dx=\frac {3 x^{3} - 76 x e^{e^{5}} + 380 x}{76 \log {\left (2 x \right )}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{76 \log ^2(2 x)} \, dx=-\frac {1}{2} \, {\rm Ei}\left (\log \left (2 \, x\right )\right ) e^{\left (e^{5}\right )} + \frac {1}{2} \, e^{\left (e^{5}\right )} \Gamma \left (-1, -\log \left (2 \, x\right )\right ) + \frac {9}{608} \, {\rm Ei}\left (3 \, \log \left (2 \, x\right )\right ) + \frac {5}{2} \, {\rm Ei}\left (\log \left (2 \, x\right )\right ) - \frac {5}{2} \, \Gamma \left (-1, -\log \left (2 \, x\right )\right ) - \frac {9}{608} \, \Gamma \left (-1, -3 \, \log \left (2 \, x\right )\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{76 \log ^2(2 x)} \, dx=\frac {3 \, x^{3}}{76 \, \log \left (2 \, x\right )} - \frac {x e^{\left (e^{5}\right )}}{\log \left (2 \, x\right )} + \frac {5 \, x}{\log \left (2 \, x\right )} \]
[In]
[Out]
Time = 10.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-380+76 e^{e^5}-3 x^2+\left (380-76 e^{e^5}+9 x^2\right ) \log (2 x)}{76 \log ^2(2 x)} \, dx=-\frac {x^2\,\left ({\mathrm {e}}^{{\mathrm {e}}^5}-5\right )-\frac {3\,x^4}{76}}{x\,\ln \left (2\,x\right )} \]
[In]
[Out]