Integrand size = 125, antiderivative size = 29 \[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx=\frac {\log (4) \left (e^x-\frac {1+\log \left ((x+\log (4))^2\right )}{x}\right )}{\log \left (\frac {x}{4}\right )} \]
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\[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx=\int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{x^2 (x+\log (4)) \log ^2\left (\frac {x}{4}\right )} \, dx \\ & = \int \frac {\log (4) \left (-\left ((x+\log (4)) \left (-1+e^x x-\log \left ((x+\log (4))^2\right )\right )\right )+\log \left (\frac {x}{4}\right ) \left (-x+e^x x^3+\log (4)+e^x x^2 \log (4)+(x+\log (4)) \log \left ((x+\log (4))^2\right )\right )\right )}{x^2 (x+\log (4)) \log ^2\left (\frac {x}{4}\right )} \, dx \\ & = \log (4) \int \frac {-\left ((x+\log (4)) \left (-1+e^x x-\log \left ((x+\log (4))^2\right )\right )\right )+\log \left (\frac {x}{4}\right ) \left (-x+e^x x^3+\log (4)+e^x x^2 \log (4)+(x+\log (4)) \log \left ((x+\log (4))^2\right )\right )}{x^2 (x+\log (4)) \log ^2\left (\frac {x}{4}\right )} \, dx \\ & = \log (4) \int \left (\frac {e^x \left (-1+x \log \left (\frac {x}{4}\right )\right )}{x \log ^2\left (\frac {x}{4}\right )}+\frac {x+\log (4)-x \log \left (\frac {x}{4}\right )+\log (4) \log \left (\frac {x}{4}\right )+x \log \left ((x+\log (4))^2\right )+\log (4) \log \left ((x+\log (4))^2\right )+x \log \left (\frac {x}{4}\right ) \log \left ((x+\log (4))^2\right )+\log (4) \log \left (\frac {x}{4}\right ) \log \left ((x+\log (4))^2\right )}{x^2 (x+\log (4)) \log ^2\left (\frac {x}{4}\right )}\right ) \, dx \\ & = \log (4) \int \frac {e^x \left (-1+x \log \left (\frac {x}{4}\right )\right )}{x \log ^2\left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {x+\log (4)-x \log \left (\frac {x}{4}\right )+\log (4) \log \left (\frac {x}{4}\right )+x \log \left ((x+\log (4))^2\right )+\log (4) \log \left ((x+\log (4))^2\right )+x \log \left (\frac {x}{4}\right ) \log \left ((x+\log (4))^2\right )+\log (4) \log \left (\frac {x}{4}\right ) \log \left ((x+\log (4))^2\right )}{x^2 (x+\log (4)) \log ^2\left (\frac {x}{4}\right )} \, dx \\ & = \frac {e^x \log (4)}{\log \left (\frac {x}{4}\right )}+\log (4) \int \frac {(x+\log (4)) \left (1+\log \left ((x+\log (4))^2\right )\right )+\log \left (\frac {x}{4}\right ) \left (-x+\log (4)+(x+\log (4)) \log \left ((x+\log (4))^2\right )\right )}{x^2 (x+\log (4)) \log ^2\left (\frac {x}{4}\right )} \, dx \\ & = \frac {e^x \log (4)}{\log \left (\frac {x}{4}\right )}+\log (4) \int \left (\frac {x+\log (4)-x \log \left (\frac {x}{4}\right )+\log (4) \log \left (\frac {x}{4}\right )}{x^2 (x+\log (4)) \log ^2\left (\frac {x}{4}\right )}+\frac {\left (1+\log \left (\frac {x}{4}\right )\right ) \log \left ((x+\log (4))^2\right )}{x^2 \log ^2\left (\frac {x}{4}\right )}\right ) \, dx \\ & = \frac {e^x \log (4)}{\log \left (\frac {x}{4}\right )}+\log (4) \int \frac {x+\log (4)-x \log \left (\frac {x}{4}\right )+\log (4) \log \left (\frac {x}{4}\right )}{x^2 (x+\log (4)) \log ^2\left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\left (1+\log \left (\frac {x}{4}\right )\right ) \log \left ((x+\log (4))^2\right )}{x^2 \log ^2\left (\frac {x}{4}\right )} \, dx \\ & = \frac {e^x \log (4)}{\log \left (\frac {x}{4}\right )}+\log (4) \int \left (\frac {1}{x^2 \log ^2\left (\frac {x}{4}\right )}+\frac {-x+\log (4)}{x^2 (x+\log (4)) \log \left (\frac {x}{4}\right )}\right ) \, dx+\log (4) \int \left (\frac {\log \left ((x+\log (4))^2\right )}{x^2 \log ^2\left (\frac {x}{4}\right )}+\frac {\log \left ((x+\log (4))^2\right )}{x^2 \log \left (\frac {x}{4}\right )}\right ) \, dx \\ & = \frac {e^x \log (4)}{\log \left (\frac {x}{4}\right )}+\log (4) \int \frac {1}{x^2 \log ^2\left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {-x+\log (4)}{x^2 (x+\log (4)) \log \left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\log \left ((x+\log (4))^2\right )}{x^2 \log ^2\left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\log \left ((x+\log (4))^2\right )}{x^2 \log \left (\frac {x}{4}\right )} \, dx \\ & = \frac {e^x \log (4)}{\log \left (\frac {x}{4}\right )}-\frac {\log (4)}{x \log \left (\frac {x}{4}\right )}-\log (4) \int \frac {1}{x^2 \log \left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {-x+\log (4)}{x^2 (x+\log (4)) \log \left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\log \left ((x+\log (4))^2\right )}{x^2 \log ^2\left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\log \left ((x+\log (4))^2\right )}{x^2 \log \left (\frac {x}{4}\right )} \, dx \\ & = \frac {e^x \log (4)}{\log \left (\frac {x}{4}\right )}-\frac {\log (4)}{x \log \left (\frac {x}{4}\right )}-\frac {1}{4} \log (4) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {x}{4}\right )\right )+\log (4) \int \frac {-x+\log (4)}{x^2 (x+\log (4)) \log \left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\log \left ((x+\log (4))^2\right )}{x^2 \log ^2\left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\log \left ((x+\log (4))^2\right )}{x^2 \log \left (\frac {x}{4}\right )} \, dx \\ & = -\frac {1}{4} \operatorname {ExpIntegralEi}\left (-\log \left (\frac {x}{4}\right )\right ) \log (4)+\frac {e^x \log (4)}{\log \left (\frac {x}{4}\right )}-\frac {\log (4)}{x \log \left (\frac {x}{4}\right )}+\log (4) \int \frac {-x+\log (4)}{x^2 (x+\log (4)) \log \left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\log \left ((x+\log (4))^2\right )}{x^2 \log ^2\left (\frac {x}{4}\right )} \, dx+\log (4) \int \frac {\log \left ((x+\log (4))^2\right )}{x^2 \log \left (\frac {x}{4}\right )} \, dx \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx=\frac {\log (4) \left (-1+e^x x-\log \left ((x+\log (4))^2\right )\right )}{x \log \left (\frac {x}{4}\right )} \]
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Time = 10.80 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48
method | result | size |
parallelrisch | \(\frac {2 x \ln \left (2\right ) {\mathrm e}^{x}-2 \ln \left (2\right ) \ln \left (4 \ln \left (2\right )^{2}+4 x \ln \left (2\right )+x^{2}\right )-2 \ln \left (2\right )}{x \ln \left (\frac {x}{4}\right )}\) | \(43\) |
risch | \(-\frac {4 \ln \left (2\right ) \ln \left (\ln \left (2\right )+\frac {x}{2}\right )}{x \ln \left (\frac {x}{4}\right )}+\frac {\ln \left (2\right ) \left (i \pi \operatorname {csgn}\left (i \left (\ln \left (2\right )+\frac {x}{2}\right )\right )^{2} \operatorname {csgn}\left (i \left (\ln \left (2\right )+\frac {x}{2}\right )^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i \left (\ln \left (2\right )+\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i \left (\ln \left (2\right )+\frac {x}{2}\right )^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i \left (\ln \left (2\right )+\frac {x}{2}\right )^{2}\right )^{3}+2 \,{\mathrm e}^{x} x -2\right )}{x \ln \left (\frac {x}{4}\right )}\) | \(115\) |
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Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx=\frac {2 \, {\left (x e^{x} \log \left (2\right ) - \log \left (2\right ) \log \left (x^{2} + 4 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right ) - \log \left (2\right )\right )}}{x \log \left (\frac {1}{4} \, x\right )} \]
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Exception generated. \[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx=-\frac {2 \, {\left (x e^{x} \log \left (2\right ) - 2 \, \log \left (2\right ) \log \left (x + 2 \, \log \left (2\right )\right ) - \log \left (2\right )\right )}}{2 \, x \log \left (2\right ) - x \log \left (x\right )} \]
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Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx=-\frac {2 \, {\left (x e^{x} \log \left (2\right ) - \log \left (2\right ) \log \left (x^{2} + 4 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right ) - \log \left (2\right )\right )}}{2 \, x \log \left (2\right ) - x \log \left (x\right )} \]
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Timed out. \[ \int \frac {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx=\int \frac {2\,x\,\ln \left (2\right )+\ln \left (x^2+4\,\ln \left (2\right )\,x+4\,{\ln \left (2\right )}^2\right )\,\left (2\,x\,\ln \left (2\right )+\ln \left (\frac {x}{4}\right )\,\left (2\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2\right )+4\,{\ln \left (2\right )}^2\right )+\ln \left (\frac {x}{4}\right )\,\left ({\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,x^3+4\,{\ln \left (2\right )}^2\,x^2\right )-2\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2\right )+4\,{\ln \left (2\right )}^2-{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,x^2+4\,{\ln \left (2\right )}^2\,x\right )}{{\ln \left (\frac {x}{4}\right )}^2\,\left (x^3+2\,\ln \left (2\right )\,x^2\right )} \,d x \]
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