Integrand size = 169, antiderivative size = 32 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=-x+(-4+\log (3 x)) \log (x+(i \pi +\log (3)) (x \log (2)-\log (x))) \]
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\[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=\int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{x^2 (-1-\log (2) (i \pi +\log (3)))+x (i \pi +\log (3)) \log (x)} \, dx \\ & = \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{x (x (-1-\log (2) (i \pi +\log (3)))+(i \pi +\log (3)) \log (x))} \, dx \\ & = \int \left (\frac {4 i}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {i x}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {(4+x) \log (2) (\pi -i \log (3))}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {4 (-\pi +i \log (3))}{x \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )}+\frac {(-\pi +i \log (3)) \log (x)}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {(\pi -i \log (3)-x (\pi \log (2)-i (1+\log (2) \log (3)))) \log (3 x)}{x \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )}+\frac {\log (x (1+\log (2) (i \pi +\log (3)))-(i \pi +\log (3)) \log (x))}{x}\right ) \, dx \\ & = i \int \frac {x}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+4 i \int \frac {1}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx-(4 (\pi -i \log (3))) \int \frac {1}{x \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )} \, dx+(\log (2) (\pi -i \log (3))) \int \frac {4+x}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+(-\pi +i \log (3)) \int \frac {\log (x)}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+\int \frac {(\pi -i \log (3)-x (\pi \log (2)-i (1+\log (2) \log (3)))) \log (3 x)}{x \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )} \, dx+\int \frac {\log (x (1+\log (2) (i \pi +\log (3)))-(i \pi +\log (3)) \log (x))}{x} \, dx \\ & = i \int \frac {x}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+4 i \int \frac {1}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx-(4 (\pi -i \log (3))) \int \frac {1}{x \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )} \, dx+(\log (2) (\pi -i \log (3))) \int \left (\frac {4}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {x}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}\right ) \, dx+(-\pi +i \log (3)) \int \left (\frac {1}{\pi -i \log (3)}+\frac {x (\pi \log (2)-i (1+\log (2) \log (3)))}{(\pi -i \log (3)) \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )}\right ) \, dx+\int \left (-\frac {i (1+\log (2) (i \pi +\log (3))) \log (3 x)}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {(\pi -i \log (3)) \log (3 x)}{x \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )}\right ) \, dx+\int \frac {\log (x (1+\log (2) (i \pi +\log (3)))-(i \pi +\log (3)) \log (x))}{x} \, dx \\ & = -x+i \int \frac {x}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+4 i \int \frac {1}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+(\pi -i \log (3)) \int \frac {\log (3 x)}{x \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )} \, dx-(4 (\pi -i \log (3))) \int \frac {1}{x \left (i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )} \, dx+(\log (2) (\pi -i \log (3))) \int \frac {x}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+(4 \log (2) (\pi -i \log (3))) \int \frac {1}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx-(i (1+\log (2) (i \pi +\log (3)))) \int \frac {\log (3 x)}{-i x (1+\log (2) (i \pi +\log (3)))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+(-\pi \log (2)+i (1+\log (2) \log (3))) \int \frac {x}{i x (1+\log (2) (i \pi +\log (3)))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)} \, dx+\int \frac {\log (x (1+\log (2) (i \pi +\log (3)))-(i \pi +\log (3)) \log (x))}{x} \, dx \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(32)=64\).
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 4.78 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=-x+i \arctan \left (\frac {\pi (-x \log (2)+\log (x))}{x+x \log (2) \log (3)-\log (3) \log (x)}\right ) (4+\log (x)-\log (3 x))+\log (x) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))-\frac {1}{2} (4+\log (x)-\log (3 x)) \log \left (x^2 \left (1+\pi ^2 \log ^2(2)+\log ^2(2) \log ^2(3)+\log (3) \log (4)\right )-x \left (\pi ^2 \log (4)+\log ^2(3) \log (4)+\log (9)\right ) \log (x)+\left (\pi ^2+\log ^2(3)\right ) \log ^2(x)\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (31 ) = 62\).
Time = 0.81 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.06
method | result | size |
risch | \(\ln \left (x \right ) \ln \left (-\left (\ln \left (3\right )+i \pi \right ) \ln \left (x \right )+x \left (\ln \left (3\right )+i \pi \right ) \ln \left (2\right )+x \right )-x +\ln \left (3\right ) \ln \left (\ln \left (x \right )-\frac {x \left (-i \ln \left (2\right ) \ln \left (3\right )+\pi \ln \left (2\right )-i\right )}{\pi -i \ln \left (3\right )}\right )-4 \ln \left (\ln \left (x \right )-\frac {x \left (-i \ln \left (2\right ) \ln \left (3\right )+\pi \ln \left (2\right )-i\right )}{\pi -i \ln \left (3\right )}\right )\) | \(98\) |
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx={\left (\log \left (3\right ) + \log \left (x\right ) - 4\right )} \log \left (i \, \pi x \log \left (2\right ) + x \log \left (3\right ) \log \left (2\right ) + {\left (-i \, \pi - \log \left (3\right )\right )} \log \left (x\right ) + x\right ) - x \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 10.66 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=- x + \log {\left (x \right )} \log {\left (x \log {\left (2 \right )} \log {\left (3 \right )} + x + i \pi x \log {\left (2 \right )} - \log {\left (3 \right )} \log {\left (x \right )} - i \pi \log {\left (x \right )} \right )} + \left (-4 + \log {\left (3 \right )}\right ) \log {\left (\frac {- x - x \log {\left (2 \right )} \log {\left (3 \right )} - i \pi x \log {\left (2 \right )}}{\log {\left (3 \right )} + i \pi } + \log {\left (x \right )} \right )} \]
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Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx={\left (\log \left (3\right ) + \log \left (x\right ) - 4\right )} \log \left ({\left (i \, \pi \log \left (2\right ) + \log \left (3\right ) \log \left (2\right ) + 1\right )} x + {\left (-i \, \pi - \log \left (3\right )\right )} \log \left (x\right )\right ) - x \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (30) = 60\).
Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.28 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx={\left (\log \left (3\right ) - 4\right )} \log \left (\pi x \log \left (2\right ) - i \, x \log \left (3\right ) \log \left (2\right ) - \pi \log \left (x\right ) + i \, \log \left (3\right ) \log \left (x\right ) - i \, x\right ) + \frac {1}{2} i \, \pi \log \left (x\right ) + \log \left (\pi x \log \left (2\right ) - i \, x \log \left (3\right ) \log \left (2\right ) - \pi \log \left (x\right ) + i \, \log \left (3\right ) \log \left (x\right ) - i \, x\right ) \log \left (x\right ) - x \]
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Timed out. \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=\int \frac {\Pi \,4{}\mathrm {i}-4\,x+4\,\ln \left (3\right )+\ln \left (x-\ln \left (x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+x\,\ln \left (2\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\right )\,\left (x-\ln \left (x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+x\,\ln \left (2\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\right )-\ln \left (3\,x\right )\,\left (\Pi \,1{}\mathrm {i}-x+\ln \left (3\right )-x\,\ln \left (2\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\right )-x^2+x\,\ln \left (x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )-\ln \left (2\right )\,\left (x^2+4\,x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}{x^2+x^2\,\ln \left (2\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )-x\,\ln \left (x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )} \,d x \]
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