Integrand size = 218, antiderivative size = 26 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=\frac {x}{e^4 (4-x) (x+\log (4)) \left (-4+\log \left (x^2\right )\right )} \]
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\[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=\int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 x^2+2 x (-4+\log (4))-24 \log (4)+\left (x^2+\log (256)\right ) \log \left (x^2\right )}{e^4 (4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2} \, dx \\ & = \frac {\int \frac {-2 x^2+2 x (-4+\log (4))-24 \log (4)+\left (x^2+\log (256)\right ) \log \left (x^2\right )}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2} \, dx}{e^4} \\ & = \frac {\int \left (\frac {2 \left (x^2-x (4-\log (4))-\log (256)\right )}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2}+\frac {x^2+\log (256)}{(-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )}\right ) \, dx}{e^4} \\ & = \frac {\int \frac {x^2+\log (256)}{(-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \, dx}{e^4}+\frac {2 \int \frac {x^2-x (4-\log (4))-\log (256)}{(4-x)^2 (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2} \, dx}{e^4} \\ & = \frac {\int \frac {x^2+\log (256)}{(-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \, dx}{e^4}+\frac {2 \int \frac {-x-\log (4)}{(4-x) (x+\log (4))^2 \left (4-\log \left (x^2\right )\right )^2} \, dx}{e^4} \\ & = \frac {\int \frac {x^2+\log (256)}{(-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \, dx}{e^4}-\frac {2 \int \frac {1}{(4-x) (x+\log (4)) \left (4-\log \left (x^2\right )\right )^2} \, dx}{e^4} \\ \end{align*}
Time = 5.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=-\frac {x \left (x^2+x (-4+\log (4))-\log (256)\right )}{e^4 (-4+x)^2 (x+\log (4))^2 \left (-4+\log \left (x^2\right )\right )} \]
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Time = 1.52 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12
method | result | size |
norman | \(-\frac {x \,{\mathrm e}^{-4}}{\left (\ln \left (x^{2}\right )-4\right ) \left (x -4\right ) \left (x +2 \ln \left (2\right )\right )}\) | \(29\) |
risch | \(-\frac {x \,{\mathrm e}^{-4}}{\left (2 x \ln \left (2\right )+x^{2}-8 \ln \left (2\right )-4 x \right ) \left (\ln \left (x^{2}\right )-4\right )}\) | \(32\) |
parallelrisch | \(-\frac {x \,{\mathrm e}^{-4}}{\left (2 x \ln \left (2\right )+x^{2}-8 \ln \left (2\right )-4 x \right ) \left (\ln \left (x^{2}\right )-4\right )}\) | \(34\) |
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=\frac {x}{8 \, {\left (x - 4\right )} e^{4} \log \left (2\right ) + 4 \, {\left (x^{2} - 4 \, x\right )} e^{4} - {\left (2 \, {\left (x - 4\right )} e^{4} \log \left (2\right ) + {\left (x^{2} - 4 \, x\right )} e^{4}\right )} \log \left (x^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=- \frac {x}{- 4 x^{2} e^{4} - 8 x e^{4} \log {\left (2 \right )} + 16 x e^{4} + \left (x^{2} e^{4} - 4 x e^{4} + 2 x e^{4} \log {\left (2 \right )} - 8 e^{4} \log {\left (2 \right )}\right ) \log {\left (x^{2} \right )} + 32 e^{4} \log {\left (2 \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=\frac {x}{2 \, {\left (2 \, x^{2} e^{4} + 4 \, x {\left (\log \left (2\right ) - 2\right )} e^{4} - 16 \, e^{4} \log \left (2\right ) - {\left (x^{2} e^{4} + 2 \, x {\left (\log \left (2\right ) - 2\right )} e^{4} - 8 \, e^{4} \log \left (2\right )\right )} \log \left (x\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=-\frac {x}{x^{2} e^{4} \log \left (x^{2}\right ) + 2 \, x e^{4} \log \left (2\right ) \log \left (x^{2}\right ) - 4 \, x^{2} e^{4} - 8 \, x e^{4} \log \left (2\right ) - 4 \, x e^{4} \log \left (x^{2}\right ) - 8 \, e^{4} \log \left (2\right ) \log \left (x^{2}\right ) + 16 \, x e^{4} + 32 \, e^{4} \log \left (2\right )} \]
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Time = 14.81 (sec) , antiderivative size = 562, normalized size of antiderivative = 21.62 \[ \int \frac {-8 x-2 x^2+(-24+2 x) \log (4)+\left (x^2+4 \log (4)\right ) \log \left (x^2\right )}{e^4 \left (256 x^2-128 x^3+16 x^4\right )+e^4 \left (512 x-256 x^2+32 x^3\right ) \log (4)+e^4 \left (256-128 x+16 x^2\right ) \log ^2(4)+\left (e^4 \left (-128 x^2+64 x^3-8 x^4\right )+e^4 \left (-256 x+128 x^2-16 x^3\right ) \log (4)+e^4 \left (-128+64 x-8 x^2\right ) \log ^2(4)\right ) \log \left (x^2\right )+\left (e^4 \left (16 x^2-8 x^3+x^4\right )+e^4 \left (32 x-16 x^2+2 x^3\right ) \log (4)+e^4 \left (16-8 x+x^2\right ) \log ^2(4)\right ) \log ^2\left (x^2\right )} \, dx=\frac {\frac {x\,\left (4\,x+24\,\ln \left (2\right )-2\,x\,\ln \left (2\right )+x^2\right )}{64\,{\mathrm {e}}^4\,{\ln \left (2\right )}^2+16\,x^2\,{\mathrm {e}}^4-8\,x^3\,{\mathrm {e}}^4+x^4\,{\mathrm {e}}^4-32\,x\,{\mathrm {e}}^4\,{\ln \left (2\right )}^2-32\,x^2\,{\mathrm {e}}^4\,\ln \left (2\right )+4\,x^3\,{\mathrm {e}}^4\,\ln \left (2\right )+4\,x^2\,{\mathrm {e}}^4\,{\ln \left (2\right )}^2+64\,x\,{\mathrm {e}}^4\,\ln \left (2\right )}-\frac {x\,\ln \left (x^2\right )\,\left (x^2+\ln \left (256\right )\right )}{2\,\left (64\,{\mathrm {e}}^4\,{\ln \left (2\right )}^2+16\,x^2\,{\mathrm {e}}^4-8\,x^3\,{\mathrm {e}}^4+x^4\,{\mathrm {e}}^4-32\,x\,{\mathrm {e}}^4\,{\ln \left (2\right )}^2-32\,x^2\,{\mathrm {e}}^4\,\ln \left (2\right )+4\,x^3\,{\mathrm {e}}^4\,\ln \left (2\right )+4\,x^2\,{\mathrm {e}}^4\,{\ln \left (2\right )}^2+64\,x\,{\mathrm {e}}^4\,\ln \left (2\right )\right )}}{\ln \left (x^2\right )-4}+\frac {\frac {{\mathrm {e}}^{-4}\,\left (32\,\ln \left (2\right )-6\,\ln \left (2\right )\,\ln \left (4\right )+3\,{\ln \left (2\right )}^2\,\ln \left (4\right )+36\,{\ln \left (2\right )}^2+2\,{\ln \left (2\right )}^3+{\ln \left (2\right )}^4+16\right )\,x^3}{2\,\left (32\,\ln \left (2\right )+24\,{\ln \left (2\right )}^2+8\,{\ln \left (2\right )}^3+{\ln \left (2\right )}^4+16\right )}+\frac {{\mathrm {e}}^{-4}\,\left (32\,\ln \left (2\right )-16\,\ln \left (4\right )+40\,\ln \left (2\right )\,\ln \left (4\right )-96\,{\ln \left (2\right )}^2\,\ln \left (4\right )+10\,{\ln \left (2\right )}^3\,\ln \left (4\right )-{\ln \left (2\right )}^4\,\ln \left (4\right )-80\,{\ln \left (2\right )}^2+192\,{\ln \left (2\right )}^3-20\,{\ln \left (2\right )}^4+2\,{\ln \left (2\right )}^5\right )\,x^2}{4\,\left (32\,\ln \left (2\right )+24\,{\ln \left (2\right )}^2+8\,{\ln \left (2\right )}^3+{\ln \left (2\right )}^4+16\right )}+\frac {2\,{\mathrm {e}}^{-4}\,\left (32\,\ln \left (2\right )-8\,\ln \left (2\right )\,\ln \left (4\right )+32\,{\ln \left (2\right )}^2\,\ln \left (4\right )-16\,{\ln \left (2\right )}^3\,\ln \left (4\right )+{\ln \left (2\right )}^4\,\ln \left (4\right )+80\,{\ln \left (2\right )}^2-16\,{\ln \left (2\right )}^3+48\,{\ln \left (2\right )}^4\right )\,x}{32\,\ln \left (2\right )+24\,{\ln \left (2\right )}^2+8\,{\ln \left (2\right )}^3+{\ln \left (2\right )}^4+16}-\frac {4\,{\mathrm {e}}^{-4}\,\left (4\,{\ln \left (2\right )}^2\,\ln \left (4\right )-20\,{\ln \left (2\right )}^3\,\ln \left (4\right )+{\ln \left (2\right )}^4\,\ln \left (4\right )-8\,{\ln \left (2\right )}^3+40\,{\ln \left (2\right )}^4-2\,{\ln \left (2\right )}^5\right )}{32\,\ln \left (2\right )+24\,{\ln \left (2\right )}^2+8\,{\ln \left (2\right )}^3+{\ln \left (2\right )}^4+16}}{x^4+\left (4\,\ln \left (2\right )-8\right )\,x^3+\left (4\,{\ln \left (2\right )}^2-32\,\ln \left (2\right )+16\right )\,x^2+\left (64\,\ln \left (2\right )-32\,{\ln \left (2\right )}^2\right )\,x+64\,{\ln \left (2\right )}^2} \]
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