Integrand size = 108, antiderivative size = 25 \[ \int \frac {4096-256 x-64 x^2}{4 x^3+x^4+\left (128 x^2+32 x^3\right ) \log (x)+\left (1024 x+256 x^2\right ) \log ^2(x)+\left (-128 x^2-32 x^3+\left (-2048 x-512 x^2\right ) \log (x)\right ) \log \left (\frac {5 x^2}{4+x}\right )+\left (1024 x+256 x^2\right ) \log ^2\left (\frac {5 x^2}{4+x}\right )} \, dx=\frac {4}{\frac {x}{16}+\log (x)-\log \left (\frac {5 x^2}{4+x}\right )} \]
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Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6820, 12, 6818} \[ \int \frac {4096-256 x-64 x^2}{4 x^3+x^4+\left (128 x^2+32 x^3\right ) \log (x)+\left (1024 x+256 x^2\right ) \log ^2(x)+\left (-128 x^2-32 x^3+\left (-2048 x-512 x^2\right ) \log (x)\right ) \log \left (\frac {5 x^2}{4+x}\right )+\left (1024 x+256 x^2\right ) \log ^2\left (\frac {5 x^2}{4+x}\right )} \, dx=\frac {64}{-16 \log \left (\frac {5 x^2}{x+4}\right )+x+16 \log (x)} \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {64 \left (64-4 x-x^2\right )}{x (4+x) \left (x+16 \log (x)-16 \log \left (\frac {5 x^2}{4+x}\right )\right )^2} \, dx \\ & = 64 \int \frac {64-4 x-x^2}{x (4+x) \left (x+16 \log (x)-16 \log \left (\frac {5 x^2}{4+x}\right )\right )^2} \, dx \\ & = \frac {64}{x+16 \log (x)-16 \log \left (\frac {5 x^2}{4+x}\right )} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {4096-256 x-64 x^2}{4 x^3+x^4+\left (128 x^2+32 x^3\right ) \log (x)+\left (1024 x+256 x^2\right ) \log ^2(x)+\left (-128 x^2-32 x^3+\left (-2048 x-512 x^2\right ) \log (x)\right ) \log \left (\frac {5 x^2}{4+x}\right )+\left (1024 x+256 x^2\right ) \log ^2\left (\frac {5 x^2}{4+x}\right )} \, dx=\frac {64}{x+16 \log (x)-16 \log \left (\frac {5 x^2}{4+x}\right )} \]
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Time = 1.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(\frac {64}{-16 \ln \left (\frac {5 x^{2}}{4+x}\right )+x +16 \ln \left (x \right )}\) | \(24\) |
default | \(-\frac {64 i}{8 \pi \,\operatorname {csgn}\left (\frac {i}{4+x}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{4+x}\right )-8 \pi \,\operatorname {csgn}\left (\frac {i}{4+x}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{4+x}\right )^{2}+8 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-16 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+8 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-8 \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{4+x}\right )^{2}+8 \pi \operatorname {csgn}\left (\frac {i x^{2}}{4+x}\right )^{3}+16 i \ln \left (5\right )+16 i \ln \left (x \right )-i x -16 i \ln \left (4+x \right )}\) | \(172\) |
risch | \(-\frac {64 i}{8 \pi \,\operatorname {csgn}\left (\frac {i}{4+x}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{4+x}\right )-8 \pi \,\operatorname {csgn}\left (\frac {i}{4+x}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{4+x}\right )^{2}+8 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-16 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+8 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-8 \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{4+x}\right )^{2}+8 \pi \operatorname {csgn}\left (\frac {i x^{2}}{4+x}\right )^{3}+16 i \ln \left (5\right )+16 i \ln \left (x \right )-i x -16 i \ln \left (4+x \right )}\) | \(172\) |
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {4096-256 x-64 x^2}{4 x^3+x^4+\left (128 x^2+32 x^3\right ) \log (x)+\left (1024 x+256 x^2\right ) \log ^2(x)+\left (-128 x^2-32 x^3+\left (-2048 x-512 x^2\right ) \log (x)\right ) \log \left (\frac {5 x^2}{4+x}\right )+\left (1024 x+256 x^2\right ) \log ^2\left (\frac {5 x^2}{4+x}\right )} \, dx=\frac {64}{x + 16 \, \log \left (x\right ) - 16 \, \log \left (\frac {5 \, x^{2}}{x + 4}\right )} \]
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Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {4096-256 x-64 x^2}{4 x^3+x^4+\left (128 x^2+32 x^3\right ) \log (x)+\left (1024 x+256 x^2\right ) \log ^2(x)+\left (-128 x^2-32 x^3+\left (-2048 x-512 x^2\right ) \log (x)\right ) \log \left (\frac {5 x^2}{4+x}\right )+\left (1024 x+256 x^2\right ) \log ^2\left (\frac {5 x^2}{4+x}\right )} \, dx=- \frac {4}{- \frac {x}{16} - \log {\left (x \right )} + \log {\left (\frac {5 x^{2}}{x + 4} \right )}} \]
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Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {4096-256 x-64 x^2}{4 x^3+x^4+\left (128 x^2+32 x^3\right ) \log (x)+\left (1024 x+256 x^2\right ) \log ^2(x)+\left (-128 x^2-32 x^3+\left (-2048 x-512 x^2\right ) \log (x)\right ) \log \left (\frac {5 x^2}{4+x}\right )+\left (1024 x+256 x^2\right ) \log ^2\left (\frac {5 x^2}{4+x}\right )} \, dx=\frac {64}{x - 16 \, \log \left (5\right ) + 16 \, \log \left (x + 4\right ) - 16 \, \log \left (x\right )} \]
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Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {4096-256 x-64 x^2}{4 x^3+x^4+\left (128 x^2+32 x^3\right ) \log (x)+\left (1024 x+256 x^2\right ) \log ^2(x)+\left (-128 x^2-32 x^3+\left (-2048 x-512 x^2\right ) \log (x)\right ) \log \left (\frac {5 x^2}{4+x}\right )+\left (1024 x+256 x^2\right ) \log ^2\left (\frac {5 x^2}{4+x}\right )} \, dx=\frac {64}{x - 16 \, \log \left (5\right ) + 16 \, \log \left (x + 4\right ) - 16 \, \log \left (x\right )} \]
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Time = 11.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {4096-256 x-64 x^2}{4 x^3+x^4+\left (128 x^2+32 x^3\right ) \log (x)+\left (1024 x+256 x^2\right ) \log ^2(x)+\left (-128 x^2-32 x^3+\left (-2048 x-512 x^2\right ) \log (x)\right ) \log \left (\frac {5 x^2}{4+x}\right )+\left (1024 x+256 x^2\right ) \log ^2\left (\frac {5 x^2}{4+x}\right )} \, dx=\frac {64}{x-16\,\ln \left (\frac {5\,x^2}{x+4}\right )+16\,\ln \left (x\right )} \]
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