Integrand size = 33, antiderivative size = 22 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (-\frac {25}{8}+\frac {2}{4+\frac {x^2}{2}+4 \log (5)}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {12, 2014, 1121, 630, 31} \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (25 x^2+8 (21+25 \log (5))\right )-\log \left (x^2+8 (1+\log (5))\right ) \]
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Rule 12
Rule 31
Rule 630
Rule 1121
Rule 2014
Rubi steps \begin{align*} \text {integral}& = 64 \int \frac {x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx \\ & = 64 \int \frac {x}{25 x^4+64 (1+\log (5)) (21+25 \log (5))+16 x^2 (23+25 \log (5))} \, dx \\ & = 32 \text {Subst}\left (\int \frac {1}{25 x^2+64 (1+\log (5)) (21+25 \log (5))+16 x (23+25 \log (5))} \, dx,x,x^2\right ) \\ & = -\left (25 \text {Subst}\left (\int \frac {1}{25 x+200 (1+\log (5))} \, dx,x,x^2\right )\right )+25 \text {Subst}\left (\int \frac {1}{25 x+8 (21+25 \log (5))} \, dx,x,x^2\right ) \\ & = -\log \left (x^2+8 (1+\log (5))\right )+\log \left (25 x^2+8 (21+25 \log (5))\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=64 \left (-\frac {1}{64} \log \left (8+x^2+8 \log (5)\right )+\frac {1}{64} \log \left (168+25 x^2+200 \log (5)\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09
method | result | size |
parallelrisch | \(-\ln \left (x^{2}+8 \ln \left (5\right )+8\right )+\ln \left (x^{2}+8 \ln \left (5\right )+\frac {168}{25}\right )\) | \(24\) |
default | \(\ln \left (25 x^{2}+200 \ln \left (5\right )+168\right )-\ln \left (x^{2}+8 \ln \left (5\right )+8\right )\) | \(26\) |
norman | \(\ln \left (25 x^{2}+200 \ln \left (5\right )+168\right )-\ln \left (x^{2}+8 \ln \left (5\right )+8\right )\) | \(26\) |
risch | \(-\ln \left (-x^{2}-8 \ln \left (5\right )-8\right )+\ln \left (25 x^{2}+200 \ln \left (5\right )+168\right )\) | \(28\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (25 \, x^{2} + 200 \, \log \left (5\right ) + 168\right ) - \log \left (x^{2} + 8 \, \log \left (5\right ) + 8\right ) \]
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Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log {\left (x^{2} + \frac {168}{25} + 8 \log {\left (5 \right )} \right )} - \log {\left (x^{2} + 8 + 8 \log {\left (5 \right )} \right )} \]
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Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (25 \, x^{2} + 200 \, \log \left (5\right ) + 168\right ) - \log \left (x^{2} + 8 \, \log \left (5\right ) + 8\right ) \]
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=\log \left (25 \, x^{2} + 200 \, \log \left (5\right ) + 168\right ) - \log \left (x^{2} + 8 \, \log \left (5\right ) + 8\right ) \]
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Time = 13.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {64 x}{1344+368 x^2+25 x^4+\left (2944+400 x^2\right ) \log (5)+1600 \log ^2(5)} \, dx=2\,\mathrm {atanh}\left (\frac {320000\,x^2}{58880000\,\ln \left (5\right )+2\,x^2\,\left (2000000\,\ln \left (5\right )+1840000\right )+32000000\,{\ln \left (5\right )}^2+26880000}\right ) \]
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