Integrand size = 70, antiderivative size = 23 \[ \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx=x+\frac {3 (-16+x) x^2 \left (4+\frac {x}{\log (x)}\right )}{4+x} \]
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\[ \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx=\int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{(4+x)^2 \log ^2(x)} \, dx \\ & = \int \left (\frac {16-1528 x-47 x^2+24 x^3}{(4+x)^2}-\frac {3 (-16+x) x^2}{(4+x) \log ^2(x)}+\frac {3 x^2 \left (-192-16 x+3 x^2\right )}{(4+x)^2 \log (x)}\right ) \, dx \\ & = -\left (3 \int \frac {(-16+x) x^2}{(4+x) \log ^2(x)} \, dx\right )+3 \int \frac {x^2 \left (-192-16 x+3 x^2\right )}{(4+x)^2 \log (x)} \, dx+\int \frac {16-1528 x-47 x^2+24 x^3}{(4+x)^2} \, dx \\ & = -\left (3 \int \frac {(-16+x) x^2}{(4+x) \log ^2(x)} \, dx\right )+3 \int \frac {x^2 \left (-192-16 x+3 x^2\right )}{(4+x)^2 \log (x)} \, dx+\int \left (-239+24 x+\frac {3840}{(4+x)^2}\right ) \, dx \\ & = -239 x+12 x^2-\frac {3840}{4+x}-3 \int \frac {(-16+x) x^2}{(4+x) \log ^2(x)} \, dx+3 \int \frac {x^2 \left (-192-16 x+3 x^2\right )}{(4+x)^2 \log (x)} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx=-239 x+12 x^2-\frac {3840}{4+x}+\frac {3 (-16+x) x^3}{(4+x) \log (x)} \]
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Time = 0.92 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74
method | result | size |
norman | \(\frac {-16 \ln \left (x \right )-48 x^{3}+3 x^{4}-191 x^{2} \ln \left (x \right )+12 x^{3} \ln \left (x \right )}{\left (4+x \right ) \ln \left (x \right )}\) | \(40\) |
risch | \(\frac {12 x^{3}-191 x^{2}-956 x -3840}{4+x}+\frac {3 x^{3} \left (x -16\right )}{\left (4+x \right ) \ln \left (x \right )}\) | \(40\) |
parallelrisch | \(\frac {-16 \ln \left (x \right )-48 x^{3}+3 x^{4}-191 x^{2} \ln \left (x \right )+12 x^{3} \ln \left (x \right )}{\left (4+x \right ) \ln \left (x \right )}\) | \(40\) |
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Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx=\frac {3 \, x^{4} - 48 \, x^{3} + {\left (12 \, x^{3} - 191 \, x^{2} - 956 \, x - 3840\right )} \log \left (x\right )}{{\left (x + 4\right )} \log \left (x\right )} \]
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Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx=12 x^{2} - 239 x + \frac {3 x^{4} - 48 x^{3}}{\left (x + 4\right ) \log {\left (x \right )}} - \frac {3840}{x + 4} \]
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Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx=\frac {3 \, x^{4} - 48 \, x^{3} + {\left (12 \, x^{3} - 191 \, x^{2} - 956 \, x - 3840\right )} \log \left (x\right )}{{\left (x + 4\right )} \log \left (x\right )} \]
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Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx=12 \, x^{2} - 239 \, x + \frac {3 \, {\left (x^{4} - 16 \, x^{3}\right )}}{x \log \left (x\right ) + 4 \, \log \left (x\right )} - \frac {3840}{x + 4} \]
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Time = 10.69 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {192 x^2+36 x^3-3 x^4+\left (-576 x^2-48 x^3+9 x^4\right ) \log (x)+\left (16-1528 x-47 x^2+24 x^3\right ) \log ^2(x)}{\left (16+8 x+x^2\right ) \log ^2(x)} \, dx=\frac {x\,\left (12\,x^2-191\,x+4\right )}{x+4}-\frac {x\,\left (48\,x^2-3\,x^3\right )}{\ln \left (x\right )\,\left (x+4\right )} \]
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