Integrand size = 77, antiderivative size = 25 \[ \int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx=\frac {\log (5)}{x^2 \left (\frac {3}{4} e^{-x} x^2+\log (6 x)\right )} \]
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\[ \int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx=\int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 e^x \log (5) \left (-4 e^x+3 (-4+x) x^2-8 e^x \log (6 x)\right )}{x^3 \left (3 x^2+4 e^x \log (6 x)\right )^2} \, dx \\ & = (4 \log (5)) \int \frac {e^x \left (-4 e^x+3 (-4+x) x^2-8 e^x \log (6 x)\right )}{x^3 \left (3 x^2+4 e^x \log (6 x)\right )^2} \, dx \\ & = (4 \log (5)) \int \left (-\frac {e^x (1+2 \log (6 x))}{x^3 \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )}+\frac {3 e^x (1-2 \log (6 x)+x \log (6 x))}{x \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )^2}\right ) \, dx \\ & = -\left ((4 \log (5)) \int \frac {e^x (1+2 \log (6 x))}{x^3 \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )} \, dx\right )+(12 \log (5)) \int \frac {e^x (1-2 \log (6 x)+x \log (6 x))}{x \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )^2} \, dx \\ & = -\left ((4 \log (5)) \int \left (\frac {2 e^x}{x^3 \left (3 x^2+4 e^x \log (6 x)\right )}+\frac {e^x}{x^3 \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )}\right ) \, dx\right )+(12 \log (5)) \int \frac {e^x (1+(-2+x) \log (6 x))}{x \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )^2} \, dx \\ & = -\left ((4 \log (5)) \int \frac {e^x}{x^3 \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )} \, dx\right )-(8 \log (5)) \int \frac {e^x}{x^3 \left (3 x^2+4 e^x \log (6 x)\right )} \, dx+(12 \log (5)) \int \left (\frac {e^x}{\left (3 x^2+4 e^x \log (6 x)\right )^2}-\frac {2 e^x}{x \left (3 x^2+4 e^x \log (6 x)\right )^2}+\frac {e^x}{x \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )^2}\right ) \, dx \\ & = -\left ((4 \log (5)) \int \frac {e^x}{x^3 \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )} \, dx\right )-(8 \log (5)) \int \frac {e^x}{x^3 \left (3 x^2+4 e^x \log (6 x)\right )} \, dx+(12 \log (5)) \int \frac {e^x}{\left (3 x^2+4 e^x \log (6 x)\right )^2} \, dx+(12 \log (5)) \int \frac {e^x}{x \log (6 x) \left (3 x^2+4 e^x \log (6 x)\right )^2} \, dx-(24 \log (5)) \int \frac {e^x}{x \left (3 x^2+4 e^x \log (6 x)\right )^2} \, dx \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx=\frac {4 e^x \log (5)}{x^2 \left (3 x^2+4 e^x \log (6 x)\right )} \]
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Time = 1.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {4 \ln \left (5\right ) {\mathrm e}^{x}}{x^{2} \left (4 \,{\mathrm e}^{x} \ln \left (6 x \right )+3 x^{2}\right )}\) | \(26\) |
parallelrisch | \(\frac {4 \ln \left (5\right ) {\mathrm e}^{x}}{x^{2} \left (4 \,{\mathrm e}^{x} \ln \left (6 x \right )+3 x^{2}\right )}\) | \(26\) |
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx=\frac {4 \, e^{x} \log \left (5\right )}{3 \, x^{4} + 4 \, x^{2} e^{x} \log \left (6 \, x\right )} \]
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Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx=- \frac {3 \log {\left (5 \right )}}{3 x^{2} \log {\left (6 x \right )} + 4 e^{x} \log {\left (6 x \right )}^{2}} + \frac {\log {\left (5 \right )}}{x^{2} \log {\left (6 x \right )}} \]
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Time = 0.37 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx=\frac {4 \, e^{x} \log \left (5\right )}{3 \, x^{4} + 4 \, {\left (x^{2} {\left (\log \left (3\right ) + \log \left (2\right )\right )} + x^{2} \log \left (x\right )\right )} e^{x}} \]
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Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx=\frac {4 \, e^{x} \log \left (5\right )}{3 \, x^{4} + 4 \, x^{2} e^{x} \log \left (6 \, x\right )} \]
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Time = 13.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {-16 e^{2 x} \log (5)+e^x \left (-48 x^2+12 x^3\right ) \log (5)-32 e^{2 x} \log (5) \log (6 x)}{9 x^7+24 e^x x^5 \log (6 x)+16 e^{2 x} x^3 \log ^2(6 x)} \, dx=\frac {4\,{\mathrm {e}}^{2\,x}\,\ln \left (5\right )\,\left (4\,{\mathrm {e}}^x+6\,x^2-3\,x^3\right )}{x\,\left (4\,\ln \left (6\,x\right )\,{\mathrm {e}}^x+3\,x^2\right )\,\left (4\,x\,{\mathrm {e}}^{2\,x}+6\,x^3\,{\mathrm {e}}^x-3\,x^4\,{\mathrm {e}}^x\right )} \]
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