Integrand size = 69, antiderivative size = 25 \[ \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left (32 x+2 e x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx=\log (x)+\left (3-\frac {3}{4 \log \left (4+\frac {1}{4} (e+4 x)\right )}\right )^2 \]
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {6, 1607, 6820, 2437, 12, 2339, 30} \[ \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left (32 x+2 e x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx=\frac {9}{16 \log ^2\left (x+\frac {16+e}{4}\right )}+\log (x)-\frac {9}{2 \log \left (x+\frac {16+e}{4}\right )} \]
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Rule 6
Rule 12
Rule 30
Rule 1607
Rule 2339
Rule 2437
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left ((32+2 e) x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx \\ & = \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{x (32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx \\ & = \int \left (\frac {1}{x}-\frac {9}{2 (16+e+4 x) \log ^3\left (4+\frac {e}{4}+x\right )}+\frac {18}{(16+e+4 x) \log ^2\left (4+\frac {e}{4}+x\right )}\right ) \, dx \\ & = \log (x)-\frac {9}{2} \int \frac {1}{(16+e+4 x) \log ^3\left (4+\frac {e}{4}+x\right )} \, dx+18 \int \frac {1}{(16+e+4 x) \log ^2\left (4+\frac {e}{4}+x\right )} \, dx \\ & = \log (x)-\frac {9}{2} \text {Subst}\left (\int \frac {4+\frac {e}{4}}{(16+e) x \log ^3(x)} \, dx,x,4+\frac {e}{4}+x\right )+18 \text {Subst}\left (\int \frac {4+\frac {e}{4}}{(16+e) x \log ^2(x)} \, dx,x,4+\frac {e}{4}+x\right ) \\ & = \log (x)-\frac {9}{8} \text {Subst}\left (\int \frac {1}{x \log ^3(x)} \, dx,x,4+\frac {e}{4}+x\right )+\frac {9}{2} \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,4+\frac {e}{4}+x\right ) \\ & = \log (x)-\frac {9}{8} \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log \left (4+\frac {e}{4}+x\right )\right )+\frac {9}{2} \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (4+\frac {e}{4}+x\right )\right ) \\ & = \log (x)+\frac {9}{16 \log ^2\left (\frac {16+e}{4}+x\right )}-\frac {9}{2 \log \left (\frac {16+e}{4}+x\right )} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left (32 x+2 e x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx=\log (x)+\frac {9}{16 \log ^2\left (4+\frac {e}{4}+x\right )}-\frac {9}{2 \log \left (4+\frac {e}{4}+x\right )} \]
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Time = 0.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08
method | result | size |
norman | \(\frac {\frac {9}{16}-\frac {9 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )}{2}}{\ln \left (\frac {{\mathrm e}}{4}+x +4\right )^{2}}+\ln \left (x \right )\) | \(27\) |
risch | \(\ln \left (x \right )-\frac {9 \left (-1+8 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )\right )}{16 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )^{2}}\) | \(28\) |
parts | \(\ln \left (x \right )+\frac {9}{16 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )^{2}}-\frac {9}{2 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )}\) | \(28\) |
derivativedivides | \(\ln \left (-4 x \right )+\frac {9}{16 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )^{2}}-\frac {9}{2 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )}\) | \(30\) |
default | \(\ln \left (-4 x \right )+\frac {9}{16 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )^{2}}-\frac {9}{2 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )}\) | \(30\) |
parallelrisch | \(\frac {32 \ln \left (x \right ) \ln \left (\frac {{\mathrm e}}{4}+x +4\right )^{2}+18-144 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )}{32 \ln \left (\frac {{\mathrm e}}{4}+x +4\right )^{2}}\) | \(39\) |
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left (32 x+2 e x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx=\frac {16 \, \log \left (x + \frac {1}{4} \, e + 4\right )^{2} \log \left (x\right ) - 72 \, \log \left (x + \frac {1}{4} \, e + 4\right ) + 9}{16 \, \log \left (x + \frac {1}{4} \, e + 4\right )^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left (32 x+2 e x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx=\frac {9 - 72 \log {\left (x + \frac {e}{4} + 4 \right )}}{16 \log {\left (x + \frac {e}{4} + 4 \right )}^{2}} + \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (19) = 38\).
Time = 0.38 (sec) , antiderivative size = 287, normalized size of antiderivative = 11.48 \[ \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left (32 x+2 e x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx=-{\left (\frac {\log \left (4 \, x + e + 16\right )}{e + 16} - \frac {\log \left (x\right )}{e + 16}\right )} e - \frac {\log \left (x + \frac {1}{4} \, e + 4\right )^{3}}{2 \, {\left (4 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) \log \left (4 \, x + e + 16\right ) + \log \left (4 \, x + e + 16\right )^{2}\right )}} + 3 \, {\left (2 \, \log \left (2\right ) - \log \left (4 \, x + e + 16\right )\right )} \log \left (-2 \, \log \left (2\right ) + \log \left (4 \, x + e + 16\right )\right ) + 3 \, \log \left (x + \frac {1}{4} \, e + 4\right ) \log \left (-2 \, \log \left (2\right ) + \log \left (4 \, x + e + 16\right )\right ) + \frac {3 \, \log \left (x + \frac {1}{4} \, e + 4\right )^{2}}{2 \, {\left (2 \, \log \left (2\right ) - \log \left (4 \, x + e + 16\right )\right )}} - \frac {16 \, \log \left (4 \, x + e + 16\right )}{e + 16} - \frac {9 \, \log \left (x + \frac {1}{4} \, e + 4\right )}{4 \, {\left (4 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) \log \left (4 \, x + e + 16\right ) + \log \left (4 \, x + e + 16\right )^{2}\right )}} + \frac {16 \, \log \left (x\right )}{e + 16} + \frac {9}{16 \, {\left (4 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) \log \left (4 \, x + e + 16\right ) + \log \left (4 \, x + e + 16\right )^{2}\right )}} + \frac {9}{4 \, {\left (2 \, \log \left (2\right ) - \log \left (4 \, x + e + 16\right )\right )}} + 3 \, \log \left (4 \, x + e + 16\right ) \]
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Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left (32 x+2 e x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx=\frac {16 \, \log \left (x + \frac {1}{4} \, e + 4\right )^{2} \log \left (x\right ) - 72 \, \log \left (x + \frac {1}{4} \, e + 4\right ) + 9}{16 \, \log \left (x + \frac {1}{4} \, e + 4\right )^{2}} \]
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Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-9 x+36 x \log \left (\frac {1}{4} (16+e+4 x)\right )+(32+2 e+8 x) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )}{\left (32 x+2 e x+8 x^2\right ) \log ^3\left (\frac {1}{4} (16+e+4 x)\right )} \, dx=\frac {16\,\ln \left (x\right )\,{\ln \left (x+\frac {\mathrm {e}}{4}+4\right )}^2-72\,\ln \left (x+\frac {\mathrm {e}}{4}+4\right )+9}{16\,{\ln \left (x+\frac {\mathrm {e}}{4}+4\right )}^2} \]
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