Integrand size = 117, antiderivative size = 23 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\left (-1+\log (x)+\log \left (4-\log (x)+\left (-e^3+\log (x)\right )^2\right )\right )^2 \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6, 12, 6820, 6818} \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\left (-\log \left (\log ^2(x)-2 e^3 \log (x)-\log (x)+e^6+4\right )-\log (x)+1\right )^2 \]
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Rule 6
Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{\left (4+e^6\right ) x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx \\ & = \text {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+2 x+e^6 x-2 e^3 x^2+x^3+3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+e^6 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6-x-2 e^3 x+x^2} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+2 x+e^6 x-2 e^3 x^2+x^3+3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+e^6 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+e^6 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+e^6 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+\left (3-2 e^3\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+\left (3-2 e^3+e^6\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+\left (3-2 e^3+e^6\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+\left (1-2 e^3\right ) x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+\left (3-2 e^3+e^6\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+\left (1-2 e^3\right ) x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {\left (3-2 e^3+e^6+\left (1-2 e^3\right ) x+x^2\right ) \left (-1+x+\log \left (4+e^6-x-2 e^3 x+x^2\right )\right )}{4+e^6-\left (1+2 e^3\right ) x+x^2} \, dx,x,\log (x)\right ) \\ & = \left (1-\log (x)-\log \left (4+e^6-\log (x)-2 e^3 \log (x)+\log ^2(x)\right )\right )^2 \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\left (-1+\log (x)+\log \left (4+e^6-\left (1+2 e^3\right ) \log (x)+\log ^2(x)\right )\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(22)=44\).
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.13
method | result | size |
risch | \(\ln \left (\ln \left (x \right )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \left (x \right )+{\mathrm e}^{6}+4\right )^{2}+2 \ln \left (x \right ) \ln \left (\ln \left (x \right )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \left (x \right )+{\mathrm e}^{6}+4\right )+\ln \left (x \right )^{2}-2 \ln \left (x \right )-2 \ln \left (\ln \left (x \right )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \left (x \right )+{\mathrm e}^{6}+4\right )\) | \(72\) |
default | \(-2 \ln \left (x \right )+\ln \left (x \right )^{2}+\left (2 \,{\mathrm e}^{3}-1\right ) \ln \left (\ln \left (x \right )^{2}-2 \ln \left (x \right ) {\mathrm e}^{3}+{\mathrm e}^{6}-\ln \left (x \right )+4\right )-\frac {4 \left (-2 \,{\mathrm e}^{6}+2 \,{\mathrm e}^{3}-7-\frac {\left (2 \,{\mathrm e}^{3}-1\right ) \left (-2 \,{\mathrm e}^{3}-1\right )}{2}\right ) \operatorname {arctanh}\left (\frac {2 \ln \left (x \right )-2 \,{\mathrm e}^{3}-1}{\sqrt {4 \,{\mathrm e}^{3}-15}}\right )}{\sqrt {4 \,{\mathrm e}^{3}-15}}+2 \ln \left (x \right ) \ln \left (\ln \left (x \right )^{2}-2 \ln \left (x \right ) {\mathrm e}^{3}+{\mathrm e}^{6}-\ln \left (x \right )+4\right )-\left (2 \,{\mathrm e}^{3}+1\right ) \ln \left (\ln \left (x \right )^{2}-2 \ln \left (x \right ) {\mathrm e}^{3}+{\mathrm e}^{6}-\ln \left (x \right )+4\right )+\frac {4 \left (-8-2 \,{\mathrm e}^{6}-\frac {\left (2 \,{\mathrm e}^{3}+1\right ) \left (-2 \,{\mathrm e}^{3}-1\right )}{2}\right ) \operatorname {arctanh}\left (\frac {2 \ln \left (x \right )-2 \,{\mathrm e}^{3}-1}{\sqrt {4 \,{\mathrm e}^{3}-15}}\right )}{\sqrt {4 \,{\mathrm e}^{3}-15}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \textit {\_Z} +4+{\mathrm e}^{6}\right )}{\sum }\left (\ln \left (\ln \left (x \right )-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\ln \left (x \right )^{2}-2 \ln \left (x \right ) {\mathrm e}^{3}+{\mathrm e}^{6}-\ln \left (x \right )+4\right )-\frac {\ln \left (\ln \left (x \right )-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{2 \left (-2 \,{\mathrm e}^{3}+2 \underline {\hspace {1.25 ex}}\alpha -1\right )}-\frac {\left (-2 \,{\mathrm e}^{3}+2 \underline {\hspace {1.25 ex}}\alpha -1\right ) \ln \left (\ln \left (x \right )-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {-2 \,{\mathrm e}^{3}+\underline {\hspace {1.25 ex}}\alpha +\ln \left (x \right )-1}{-2 \,{\mathrm e}^{3}+2 \underline {\hspace {1.25 ex}}\alpha -1}\right )}{-4 \,{\mathrm e}^{3}+15}-\frac {\left (-2 \,{\mathrm e}^{3}+2 \underline {\hspace {1.25 ex}}\alpha -1\right ) \operatorname {dilog}\left (\frac {-2 \,{\mathrm e}^{3}+\underline {\hspace {1.25 ex}}\alpha +\ln \left (x \right )-1}{-2 \,{\mathrm e}^{3}+2 \underline {\hspace {1.25 ex}}\alpha -1}\right )}{-4 \,{\mathrm e}^{3}+15}\right )\right )\) | \(414\) |
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=2 \, {\left (\log \left (x\right ) - 1\right )} \log \left (-{\left (2 \, e^{3} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} + 4\right ) + \log \left (-{\left (2 \, e^{3} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} + 4\right )^{2} + \log \left (x\right )^{2} - 2 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (20) = 40\).
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.83 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\log {\left (x \right )}^{2} + 2 \log {\left (x \right )} \log {\left (\log {\left (x \right )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\left (x \right )} + 4 + e^{6} \right )} - 2 \log {\left (x \right )} + \log {\left (\log {\left (x \right )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\left (x \right )} + 4 + e^{6} \right )}^{2} - 2 \log {\left (\log {\left (x \right )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\left (x \right )} + 4 + e^{6} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=2 \, {\left (\log \left (x\right ) - 1\right )} \log \left (-{\left (2 \, e^{3} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} + 4\right ) + \log \left (-{\left (2 \, e^{3} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} + 4\right )^{2} + \log \left (x\right )^{2} - 2 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (22) = 44\).
Time = 0.36 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx=\log \left (-2 \, e^{3} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} - \log \left (x\right ) + 4\right )^{2} + 2 \, \log \left (-2 \, e^{3} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} - \log \left (x\right ) + 4\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \log \left (-2 \, e^{3} \log \left (x\right ) + \log \left (x\right )^{2} + e^{6} - \log \left (x\right ) + 4\right ) - 2 \, \log \left (x\right ) \]
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Time = 10.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{4 x+e^6 x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx={\ln \left ({\mathrm {e}}^6-\ln \left (x\right )+{\ln \left (x\right )}^2-2\,{\mathrm {e}}^3\,\ln \left (x\right )+4\right )}^2+2\,\ln \left ({\mathrm {e}}^6-\ln \left (x\right )+{\ln \left (x\right )}^2-2\,{\mathrm {e}}^3\,\ln \left (x\right )+4\right )\,\ln \left (x\right )-2\,\ln \left ({\mathrm {e}}^6-\ln \left (x\right )+{\ln \left (x\right )}^2-2\,{\mathrm {e}}^3\,\ln \left (x\right )+4\right )+{\ln \left (x\right )}^2-2\,\ln \left (x\right ) \]
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