Integrand size = 166, antiderivative size = 27 \[ \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx=e^{\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}}-x \]
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\[ \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx=\int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{x \left (3-x-\log \left (\frac {4}{x^3}\right )\right )^2} \, dx \\ & = \int \left (-\frac {9}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}+\frac {6 x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}-\frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}-\frac {2 (-3+x) \log \left (\frac {4}{x^3}\right )}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}-\frac {\log ^2\left (\frac {4}{x^3}\right )}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}+\frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) \left (3-37 x-12 x^2+8 x^3+12 x \log \left (\frac {4}{x^3}\right )+8 x^2 \log \left (\frac {4}{x^3}\right )\right )}{x \left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}\right ) \, dx \\ & = -\left (2 \int \frac {(-3+x) \log \left (\frac {4}{x^3}\right )}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx\right )+6 \int \frac {x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-9 \int \frac {1}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-\int \frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-\int \frac {\log ^2\left (\frac {4}{x^3}\right )}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx+\int \frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) \left (3-37 x-12 x^2+8 x^3+12 x \log \left (\frac {4}{x^3}\right )+8 x^2 \log \left (\frac {4}{x^3}\right )\right )}{x \left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx \\ & = -\left (2 \int \left (-\frac {(-3+x)^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}+\frac {-3+x}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) \, dx\right )+6 \int \frac {x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-9 \int \frac {1}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-\int \frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-\int \left (1+\frac {(-3+x)^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}-\frac {2 (-3+x)}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) \, dx+\int \left (\frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) (3-x)}{x \left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}+\frac {4 \exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) (3+2 x)}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) \, dx \\ & = -x+2 \int \frac {(-3+x)^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx+4 \int \frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) (3+2 x)}{-3+x+\log \left (\frac {4}{x^3}\right )} \, dx+6 \int \frac {x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-9 \int \frac {1}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-\int \frac {(-3+x)^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx+\int \frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) (3-x)}{x \left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-\int \frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx \\ & = -x+2 \int \left (\frac {9}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}-\frac {6 x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}+\frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}\right ) \, dx+4 \int \left (\frac {3 \exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right )}{-3+x+\log \left (\frac {4}{x^3}\right )}+\frac {2 \exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) x}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) \, dx+6 \int \frac {x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-9 \int \frac {1}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-\int \frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx+\int \left (-\frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right )}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}+\frac {3 \exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right )}{x \left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}\right ) \, dx-\int \left (\frac {9}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}-\frac {6 x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}+\frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2}\right ) \, dx \\ & = -x+2 \int \frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx+3 \int \frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right )}{x \left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx+2 \left (6 \int \frac {x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx\right )+8 \int \frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right ) x}{-3+x+\log \left (\frac {4}{x^3}\right )} \, dx-2 \left (9 \int \frac {1}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx\right )-12 \int \frac {x}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx+12 \int \frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right )}{-3+x+\log \left (\frac {4}{x^3}\right )} \, dx+18 \int \frac {1}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-\int \frac {\exp \left (9+12 x+4 x^2+\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}\right )}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx-2 \int \frac {x^2}{\left (-3+x+\log \left (\frac {4}{x^3}\right )\right )^2} \, dx \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx=e^{\frac {e^{(3+2 x)^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}}-x \]
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Time = 44.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11
method | result | size |
parallelrisch | \(-6-x +{\mathrm e}^{\frac {{\mathrm e}^{4 x^{2}+12 x +9}}{\ln \left (\frac {4}{x^{3}}\right )+x -3}}\) | \(30\) |
risch | \(-x +{\mathrm e}^{\frac {2 \,{\mathrm e}^{\left (3+2 x \right )^{2}}}{i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+i \pi \operatorname {csgn}\left (i x^{3}\right )^{3}+4 \ln \left (2\right )-6 \ln \left (x \right )+2 x -6}}\) | \(154\) |
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Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx=-x + e^{\left (\frac {e^{\left (4 \, x^{2} + 12 \, x + 9\right )}}{x + \log \left (\frac {4}{x^{3}}\right ) - 3}\right )} \]
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Time = 0.71 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx=- x + e^{\frac {e^{4 x^{2} + 12 x + 9}}{x + \log {\left (\frac {4}{x^{3}} \right )} - 3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.19 \[ \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx=-{\left (x e^{\left (-\frac {e^{\left (4 \, x^{2} + 12 \, x + 9\right )}}{x + 2 \, \log \left (2\right ) - 3 \, \log \left (x\right ) - 3}\right )} - 1\right )} e^{\left (\frac {e^{\left (4 \, x^{2} + 12 \, x + 9\right )}}{x + 2 \, \log \left (2\right ) - 3 \, \log \left (x\right ) - 3}\right )} \]
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\[ \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx=\int { -\frac {x^{3} + x \log \left (\frac {4}{x^{3}}\right )^{2} - 6 \, x^{2} - {\left (4 \, {\left (2 \, x^{2} + 3 \, x\right )} e^{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac {4}{x^{3}}\right ) + {\left (8 \, x^{3} - 12 \, x^{2} - 37 \, x + 3\right )} e^{\left (4 \, x^{2} + 12 \, x + 9\right )}\right )} e^{\left (\frac {e^{\left (4 \, x^{2} + 12 \, x + 9\right )}}{x + \log \left (\frac {4}{x^{3}}\right ) - 3}\right )} + 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (\frac {4}{x^{3}}\right ) + 9 \, x}{x^{3} + x \log \left (\frac {4}{x^{3}}\right )^{2} - 6 \, x^{2} + 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (\frac {4}{x^{3}}\right ) + 9 \, x} \,d x } \]
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Time = 13.83 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {-9 x+6 x^2-x^3+\left (6 x-2 x^2\right ) \log \left (\frac {4}{x^3}\right )-x \log ^2\left (\frac {4}{x^3}\right )+e^{\frac {e^{9+12 x+4 x^2}}{-3+x+\log \left (\frac {4}{x^3}\right )}} \left (e^{9+12 x+4 x^2} \left (3-37 x-12 x^2+8 x^3\right )+e^{9+12 x+4 x^2} \left (12 x+8 x^2\right ) \log \left (\frac {4}{x^3}\right )\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log \left (\frac {4}{x^3}\right )+x \log ^2\left (\frac {4}{x^3}\right )} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{12\,x}\,{\mathrm {e}}^9\,{\mathrm {e}}^{4\,x^2}}{x+\ln \left (\frac {4}{x^3}\right )-3}}-x \]
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