Integrand size = 100, antiderivative size = 25 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\left (20-\frac {3^{\frac {1}{x}} x \left (x^4\right )^{-1/x}}{\log (x)}\right )^2 \]
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\[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2\ 3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (-x+\log (x) \left (-4+x+\log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \\ & = 2 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (-x+\log (x) \left (-4+x+\log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \\ & = 2 \int \left (\frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{\log ^3(x)}-\frac {20\ 3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{x \log ^2(x)}\right ) \, dx \\ & = 2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{\log ^3(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{x \log ^2(x)} \, dx \\ & = 2 \int \left (\frac {3^{2/x} \left (x^4\right )^{-2/x} (-x-4 \log (x)+x \log (x))}{\log ^3(x)}+\frac {3^{2/x} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)}\right ) \, dx-40 \int \left (\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} (-x-4 \log (x)+x \log (x))}{x \log ^2(x)}+\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)}\right ) \, dx \\ & = 2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} (-x-4 \log (x)+x \log (x))}{\log ^3(x)} \, dx+2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} (-x-4 \log (x)+x \log (x))}{x \log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx \\ & = 2 \int \left (-\frac {3^{2/x} x \left (x^4\right )^{-2/x}}{\log ^3(x)}+\frac {3^{2/x} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)}\right ) \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \left (-\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)}+\frac {3^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-1/x}}{x \log (x)}\right ) \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx \\ & = -\left (2 \int \frac {3^{2/x} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {3^{2/x} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-1/x}}{x \log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx \\ & = -\left (2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {9^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \left (\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)}-\frac {4\ 3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)}\right ) \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx \\ & = 2 \int \left (-\frac {4\ 9^{\frac {1}{x}} \left (x^4\right )^{-2/x}}{\log ^2(x)}+\frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^2(x)}\right ) \, dx-2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx+160 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)} \, dx \\ & = -\left (2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-8 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx+160 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)} \, dx \\ \end{align*}
\[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \]
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Time = 1.77 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68
method | result | size |
default | \(-\frac {40 x \,{\mathrm e}^{-\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )}+\frac {x^{2} {\mathrm e}^{-\frac {2 \ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )^{2}}\) | \(42\) |
parts | \(-\frac {40 x \,{\mathrm e}^{-\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )}+\frac {x^{2} {\mathrm e}^{-\frac {2 \ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )^{2}}\) | \(42\) |
parallelrisch | \(\frac {\left (-320 \ln \left (x \right ) {\mathrm e}^{\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}} x^{2}+8 x^{3}\right ) {\mathrm e}^{-\frac {2 \ln \left (\frac {x^{4}}{3}\right )}{x}}}{8 x \ln \left (x \right )^{2}}\) | \(47\) |
risch | \(-\frac {40 x 3^{\frac {1}{x}} x^{-\frac {4}{x}} {\mathrm e}^{\frac {i \pi \left (\operatorname {csgn}\left (i x^{2}\right )^{3}-2 \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )-\operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x^{2}\right )-\operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{3}\right )^{3}-\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{4}\right )^{3}\right )}{2 x}}}{\ln \left (x \right )}+\frac {x^{2} 3^{\frac {2}{x}} x^{-\frac {8}{x}} {\mathrm e}^{\frac {i \pi \left (\operatorname {csgn}\left (i x^{2}\right )^{3}-2 \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )-\operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x^{2}\right )-\operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{3}\right )^{3}-\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{4}\right )^{3}\right )}{x}}}{\ln \left (x \right )^{2}}\) | \(399\) |
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=-\frac {{\left (40 \, x e^{\left (-\frac {\log \left (3\right ) - 4 \, \log \left (x\right )}{x}\right )} \log \left (x\right ) - x^{2}\right )} e^{\left (\frac {2 \, {\left (\log \left (3\right ) - 4 \, \log \left (x\right )\right )}}{x}\right )}}{\log \left (x\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\frac {x^{2} e^{- \frac {2 \cdot \left (4 \log {\left (x \right )} - \log {\left (3 \right )}\right )}{x}} \log {\left (x \right )} - 40 x e^{- \frac {4 \log {\left (x \right )} - \log {\left (3 \right )}}{x}} \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\frac {x^{2} e^{\left (\frac {2 \, \log \left (3\right )}{x} - \frac {8 \, \log \left (x\right )}{x}\right )} - 40 \, x e^{\left (\frac {\log \left (3\right )}{x} - \frac {4 \, \log \left (x\right )}{x}\right )} \log \left (x\right )}{\log \left (x\right )^{2}} \]
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\[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\int { \frac {2 \, {\left (x \log \left (\frac {1}{3} \, x^{4}\right ) \log \left (x\right ) - 20 \, {\left ({\left (x - 4\right )} \log \left (x\right )^{2} + \log \left (\frac {1}{3} \, x^{4}\right ) \log \left (x\right )^{2} - x \log \left (x\right )\right )} \left (\frac {1}{3} \, x^{4}\right )^{\left (\frac {1}{x}\right )} - x^{2} + {\left (x^{2} - 4 \, x\right )} \log \left (x\right )\right )}}{\left (\frac {1}{3} \, x^{4}\right )^{\frac {2}{x}} x \log \left (x\right )^{3}} \,d x } \]
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Time = 9.65 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\frac {3^{1/x}\,x\,\left (3^{1/x}\,x-40\,\ln \left (x\right )\,{\left (x^4\right )}^{1/x}\right )}{{\ln \left (x\right )}^2\,{\left (x^4\right )}^{2/x}} \]
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