Integrand size = 132, antiderivative size = 33 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=\frac {3+\log (3)-\log \left (\frac {x}{6+3 e^{-2 e^{-x}}-x^2}\right )}{x} \]
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Time = 3.76 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58, number of steps used = 24, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6874, 6820, 6857, 464, 212, 2631, 213} \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=-\frac {\log \left (\frac {e^{2 e^{-x}} x}{e^{2 e^{-x}} \left (6-x^2\right )+3}\right )}{x}-\frac {1}{x}+\frac {4+\log (3)}{x} \]
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Rule 212
Rule 213
Rule 464
Rule 2631
Rule 6820
Rule 6857
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {6 e^{-x}}{x \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}+\frac {-12 \left (1+\frac {\log (3)}{4}\right )-24 e^{2 e^{-x}} \left (1+\frac {\log (3)}{4}\right )+2 e^{2 e^{-x}} x^2 \left (1+\frac {\log (3)}{2}\right )+3 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )+6 e^{2 e^{-x}} \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )-e^{2 e^{-x}} x^2 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (3+6 e^{2 e^{-x}}-e^{2 e^{-x}} x^2\right )}\right ) \, dx \\ & = -\left (6 \int \frac {e^{-x}}{x \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx\right )+\int \frac {-12 \left (1+\frac {\log (3)}{4}\right )-24 e^{2 e^{-x}} \left (1+\frac {\log (3)}{4}\right )+2 e^{2 e^{-x}} x^2 \left (1+\frac {\log (3)}{2}\right )+3 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )+6 e^{2 e^{-x}} \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )-e^{2 e^{-x}} x^2 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (3+6 e^{2 e^{-x}}-e^{2 e^{-x}} x^2\right )} \, dx \\ & = -\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )+\int \frac {-12 \left (1+\frac {\log (3)}{4}\right )-e^{2 e^{-x}} \left (24-x^2 (2+\log (3))+\log (729)\right )-\left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (3-e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx \\ & = -\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )+\int \left (\frac {6}{\left (-6+x^2\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}+\frac {-24 \left (1+\frac {\log (3)}{4}\right )+2 x^2 \left (1+\frac {\log (3)}{2}\right )+6 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )-x^2 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (6-x^2\right )}\right ) \, dx \\ & = 6 \int \frac {1}{\left (-6+x^2\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx+\int \frac {-24 \left (1+\frac {\log (3)}{4}\right )+2 x^2 \left (1+\frac {\log (3)}{2}\right )+6 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )-x^2 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (6-x^2\right )} \, dx \\ & = -\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )+6 \int \left (-\frac {1}{2 \sqrt {6} \left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}-\frac {1}{2 \sqrt {6} \left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}\right ) \, dx+\int \frac {x^2 (2+\log (3))-6 (4+\log (3))-\left (-6+x^2\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (6-x^2\right )} \, dx \\ & = -\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx+\int \left (\frac {x^2 (2+\log (3))-6 (4+\log (3))}{x^2 \left (6-x^2\right )}+\frac {\log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2}\right ) \, dx \\ & = -\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx+\int \frac {x^2 (2+\log (3))-6 (4+\log (3))}{x^2 \left (6-x^2\right )} \, dx+\int \frac {\log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2} \, dx \\ & = \frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-2 \int \frac {1}{6-x^2} \, dx-6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx+\int \frac {e^{-x} \left (3 e^x-6 x+e^{2 e^{-x}+x} \left (6+x^2\right )\right )}{x^2 \left (3-e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx \\ & = -\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx+\int \left (\frac {6 e^{-x}}{x \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}-\frac {3+6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2}{x^2 \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}\right ) \, dx \\ & = -\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}+6 \int \frac {e^{-x}}{x \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\int \frac {3+6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2}{x^2 \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx \\ & = -\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\int \frac {-3-e^{2 e^{-x}} \left (6+x^2\right )}{x^2 \left (3-e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx \\ & = -\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\int \left (\frac {6+x^2}{x^2 \left (-6+x^2\right )}+\frac {6}{\left (-6+x^2\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}\right ) \, dx \\ & = -\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-6 \int \frac {1}{\left (-6+x^2\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\int \frac {6+x^2}{x^2 \left (-6+x^2\right )} \, dx \\ & = -\frac {1}{x}-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-2 \int \frac {1}{-6+x^2} \, dx-6 \int \left (-\frac {1}{2 \sqrt {6} \left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}-\frac {1}{2 \sqrt {6} \left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}\right ) \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx \\ & = -\frac {1}{x}+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=\frac {18+\log (729)-6 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{6 x} \]
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Time = 87.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76
method | result | size |
parallelrisch | \(\frac {\left (\ln \left (3\right ) {\mathrm e}^{x}-{\mathrm e}^{x} \ln \left (-\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )+3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x}\) | \(58\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 \,{\mathrm e}^{-x}}\right )}{x}+\frac {-i \pi \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{-x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{-x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )+2 i \pi \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \operatorname {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )-2 i \pi +6+2 \ln \left (3\right )-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3\right )}{2 x}\) | \(587\) |
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Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=\frac {\log \left (3\right ) - \log \left (-\frac {x e^{\left ({\left (x e^{x} + 2\right )} e^{\left (-x\right )}\right )}}{{\left (x^{2} - 6\right )} e^{\left ({\left (x e^{x} + 2\right )} e^{\left (-x\right )}\right )} - 3 \, e^{x}}\right ) + 3}{x} \]
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Time = 0.58 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=- \frac {\log {\left (- \frac {x e^{2 e^{- x}}}{\left (x^{2} - 6\right ) e^{2 e^{- x}} - 3} \right )}}{x} - \frac {-3 - \log {\left (3 \right )}}{x} \]
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Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=-\frac {2 \, e^{\left (-x\right )} - \log \left (3\right ) - \log \left (-{\left (x^{2} - 6\right )} e^{\left (2 \, e^{\left (-x\right )}\right )} + 3\right ) + \log \left (x\right ) - 3}{x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=-\frac {2 \, x {\rm Ei}\left (-x\right ) + 2 \, e^{\left (-x\right )} - \log \left (3\right ) - \log \left (x^{2} e^{\left (2 \, e^{\left (-x\right )}\right )} - 6 \, e^{\left (2 \, e^{\left (-x\right )}\right )} - 3\right ) + \log \left (-x\right ) - 3}{x} + 2 \, {\rm Ei}\left (-x\right ) \]
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Time = 11.66 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} \left (24-2 x^2+\left (6-x^2\right ) \log (3)\right )+\left (-3 e^x+e^{2 e^{-x}+x} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{-3 e^x x^2+e^{2 e^{-x}+x} \left (-6 x^2+x^4\right )} \, dx=-\frac {\ln \left (-\frac {x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-x}}}{3\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{-x}}\,\left (x^2-6\right )-3\right )}\right )-3}{x} \]
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