Integrand size = 120, antiderivative size = 27 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (\frac {3}{x}-x+\frac {x}{1-e^x}+\log (3)\right )} \]
[Out]
\[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (1-e^x\right ) x \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx \\ & = \int \left (\frac {5}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}-\frac {5 \left (3+x^2\right )}{x \left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {5 \left (9-6 x (1-\log (3))-x^3 \log (3)-x^2 \left (3+\log (3)-\log ^2(3)\right )\right )}{\left (3-x^2+x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-5 \int \frac {3+x^2}{x \left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {9-6 x (1-\log (3))-x^3 \log (3)-x^2 \left (3+\log (3)-\log ^2(3)\right )}{\left (3-x^2+x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = -\left (5 \int \left (-\frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {2 x-\log (3)}{\left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx\right )+5 \int \left (\frac {3 \left (1+\frac {\log (3)}{3}\right )}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {x \log (3)}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {-x \left (6+\log ^2(3)\right )-\log (27)}{\left (3-x^2+x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx+5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-5 \int \frac {2 x-\log (3)}{\left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {-x \left (6+\log ^2(3)\right )-\log (27)}{\left (3-x^2+x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = -\left (5 \int \left (\frac {2 x}{\left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {\log (3)}{\left (3-x^2+x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx\right )+5 \int \left (\frac {x \left (6+\log ^2(3)\right )}{\left (-3+x^2-x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {\log (27)}{\left (-3+x^2-x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx+5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-10 \int \frac {x}{\left (-3+x^2-x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-(5 \log (3)) \int \frac {1}{\left (3-x^2+x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+\left (5 \left (6+\log ^2(3)\right )\right ) \int \frac {x}{\left (-3+x^2-x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (27)) \int \frac {1}{\left (-3+x^2-x \log (3)\right ) \left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-10 \int \left (\frac {1+\frac {\log (3)}{\sqrt {12+\log ^2(3)}}}{\left (2 x-\log (3)-\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {1-\frac {\log (3)}{\sqrt {12+\log ^2(3)}}}{\left (2 x-\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx-(5 \log (3)) \int \left (\frac {2}{\sqrt {12+\log ^2(3)} \left (-2 x+\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {4}{\sqrt {12+\log ^2(3)} \left (4 x+2 \sqrt {12+\log ^2(3)}-\log (9)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+\left (5 \left (6+\log ^2(3)\right )\right ) \int \left (\frac {1+\frac {\log (3)}{\sqrt {12+\log ^2(3)}}}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (2 x-\log (3)-\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}+\frac {1-\frac {\log (3)}{\sqrt {12+\log ^2(3)}}}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (2 x-\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx+(5 \log (27)) \int \left (-\frac {2}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \sqrt {12+\log ^2(3)} \left (-2 x+\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}-\frac {4}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \sqrt {12+\log ^2(3)} \left (4 x+2 \sqrt {12+\log ^2(3)}-\log (9)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )}\right ) \, dx \\ & = 5 \int \frac {1}{\left (-1+e^x\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+5 \int \frac {1}{x \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 \log (3)) \int \frac {x}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+(5 (3+\log (3))) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-\frac {(10 \log (3)) \int \frac {1}{\left (-2 x+\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx}{\sqrt {12+\log ^2(3)}}-\frac {(20 \log (3)) \int \frac {1}{\left (4 x+2 \sqrt {12+\log ^2(3)}-\log (9)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx}{\sqrt {12+\log ^2(3)}}-\left (10 \left (1-\frac {\log (3)}{\sqrt {12+\log ^2(3)}}\right )\right ) \int \frac {1}{\left (2 x-\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+\left (5 \left (6+\log ^2(3)\right ) \left (1-\frac {\log (3)}{\sqrt {12+\log ^2(3)}}\right )\right ) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (2 x-\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-\left (10 \left (1+\frac {\log (3)}{\sqrt {12+\log ^2(3)}}\right )\right ) \int \frac {1}{\left (2 x-\log (3)-\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx+\left (5 \left (6+\log ^2(3)\right ) \left (1+\frac {\log (3)}{\sqrt {12+\log ^2(3)}}\right )\right ) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (2 x-\log (3)-\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx-\frac {(10 \log (27)) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (-2 x+\log (3)+\sqrt {12+\log ^2(3)}\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx}{\sqrt {12+\log ^2(3)}}-\frac {(20 \log (27)) \int \frac {1}{\left (3-3 e^x+e^x x^2+x \log (3)-e^x x \log (3)\right ) \left (4 x+2 \sqrt {12+\log ^2(3)}-\log (9)\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \, dx}{\sqrt {12+\log ^2(3)}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{\left (-1+e^x\right ) x}\right )} \]
[In]
[Out]
Time = 9.75 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37
method | result | size |
parallelrisch | \(\frac {5}{\ln \left (\frac {\left (x \ln \left (3\right )-x^{2}+3\right ) {\mathrm e}^{x}-x \ln \left (3\right )-3}{\left ({\mathrm e}^{x}-1\right ) x}\right )}\) | \(37\) |
risch | \(\frac {10 i}{\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}-1}\right ) \operatorname {csgn}\left (i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )\right ) \operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{x}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right )}^{3}-\pi \,\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{x \left ({\mathrm e}^{x}-1\right )}\right )}^{2}+\pi \,\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{{\mathrm e}^{x}-1}\right ) \operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{x \left ({\mathrm e}^{x}-1\right )}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+\pi {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{x \left ({\mathrm e}^{x}-1\right )}\right )}^{3}-\pi {\operatorname {csgn}\left (\frac {i \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}{x \left ({\mathrm e}^{x}-1\right )}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-2 i \ln \left (x \right )-2 i \ln \left ({\mathrm e}^{x}-1\right )+2 i \ln \left (-{\mathrm e}^{x} x^{2}+\ln \left (3\right ) \left ({\mathrm e}^{x}-1\right ) x +3 \,{\mathrm e}^{x}-3\right )}\) | \(483\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (-\frac {{\left (x^{2} - x \log \left (3\right ) - 3\right )} e^{x} + x \log \left (3\right ) + 3}{x e^{x} - x}\right )} \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log {\left (\frac {- x \log {\left (3 \right )} + \left (- x^{2} + x \log {\left (3 \right )} + 3\right ) e^{x} - 3}{x e^{x} - x} \right )}} \]
[In]
[Out]
none
Time = 1.53 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (-{\left (x^{2} - x \log \left (3\right ) - 3\right )} e^{x} - x \log \left (3\right ) - 3\right ) - \log \left (x\right ) - \log \left (e^{x} - 1\right )} \]
[In]
[Out]
none
Time = 0.60 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\log \left (-x^{2} e^{x} + x e^{x} \log \left (3\right ) - x \log \left (3\right ) + 3 \, e^{x} - 3\right ) - \log \left (x e^{x} - x\right )} \]
[In]
[Out]
Time = 10.53 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {15+e^{2 x} \left (15+5 x^2\right )+e^x \left (-30-5 x^2-5 x^3\right )}{\left (3 x+x^2 \log (3)+e^x \left (-6 x+x^3-2 x^2 \log (3)\right )+e^{2 x} \left (3 x-x^3+x^2 \log (3)\right )\right ) \log ^2\left (\frac {-3-x \log (3)+e^x \left (3-x^2+x \log (3)\right )}{-x+e^x x}\right )} \, dx=\frac {5}{\ln \left (\frac {x\,\ln \left (3\right )-{\mathrm {e}}^x\,\left (-x^2+\ln \left (3\right )\,x+3\right )+3}{x-x\,{\mathrm {e}}^x}\right )} \]
[In]
[Out]