Integrand size = 293, antiderivative size = 31 \[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx=\frac {x}{\left (2+x+e^{x+\frac {\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} x\right )^2} \]
[Out]
\[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx=\int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {-\left (\left (2+e^2 x\right ) \left (-2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x+2 e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x^2\right )\right )+8 e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} \log \left (e^2+\frac {2}{x}\right )+2 e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx \\ & = \int \left (\frac {-2 x-4 \left (1+\frac {e^2}{4}\right ) x^2-2 e^2 x^3+8 \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2}+\frac {2 \left (4 x+4 \left (1+\frac {e^2}{2}\right ) x^2+2 \left (1+e^2\right ) x^3+e^2 x^4-8 \log \left (e^2+\frac {2}{x}\right )-4 x \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )-e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3}\right ) \, dx \\ & = 2 \int \frac {4 x+4 \left (1+\frac {e^2}{2}\right ) x^2+2 \left (1+e^2\right ) x^3+e^2 x^4-8 \log \left (e^2+\frac {2}{x}\right )-4 x \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )-e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx+\int \frac {-2 x-4 \left (1+\frac {e^2}{4}\right ) x^2-2 e^2 x^3+8 \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx \\ & = 2 \int \frac {x \left (2+e^2 x\right ) \left (2+2 x+x^2\right )-4 (2+x) \log \left (e^2+\frac {2}{x}\right )-(2+x) \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx+\int \frac {-x (1+2 x) \left (2+e^2 x\right )+8 \log \left (e^2+\frac {2}{x}\right )+2 \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx \\ & = 2 \int \left (\frac {4 x+4 \left (1+\frac {e^2}{2}\right ) x^2+2 \left (1+e^2\right ) x^3+e^2 x^4-8 \log \left (e^2+\frac {2}{x}\right )-4 x \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )-e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )}{2 x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3}+\frac {e^2 \left (-4 x-4 \left (1+\frac {e^2}{2}\right ) x^2-2 \left (1+e^2\right ) x^3-e^2 x^4+8 \log \left (e^2+\frac {2}{x}\right )+4 x \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )+e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )\right )}{2 \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3}\right ) \, dx+\int \left (\frac {e^2 \left (2 x+4 \left (1+\frac {e^2}{4}\right ) x^2+2 e^2 x^3-8 \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )\right )}{2 \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2}+\frac {-2 x-4 \left (1+\frac {e^2}{4}\right ) x^2-2 e^2 x^3+8 \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{2 x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-2 x-4 \left (1+\frac {e^2}{4}\right ) x^2-2 e^2 x^3+8 \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx+\frac {1}{2} e^2 \int \frac {2 x+4 \left (1+\frac {e^2}{4}\right ) x^2+2 e^2 x^3-8 \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx+e^2 \int \frac {-4 x-4 \left (1+\frac {e^2}{2}\right ) x^2-2 \left (1+e^2\right ) x^3-e^2 x^4+8 \log \left (e^2+\frac {2}{x}\right )+4 x \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )+e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx+\int \frac {4 x+4 \left (1+\frac {e^2}{2}\right ) x^2+2 \left (1+e^2\right ) x^3+e^2 x^4-8 \log \left (e^2+\frac {2}{x}\right )-4 x \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )-e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx \\ & = \frac {1}{2} \int \frac {-x (1+2 x) \left (2+e^2 x\right )+8 \log \left (e^2+\frac {2}{x}\right )+2 \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx+\frac {1}{2} e^2 \int \frac {x (1+2 x) \left (2+e^2 x\right )-8 \log \left (e^2+\frac {2}{x}\right )-2 \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx+e^2 \int \frac {-x \left (2+e^2 x\right ) \left (2+2 x+x^2\right )+4 (2+x) \log \left (e^2+\frac {2}{x}\right )+(2+x) \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx+\int \frac {x \left (2+e^2 x\right ) \left (2+2 x+x^2\right )-4 (2+x) \log \left (e^2+\frac {2}{x}\right )-(2+x) \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx=\frac {x}{\left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \]
[In]
[Out]
Time = 9.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {x}{\left (x \,{\mathrm e}^{\frac {\ln \left (\frac {{\mathrm e}^{2} x +2}{x}\right )^{2}+x^{2}}{x}}+x +2\right )^{2}}\) | \(32\) |
parallelrisch | \(\frac {x}{{\mathrm e}^{\frac {2 \ln \left (\frac {{\mathrm e}^{2} x +2}{x}\right )^{2}+2 x^{2}}{x}} x^{2}+2 \,{\mathrm e}^{\frac {\ln \left (\frac {{\mathrm e}^{2} x +2}{x}\right )^{2}+x^{2}}{x}} x^{2}+x^{2}+4 x \,{\mathrm e}^{\frac {\ln \left (\frac {{\mathrm e}^{2} x +2}{x}\right )^{2}+x^{2}}{x}}+4 x +4}\) | \(93\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.26 \[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx=\frac {x}{x^{2} e^{\left (\frac {2 \, {\left (x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}\right )}}{x}\right )} + x^{2} + 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}}{x}\right )} + 4 \, x + 4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx=\frac {x}{x^{2} e^{\frac {2 \left (x^{2} + \log {\left (\frac {x e^{2} + 2}{x} \right )}^{2}\right )}{x}} + x^{2} + 4 x + \left (2 x^{2} + 4 x\right ) e^{\frac {x^{2} + \log {\left (\frac {x e^{2} + 2}{x} \right )}^{2}}{x}} + 4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (29) = 58\).
Time = 5.85 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.97 \[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx=\frac {x e^{\left (\frac {4 \, \log \left (x e^{2} + 2\right ) \log \left (x\right )}{x}\right )}}{x^{2} e^{\left (2 \, x + \frac {2 \, \log \left (x e^{2} + 2\right )^{2}}{x} + \frac {2 \, \log \left (x\right )^{2}}{x}\right )} + 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x + \frac {\log \left (x e^{2} + 2\right )^{2}}{x} + \frac {2 \, \log \left (x e^{2} + 2\right ) \log \left (x\right )}{x} + \frac {\log \left (x\right )^{2}}{x}\right )} + {\left (x^{2} + 4 \, x + 4\right )} e^{\left (\frac {4 \, \log \left (x e^{2} + 2\right ) \log \left (x\right )}{x}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (29) = 58\).
Time = 92.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.94 \[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx=\frac {x}{x^{2} e^{\left (\frac {2 \, {\left (x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}\right )}}{x}\right )} + 2 \, x^{2} e^{\left (\frac {x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}}{x}\right )} + x^{2} + 4 \, x e^{\left (\frac {x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}}{x}\right )} + 4 \, x + 4} \]
[In]
[Out]
Time = 19.62 (sec) , antiderivative size = 452, normalized size of antiderivative = 14.58 \[ \int \frac {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx=-\frac {{\left ({\mathrm {e}}^2\,x^2+2\,x\right )}^2\,\left (4\,x-8\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )-4\,x\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )-2\,x\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2+2\,x^2\,{\mathrm {e}}^2+2\,x^3\,{\mathrm {e}}^2+x^4\,{\mathrm {e}}^2-4\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2+4\,x^2+2\,x^3-x^2\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^2-2\,x\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^2\right )}{\left (x\,{\mathrm {e}}^2+2\right )\,\left ({\left (x+2\right )}^2+x^2\,{\mathrm {e}}^{2\,x+\frac {2\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2}{x}}+2\,x\,{\mathrm {e}}^{x+\frac {{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2}{x}}\,\left (x+2\right )\right )\,\left (16\,x\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )+8\,x\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2+8\,x^2\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )-8\,x^3\,{\mathrm {e}}^2-8\,x^4\,{\mathrm {e}}^2-4\,x^5\,{\mathrm {e}}^2-2\,x^4\,{\mathrm {e}}^4-2\,x^5\,{\mathrm {e}}^4-x^6\,{\mathrm {e}}^4+4\,x^2\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2-8\,x^2-8\,x^3-4\,x^4+8\,x^2\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^2+4\,x^3\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^2+2\,x^3\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^4+x^4\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^4+8\,x^2\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )\,{\mathrm {e}}^2+4\,x^3\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )\,{\mathrm {e}}^2\right )} \]
[In]
[Out]