Integrand size = 137, antiderivative size = 37 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{\frac {3 \left (5-\frac {x}{1+e^{\frac {3+e^x x}{x}}}\right )}{x}}-x}{x} \]
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\[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\int \frac {\exp \left (\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}\right ) \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right ) \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx \\ & = \int \left (-\frac {15 \exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3}-\frac {39 \exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3}+\frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right ) (-15-x)}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2}-\frac {2 \exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2}+\frac {3 \exp \left (e^x+\frac {3}{x}+x+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x}\right ) \, dx \\ & = -\left (2 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\right )+3 \int \frac {\exp \left (e^x+\frac {3}{x}+x+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x} \, dx-15 \int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-39 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx+\int \frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right ) (-15-x)}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-\int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx \\ & = -\left (2 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\right )+3 \int \frac {\exp \left (e^x+\frac {3}{x}+x+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x} \, dx-15 \int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-39 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx+\int \left (-\frac {15 \exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3}-\frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2}\right ) \, dx-\int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx \\ & = -\left (2 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\right )+3 \int \frac {\exp \left (e^x+\frac {3}{x}+x+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x} \, dx-15 \int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-15 \int \frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-39 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-\int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx-\int \frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{-\frac {3}{1+e^{e^x+\frac {3}{x}}}+\frac {15}{x}}}{x} \]
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Time = 3.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14
method | result | size |
risch | \(\frac {{\mathrm e}^{-\frac {3 \left (-5 \,{\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}-5+x \right )}{x \left ({\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}+1\right )}}}{x}\) | \(42\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {15 \,{\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}+15-3 x}{x \left ({\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}+1\right )}}}{x}\) | \(44\) |
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Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{\left (-\frac {3 \, {\left (x - 5 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} - 5\right )}}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x}\right )}}{x} \]
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Time = 0.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{\frac {- 3 x + 15 e^{\frac {x e^{x} + 3}{x}} + 15}{x e^{\frac {x e^{x} + 3}{x}} + x}}}{x} \]
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\[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\int { -\frac {{\left ({\left (x + 15\right )} e^{\left (\frac {2 \, {\left (x e^{x} + 3\right )}}{x}\right )} - {\left (3 \, x^{2} e^{x} - 2 \, x - 39\right )} e^{\left (\frac {x e^{x} + 3}{x}\right )} + x + 15\right )} e^{\left (-\frac {3 \, {\left (x - 5 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} - 5\right )}}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x}\right )}}{x^{3} e^{\left (\frac {2 \, {\left (x e^{x} + 3\right )}}{x}\right )} + 2 \, x^{3} e^{\left (\frac {x e^{x} + 3}{x}\right )} + x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (34) = 68\).
Time = 0.61 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.95 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{\left (\frac {x e^{\left (x + \frac {x e^{x} + 3}{x}\right )} + x e^{x} - 3 \, x + 18 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} + 18}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x} - \frac {x e^{x} + 3}{x}\right )}}{x} \]
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Time = 9.79 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.78 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {{\mathrm {e}}^{\frac {15\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}{x+x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}}\,{\mathrm {e}}^{\frac {15}{x+x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}}\,{\mathrm {e}}^{-\frac {3}{{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}+1}}}{x} \]
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