Integrand size = 116, antiderivative size = 25 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2}{\left (1+e^{e^{2 e^x+6 (8+x)} x}\right )^2} \]
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\[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx \\ & = \int \left (\frac {2 e^{2 \left (24+e^x+3 x\right )} x^2 \left (1+6 x+2 e^x x\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3}-\frac {2 x \left (-1+e^{48+2 e^x+6 x} x+6 e^{48+2 e^x+6 x} x^2+2 e^{48+2 e^x+7 x} x^2\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}\right ) \, dx \\ & = 2 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^2 \left (1+6 x+2 e^x x\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx-2 \int \frac {x \left (-1+e^{48+2 e^x+6 x} x+6 e^{48+2 e^x+6 x} x^2+2 e^{48+2 e^x+7 x} x^2\right )}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx \\ & = 2 \int \left (\frac {e^{2 \left (24+e^x+3 x\right )} x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3}+\frac {6 e^{2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3}+\frac {2 e^{x+2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3}\right ) \, dx-2 \int \left (-\frac {x}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}+\frac {e^{2 \left (24+e^x+3 x\right )} x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}+\frac {6 e^{2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}+\frac {2 e^{48+2 e^x+7 x} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2}\right ) \, dx \\ & = 2 \int \frac {x}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx+2 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx-2 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx+4 \int \frac {e^{x+2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx-4 \int \frac {e^{48+2 e^x+7 x} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx+12 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^3} \, dx-12 \int \frac {e^{2 \left (24+e^x+3 x\right )} x^3}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \, dx \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2}{\left (1+e^{e^{48+2 e^x+6 x} x}\right )^2} \]
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Time = 1.77 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {x^{2}}{\left (1+{\mathrm e}^{x \,{\mathrm e}^{2 \,{\mathrm e}^{x}+6 x +48}}\right )^{2}}\) | \(22\) |
parallelrisch | \(\frac {x^{2}}{{\mathrm e}^{2 x \,{\mathrm e}^{2 \,{\mathrm e}^{x}+6 x +48}}+2 \,{\mathrm e}^{x \,{\mathrm e}^{2 \,{\mathrm e}^{x}+6 x +48}}+1}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{2 x e^{6 x + 2 e^{x} + 48}} + 2 e^{x e^{6 x + 2 e^{x} + 48}} + 1} \]
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]
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Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^{2}}{e^{\left (2 \, x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 2 \, e^{\left (x e^{\left (6 \, x + 2 \, e^{x} + 48\right )}\right )} + 1} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {2 x+e^{e^{48+2 e^x+6 x} x} \left (2 x+e^{48+2 e^x+6 x} \left (-2 x^2-12 x^3-4 e^x x^3\right )\right )}{1+3 e^{e^{48+2 e^x+6 x} x}+3 e^{2 e^{48+2 e^x+6 x} x}+e^{3 e^{48+2 e^x+6 x} x}} \, dx=\frac {x^2\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{48}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}}{4\,{\mathrm {cosh}\left (\frac {x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{48}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{2}\right )}^2} \]
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