Integrand size = 123, antiderivative size = 31 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {2}{-3-e^{2+2 x}+e^{3-\left (-8+e^x\right )^2+x} x} \]
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\[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 \left (61+e^{2 x}\right )} \left (4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )\right )}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx \\ & = \int \left (\frac {4 e^{-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )} \left (e^{126+2 e^{2 x}}-8 e^{63+16 e^x+e^{2 x}} x+e^{63+16 e^x+e^{2 x}+x} x+e^{32 e^x} x^2\right )}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x}+\frac {2 e^{-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )} \left (-6 e^{187+3 e^{2 x}}-e^{126+16 e^x+2 e^{2 x}+x}+48 e^{124+16 e^x+2 e^{2 x}} x-6 e^{124+16 e^x+2 e^{2 x}+x} \left (1-\frac {e^2}{6}\right ) x-6 e^{61+32 e^x+e^{2 x}} x^2-16 e^{63+32 e^x+e^{2 x}+x} x^2+2 e^{48 e^x+x} x^3\right )}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}\right ) \, dx \\ & = 2 \int \frac {e^{-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )} \left (-6 e^{187+3 e^{2 x}}-e^{126+16 e^x+2 e^{2 x}+x}+48 e^{124+16 e^x+2 e^{2 x}} x-6 e^{124+16 e^x+2 e^{2 x}+x} \left (1-\frac {e^2}{6}\right ) x-6 e^{61+32 e^x+e^{2 x}} x^2-16 e^{63+32 e^x+e^{2 x}+x} x^2+2 e^{48 e^x+x} x^3\right )}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+4 \int \frac {e^{-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )} \left (e^{126+2 e^{2 x}}-8 e^{63+16 e^x+e^{2 x}} x+e^{63+16 e^x+e^{2 x}+x} x+e^{32 e^x} x^2\right )}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \, dx \\ & = 2 \int \left (-\frac {6 e^{2 \left (61+e^{2 x}\right )}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}-\frac {e^{-61+16 e^x-e^{2 x}+2 \left (61+e^{2 x}\right )+x}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}+\frac {48 \exp \left (-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )+2 \left (62+8 e^x+e^{2 x}\right )\right ) x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}+\frac {e^{-63+16 e^x-e^{2 x}+2 \left (61+e^{2 x}\right )+x} \left (-6+e^2\right ) x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}-\frac {6 e^{-126+32 e^x-2 e^{2 x}+2 \left (61+e^{2 x}\right )} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}-\frac {16 e^{-124+32 e^x-2 e^{2 x}+2 \left (61+e^{2 x}\right )+x} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}+\frac {2 e^{-187+48 e^x-3 e^{2 x}+2 \left (61+e^{2 x}\right )+x} x^3}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2}\right ) \, dx+4 \int \frac {e^{61+e^{2 x}}-8 e^{-2+16 e^x} x+e^{-2+16 e^x+x} x+e^{-65+32 e^x-e^{2 x}} x^2}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \, dx \\ & = -\left (2 \int \frac {e^{-61+16 e^x-e^{2 x}+2 \left (61+e^{2 x}\right )+x}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx\right )+4 \int \frac {e^{-187+48 e^x-3 e^{2 x}+2 \left (61+e^{2 x}\right )+x} x^3}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+4 \int \left (\frac {e^{61+e^{2 x}}}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x}+\frac {e^{-65+32 e^x-e^{2 x}} x^2}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x}+\frac {8 e^{16 e^x} x}{-3 e^{63+e^{2 x}}-e^{65+e^{2 x}+2 x}+e^{2+16 e^x+x} x}-\frac {e^{16 e^x+x} x}{-3 e^{63+e^{2 x}}-e^{65+e^{2 x}+2 x}+e^{2+16 e^x+x} x}\right ) \, dx-12 \int \frac {e^{2 \left (61+e^{2 x}\right )}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-12 \int \frac {e^{-126+32 e^x-2 e^{2 x}+2 \left (61+e^{2 x}\right )} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-32 \int \frac {e^{-124+32 e^x-2 e^{2 x}+2 \left (61+e^{2 x}\right )+x} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+96 \int \frac {\exp \left (-187-3 e^{2 x}+2 \left (61+e^{2 x}\right )+2 \left (62+8 e^x+e^{2 x}\right )\right ) x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-\left (2 \left (6-e^2\right )\right ) \int \frac {e^{-63+16 e^x-e^{2 x}+2 \left (61+e^{2 x}\right )+x} x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx \\ & = -\left (2 \int \frac {e^{61+16 e^x+e^{2 x}+x}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx\right )+4 \int \frac {e^{-65+48 e^x-e^{2 x}+x} x^3}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+4 \int \frac {e^{61+e^{2 x}}}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \, dx+4 \int \frac {e^{-65+32 e^x-e^{2 x}} x^2}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \, dx-4 \int \frac {e^{16 e^x+x} x}{-3 e^{63+e^{2 x}}-e^{65+e^{2 x}+2 x}+e^{2+16 e^x+x} x} \, dx-12 \int \frac {e^{2 \left (61+e^{2 x}\right )}}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-12 \int \frac {e^{4 \left (-1+8 e^x\right )} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-32 \int \frac {e^{-2+32 e^x+x} x^2}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx+32 \int \frac {e^{16 e^x} x}{-3 e^{63+e^{2 x}}-e^{65+e^{2 x}+2 x}+e^{2+16 e^x+x} x} \, dx+96 \int \frac {e^{59+16 e^x+e^{2 x}} x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx-\left (2 \left (6-e^2\right )\right ) \int \frac {e^{59+16 e^x+e^{2 x}+x} x}{\left (3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x\right )^2} \, dx \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=-\frac {2 e^{61+e^{2 x}}}{3 e^{61+e^{2 x}}+e^{63+e^{2 x}+2 x}-e^{16 e^x+x} x} \]
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Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {2}{{\mathrm e}^{2+2 x}-{\mathrm e}^{-{\mathrm e}^{2 x}+16 \,{\mathrm e}^{x}+x -61} x +3}\) | \(30\) |
parallelrisch | \(\frac {2}{{\mathrm e}^{-{\mathrm e}^{2 x}+16 \,{\mathrm e}^{x}+x -61} x -{\mathrm e}^{2+2 x}-3}\) | \(31\) |
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Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {2}{x e^{\left (x - e^{\left (2 \, x\right )} + 16 \, e^{x} - 61\right )} - e^{\left (2 \, x + 2\right )} - 3} \]
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Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {2}{x e^{x - e^{2 x} + 16 e^{x} - 61} - e^{2} e^{2 x} - 3} \]
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Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {2 \, e^{\left (e^{\left (2 \, x\right )} + 61\right )}}{x e^{\left (x + 16 \, e^{x}\right )} - {\left (3 \, e^{61} + e^{\left (2 \, x + 63\right )}\right )} e^{\left (e^{\left (2 \, x\right )}\right )}} \]
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\[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\int { \frac {2 \, {\left ({\left (2 \, x e^{\left (2 \, x\right )} - 16 \, x e^{x} - x - 1\right )} e^{\left (x - e^{\left (2 \, x\right )} + 16 \, e^{x} - 61\right )} + 2 \, e^{\left (2 \, x + 2\right )}\right )}}{x^{2} e^{\left (2 \, x - 2 \, e^{\left (2 \, x\right )} + 32 \, e^{x} - 122\right )} - 2 \, {\left (x e^{\left (2 \, x + 2\right )} + 3 \, x\right )} e^{\left (x - e^{\left (2 \, x\right )} + 16 \, e^{x} - 61\right )} + e^{\left (4 \, x + 4\right )} + 6 \, e^{\left (2 \, x + 2\right )} + 9} \,d x } \]
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Time = 10.56 (sec) , antiderivative size = 166, normalized size of antiderivative = 5.35 \[ \int \frac {4 e^{2+2 x}+e^{-61+16 e^x-e^{2 x}+x} \left (-2-2 x-32 e^x x+4 e^{2 x} x\right )}{9+6 e^{2+2 x}+e^{4+4 x}+e^{-122+32 e^x-2 e^{2 x}+2 x} x^2+e^{-61+16 e^x-e^{2 x}+x} \left (-6 x-2 e^{2+2 x} x\right )} \, dx=\frac {6\,x+96\,x^2\,{\mathrm {e}}^x+{\mathrm {e}}^{2\,x+2}\,\left (2\,x-2\,x^2\right )-12\,x^2\,{\mathrm {e}}^{2\,x}+32\,x^2\,{\mathrm {e}}^{3\,x+2}-4\,x^2\,{\mathrm {e}}^{4\,x+2}+6\,x^2}{\left ({\mathrm {e}}^{x-{\mathrm {e}}^{2\,x}+16\,{\mathrm {e}}^x-61}-\frac {{\mathrm {e}}^{2\,x+2}+3}{x}\right )\,\left (48\,x^3\,{\mathrm {e}}^x-6\,x^3\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{2\,x+2}-x^3\,{\mathrm {e}}^{2\,x+2}+16\,x^3\,{\mathrm {e}}^{3\,x+2}-2\,x^3\,{\mathrm {e}}^{4\,x+2}+3\,x^2+3\,x^3\right )} \]
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