Integrand size = 93, antiderivative size = 23 \[ \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx=e^{\frac {4 e^3}{2+e^x}+x+x^2}+12 x \]
[Out]
\[ \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx=\int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{\left (2+e^x\right )^2} \, dx \\ & = \int \left (12+\frac {e^{\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}} \left (4+e^{2 x}+4 e^x \left (1-e^3\right )+8 x+8 e^x x+2 e^{2 x} x\right )}{\left (2+e^x\right )^2}\right ) \, dx \\ & = 12 x+\int \frac {e^{\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}} \left (4+e^{2 x}+4 e^x \left (1-e^3\right )+8 x+8 e^x x+2 e^{2 x} x\right )}{\left (2+e^x\right )^2} \, dx \\ & = 12 x+\int \left (e^{\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}}+\frac {8 \exp \left (3+\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}\right )}{\left (2+e^x\right )^2}-\frac {4 \exp \left (3+\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}\right )}{2+e^x}+2 e^{\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}} x\right ) \, dx \\ & = 12 x+2 \int e^{\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}} x \, dx-4 \int \frac {\exp \left (3+\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}\right )}{2+e^x} \, dx+8 \int \frac {\exp \left (3+\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}\right )}{\left (2+e^x\right )^2} \, dx+\int e^{\frac {4 e^3+2 x+e^x x+2 x^2+e^x x^2}{2+e^x}} \, dx \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx=e^{\frac {4 e^3}{2+e^x}+x+x^2}+12 x \]
[In]
[Out]
Time = 0.53 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48
method | result | size |
parallelrisch | \(12 x +{\mathrm e}^{\frac {\left (x^{2}+x \right ) {\mathrm e}^{x}+4 \,{\mathrm e}^{3}+2 x^{2}+2 x}{{\mathrm e}^{x}+2}}\) | \(34\) |
risch | \(12 x +{\mathrm e}^{\frac {{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x +2 x^{2}+4 \,{\mathrm e}^{3}+2 x}{{\mathrm e}^{x}+2}}\) | \(36\) |
norman | \(\frac {{\mathrm e}^{x} {\mathrm e}^{\frac {\left (x^{2}+x \right ) {\mathrm e}^{x}+4 \,{\mathrm e}^{3}+2 x^{2}+2 x}{{\mathrm e}^{x}+2}}+24 x +12 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{\frac {\left (x^{2}+x \right ) {\mathrm e}^{x}+4 \,{\mathrm e}^{3}+2 x^{2}+2 x}{{\mathrm e}^{x}+2}}}{{\mathrm e}^{x}+2}\) | \(80\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx=12 \, x + e^{\left (\frac {2 \, x^{2} + {\left (x^{2} + x\right )} e^{x} + 2 \, x + 4 \, e^{3}}{e^{x} + 2}\right )} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx=12 x + e^{\frac {2 x^{2} + 2 x + \left (x^{2} + x\right ) e^{x} + 4 e^{3}}{e^{x} + 2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx=12 \, x + e^{\left (\frac {x^{2} e^{x}}{e^{x} + 2} + \frac {2 \, x^{2}}{e^{x} + 2} + \frac {x e^{x}}{e^{x} + 2} + \frac {2 \, x}{e^{x} + 2} + \frac {4 \, e^{3}}{e^{x} + 2}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.49 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx={\left (12 \, x e^{3} + e^{\left (\frac {x^{2} e^{x} + 2 \, x^{2} + x e^{x} + 2 \, x - 2 \, e^{\left (x + 3\right )}}{e^{x} + 2} + 2 \, e^{3} + 3\right )}\right )} e^{\left (-3\right )} \]
[In]
[Out]
Time = 10.89 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.70 \[ \int \frac {48+48 e^x+12 e^{2 x}+e^{\frac {4 e^3+2 x+2 x^2+e^x \left (x+x^2\right )}{2+e^x}} \left (4+8 x+e^{2 x} (1+2 x)+e^x \left (4-4 e^3+8 x\right )\right )}{4+4 e^x+e^{2 x}} \, dx=12\,x+{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^x}{{\mathrm {e}}^x+2}}\,{\mathrm {e}}^{\frac {2\,x^2}{{\mathrm {e}}^x+2}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{{\mathrm {e}}^x+2}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^3}{{\mathrm {e}}^x+2}}\,{\mathrm {e}}^{\frac {2\,x}{{\mathrm {e}}^x+2}} \]
[In]
[Out]