Integrand size = 37, antiderivative size = 14 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=1+x-\frac {12}{-e^x+x} \]
[Out]
\[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{\left (e^x-x\right )^2} \, dx \\ & = \int \left (1-\frac {12}{e^x-x}-\frac {12 (-1+x)}{\left (e^x-x\right )^2}\right ) \, dx \\ & = x-12 \int \frac {1}{e^x-x} \, dx-12 \int \frac {-1+x}{\left (e^x-x\right )^2} \, dx \\ & = x-12 \int \frac {1}{e^x-x} \, dx-12 \int \left (-\frac {1}{\left (e^x-x\right )^2}+\frac {x}{\left (e^x-x\right )^2}\right ) \, dx \\ & = x+12 \int \frac {1}{\left (e^x-x\right )^2} \, dx-12 \int \frac {1}{e^x-x} \, dx-12 \int \frac {x}{\left (e^x-x\right )^2} \, dx \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\frac {12+e^x x-x^2}{e^x-x} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
risch | \(x -\frac {12}{x -{\mathrm e}^{x}}\) | \(13\) |
norman | \(\frac {-12+x^{2}-{\mathrm e}^{x} x}{x -{\mathrm e}^{x}}\) | \(20\) |
parallelrisch | \(\frac {-12+x^{2}-{\mathrm e}^{x} x}{x -{\mathrm e}^{x}}\) | \(20\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\frac {x^{2} - x e^{x} - 12}{x - e^{x}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=x + \frac {12}{- x + e^{x}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\frac {x^{2} - x e^{x} - 12}{x - e^{x}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\frac {x^{2} - x e^{x} - 12}{x - e^{x}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=x-\frac {12}{x-{\mathrm {e}}^x} \]
[In]
[Out]