Integrand size = 70, antiderivative size = 23 \[ \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 x^2-4 x^3+2 x^4} \, dx=e^{3+\frac {2 e}{x (-4+4 x)}}+x-\log (x) \]
[Out]
Time = 0.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1608, 27, 12, 6820, 6874, 45, 6838} \[ \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 x^2-4 x^3+2 x^4} \, dx=x+e^{3-\frac {e}{2 (1-x) x}}-\log (x) \]
[In]
[Out]
Rule 12
Rule 27
Rule 45
Rule 1608
Rule 6820
Rule 6838
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{x^2 \left (2-4 x+2 x^2\right )} \, dx \\ & = \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 (-1+x)^2 x^2} \, dx \\ & = \frac {1}{2} \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{(-1+x)^2 x^2} \, dx \\ & = \frac {1}{2} \int \frac {e^{4+\frac {e}{2 (-1+x) x}} (1-2 x)+2 (-1+x)^3 x}{(1-x)^2 x^2} \, dx \\ & = \frac {1}{2} \int \left (\frac {2 (-1+x)}{x}-\frac {e^{4+\frac {e}{2 (-1+x) x}} (-1+2 x)}{(-1+x)^2 x^2}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {e^{4+\frac {e}{2 (-1+x) x}} (-1+2 x)}{(-1+x)^2 x^2} \, dx\right )+\int \frac {-1+x}{x} \, dx \\ & = e^{3-\frac {e}{2 (1-x) x}}+\int \left (1-\frac {1}{x}\right ) \, dx \\ & = e^{3-\frac {e}{2 (1-x) x}}+x-\log (x) \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 x^2-4 x^3+2 x^4} \, dx=e^{3+\frac {e}{2 (-1+x)}-\frac {e}{2 x}}+x-\log (x) \]
[In]
[Out]
Time = 1.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26
method | result | size |
risch | \(x -\ln \left (x \right )+{\mathrm e}^{\frac {{\mathrm e}+6 x^{2}-6 x}{2 x \left (-1+x \right )}}\) | \(29\) |
parallelrisch | \(x -\ln \left (x \right )+{\mathrm e}^{\frac {{\mathrm e}+6 x^{2}-6 x}{2 x \left (-1+x \right )}}+4\) | \(30\) |
parts | \(x -\ln \left (x \right )+\frac {x^{2} {\mathrm e}^{\frac {{\mathrm e}+6 x^{2}-6 x}{2 x^{2}-2 x}}-x \,{\mathrm e}^{\frac {{\mathrm e}+6 x^{2}-6 x}{2 x^{2}-2 x}}}{\left (-1+x \right ) x}\) | \(72\) |
norman | \(\frac {x^{3}+x^{2} {\mathrm e}^{\frac {{\mathrm e}+6 x^{2}-6 x}{2 x^{2}-2 x}}-x -x \,{\mathrm e}^{\frac {{\mathrm e}+6 x^{2}-6 x}{2 x^{2}-2 x}}}{x \left (-1+x \right )}-\ln \left (x \right )\) | \(77\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 x^2-4 x^3+2 x^4} \, dx={\left (x e - e \log \left (x\right ) + e^{\left (\frac {8 \, x^{2} - 8 \, x + e}{2 \, {\left (x^{2} - x\right )}}\right )}\right )} e^{\left (-1\right )} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 x^2-4 x^3+2 x^4} \, dx=x + e^{\frac {6 x^{2} - 6 x + e}{2 x^{2} - 2 x}} - \log {\left (x \right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 x^2-4 x^3+2 x^4} \, dx=x + e^{\left (\frac {e}{2 \, {\left (x - 1\right )}} - \frac {e}{2 \, x} + 3\right )} - \log \left (x\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (21) = 42\).
Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 x^2-4 x^3+2 x^4} \, dx=x + e^{\left (\frac {4 \, x^{2}}{x^{2} - x} - \frac {4 \, x}{x^{2} - x} + \frac {e}{2 \, {\left (x^{2} - x\right )}} - 1\right )} - \log \left (x\right ) \]
[In]
[Out]
Time = 17.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {e^{1+\frac {e-6 x+6 x^2}{-2 x+2 x^2}} (1-2 x)-2 x+6 x^2-6 x^3+2 x^4}{2 x^2-4 x^3+2 x^4} \, dx=x-\ln \left (x\right )+{\mathrm {e}}^{\frac {6\,x}{2\,x-2\,x^2}}\,{\mathrm {e}}^{-\frac {6\,x^2}{2\,x-2\,x^2}}\,{\mathrm {e}}^{-\frac {\mathrm {e}}{2\,x-2\,x^2}} \]
[In]
[Out]