Integrand size = 75, antiderivative size = 27 \[ \int \frac {e^{\frac {3-3 x-3 x^2+4 x^3+e^x \left (-1+x^2\right )}{-1+x^2}} \left (-3+9 x^2-4 x^4+e^x \left (-1+2 x^2-x^4\right )\right )}{1-2 x^2+x^4} \, dx=9-e^{-3+e^x+4 x+\frac {x^2}{-x+x^3}} \] Output:
9-exp(exp(x)+4*x+x^2/(x^3-x)-3)
Time = 3.59 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\frac {3-3 x-3 x^2+4 x^3+e^x \left (-1+x^2\right )}{-1+x^2}} \left (-3+9 x^2-4 x^4+e^x \left (-1+2 x^2-x^4\right )\right )}{1-2 x^2+x^4} \, dx=-e^{-3+e^x+4 x+\frac {x}{-1+x^2}} \] Input:
Integrate[(E^((3 - 3*x - 3*x^2 + 4*x^3 + E^x*(-1 + x^2))/(-1 + x^2))*(-3 + 9*x^2 - 4*x^4 + E^x*(-1 + 2*x^2 - x^4)))/(1 - 2*x^2 + x^4),x]
Output:
-E^(-3 + E^x + 4*x + x/(-1 + x^2))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (-4 x^4+9 x^2+e^x \left (-x^4+2 x^2-1\right )-3\right ) \exp \left (\frac {4 x^3-3 x^2+e^x \left (x^2-1\right )-3 x+3}{x^2-1}\right )}{x^4-2 x^2+1} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int -\frac {\left (4 x^4-9 x^2+e^x \left (x^4-2 x^2+1\right )+3\right ) \exp \left (-\frac {4 x^3-3 x^2-e^x \left (1-x^2\right )-3 x+3}{1-x^2}\right )}{\left (1-x^2\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\exp \left (-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right ) \left (4 x^4-9 x^2+e^x \left (x^4-2 x^2+1\right )+3\right )}{\left (1-x^2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\int \left (\frac {4 \exp \left (-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right ) x^4}{\left (x^2-1\right )^2}-\frac {9 \exp \left (-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right ) x^2}{\left (x^2-1\right )^2}+\exp \left (x-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right )+\frac {3 \exp \left (-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right )}{\left (x^2-1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \exp \left (-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right )dx+\frac {3}{2} \int \frac {\exp \left (-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right )}{(1-x)^2}dx-\int \frac {\exp \left (-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right )}{(x-1)^2}dx+\frac {1}{2} \int \frac {\exp \left (-\frac {4 x^3-3 x^2-3 x-e^x \left (1-x^2\right )+3}{1-x^2}\right )}{(x+1)^2}dx-\int e^{\frac {5 x^3+e^x x^2-3 x^2-4 x-e^x+3}{x^2-1}}dx\) |
Input:
Int[(E^((3 - 3*x - 3*x^2 + 4*x^3 + E^x*(-1 + x^2))/(-1 + x^2))*(-3 + 9*x^2 - 4*x^4 + E^x*(-1 + 2*x^2 - x^4)))/(1 - 2*x^2 + x^4),x]
Output:
$Aborted
Time = 2.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37
method | result | size |
parallelrisch | \(-{\mathrm e}^{\frac {{\mathrm e}^{x} x^{2}+4 x^{3}-3 x^{2}-{\mathrm e}^{x}-3 x +3}{x^{2}-1}}\) | \(37\) |
risch | \(-{\mathrm e}^{\frac {{\mathrm e}^{x} x^{2}+4 x^{3}-3 x^{2}-{\mathrm e}^{x}-3 x +3}{\left (-1+x \right ) \left (1+x \right )}}\) | \(40\) |
norman | \(\frac {-x^{2} {\mathrm e}^{\frac {\left (x^{2}-1\right ) {\mathrm e}^{x}+4 x^{3}-3 x^{2}-3 x +3}{x^{2}-1}}+{\mathrm e}^{\frac {\left (x^{2}-1\right ) {\mathrm e}^{x}+4 x^{3}-3 x^{2}-3 x +3}{x^{2}-1}}}{x^{2}-1}\) | \(79\) |
Input:
int(((-x^4+2*x^2-1)*exp(x)-4*x^4+9*x^2-3)*exp(((x^2-1)*exp(x)+4*x^3-3*x^2- 3*x+3)/(x^2-1))/(x^4-2*x^2+1),x,method=_RETURNVERBOSE)
Output:
-exp((exp(x)*x^2+4*x^3-3*x^2-exp(x)-3*x+3)/(x^2-1))
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {3-3 x-3 x^2+4 x^3+e^x \left (-1+x^2\right )}{-1+x^2}} \left (-3+9 x^2-4 x^4+e^x \left (-1+2 x^2-x^4\right )\right )}{1-2 x^2+x^4} \, dx=-e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + {\left (x^{2} - 1\right )} e^{x} - 3 \, x + 3}{x^{2} - 1}\right )} \] Input:
integrate(((-x^4+2*x^2-1)*exp(x)-4*x^4+9*x^2-3)*exp(((x^2-1)*exp(x)+4*x^3- 3*x^2-3*x+3)/(x^2-1))/(x^4-2*x^2+1),x, algorithm="fricas")
Output:
-e^((4*x^3 - 3*x^2 + (x^2 - 1)*e^x - 3*x + 3)/(x^2 - 1))
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {3-3 x-3 x^2+4 x^3+e^x \left (-1+x^2\right )}{-1+x^2}} \left (-3+9 x^2-4 x^4+e^x \left (-1+2 x^2-x^4\right )\right )}{1-2 x^2+x^4} \, dx=- e^{\frac {4 x^{3} - 3 x^{2} - 3 x + \left (x^{2} - 1\right ) e^{x} + 3}{x^{2} - 1}} \] Input:
integrate(((-x**4+2*x**2-1)*exp(x)-4*x**4+9*x**2-3)*exp(((x**2-1)*exp(x)+4 *x**3-3*x**2-3*x+3)/(x**2-1))/(x**4-2*x**2+1),x)
Output:
-exp((4*x**3 - 3*x**2 - 3*x + (x**2 - 1)*exp(x) + 3)/(x**2 - 1))
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {3-3 x-3 x^2+4 x^3+e^x \left (-1+x^2\right )}{-1+x^2}} \left (-3+9 x^2-4 x^4+e^x \left (-1+2 x^2-x^4\right )\right )}{1-2 x^2+x^4} \, dx=-e^{\left (4 \, x + \frac {1}{2 \, {\left (x + 1\right )}} + \frac {1}{2 \, {\left (x - 1\right )}} + e^{x} - 3\right )} \] Input:
integrate(((-x^4+2*x^2-1)*exp(x)-4*x^4+9*x^2-3)*exp(((x^2-1)*exp(x)+4*x^3- 3*x^2-3*x+3)/(x^2-1))/(x^4-2*x^2+1),x, algorithm="maxima")
Output:
-e^(4*x + 1/2/(x + 1) + 1/2/(x - 1) + e^x - 3)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.63 \[ \int \frac {e^{\frac {3-3 x-3 x^2+4 x^3+e^x \left (-1+x^2\right )}{-1+x^2}} \left (-3+9 x^2-4 x^4+e^x \left (-1+2 x^2-x^4\right )\right )}{1-2 x^2+x^4} \, dx=-e^{\left (\frac {4 \, x^{3}}{x^{2} - 1} + \frac {x^{2} e^{x}}{x^{2} - 1} - \frac {3 \, x^{2}}{x^{2} - 1} - \frac {3 \, x}{x^{2} - 1} - \frac {e^{x}}{x^{2} - 1} + \frac {3}{x^{2} - 1}\right )} \] Input:
integrate(((-x^4+2*x^2-1)*exp(x)-4*x^4+9*x^2-3)*exp(((x^2-1)*exp(x)+4*x^3- 3*x^2-3*x+3)/(x^2-1))/(x^4-2*x^2+1),x, algorithm="giac")
Output:
-e^(4*x^3/(x^2 - 1) + x^2*e^x/(x^2 - 1) - 3*x^2/(x^2 - 1) - 3*x/(x^2 - 1) - e^x/(x^2 - 1) + 3/(x^2 - 1))
Time = 3.91 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \[ \int \frac {e^{\frac {3-3 x-3 x^2+4 x^3+e^x \left (-1+x^2\right )}{-1+x^2}} \left (-3+9 x^2-4 x^4+e^x \left (-1+2 x^2-x^4\right )\right )}{1-2 x^2+x^4} \, dx=-{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^x}{x^2-1}}\,{\mathrm {e}}^{-\frac {3\,x^2}{x^2-1}}\,{\mathrm {e}}^{\frac {4\,x^3}{x^2-1}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{x^2-1}}\,{\mathrm {e}}^{\frac {3}{x^2-1}}\,{\mathrm {e}}^{-\frac {3\,x}{x^2-1}} \] Input:
int(-(exp((exp(x)*(x^2 - 1) - 3*x - 3*x^2 + 4*x^3 + 3)/(x^2 - 1))*(exp(x)* (x^4 - 2*x^2 + 1) - 9*x^2 + 4*x^4 + 3))/(x^4 - 2*x^2 + 1),x)
Output:
-exp((x^2*exp(x))/(x^2 - 1))*exp(-(3*x^2)/(x^2 - 1))*exp((4*x^3)/(x^2 - 1) )*exp(-exp(x)/(x^2 - 1))*exp(3/(x^2 - 1))*exp(-(3*x)/(x^2 - 1))
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\frac {3-3 x-3 x^2+4 x^3+e^x \left (-1+x^2\right )}{-1+x^2}} \left (-3+9 x^2-4 x^4+e^x \left (-1+2 x^2-x^4\right )\right )}{1-2 x^2+x^4} \, dx=-\frac {e^{\frac {e^{x} x^{2}-e^{x}+4 x^{3}-3 x}{x^{2}-1}}}{e^{3}} \] Input:
int(((-x^4+2*x^2-1)*exp(x)-4*x^4+9*x^2-3)*exp(((x^2-1)*exp(x)+4*x^3-3*x^2- 3*x+3)/(x^2-1))/(x^4-2*x^2+1),x)
Output:
( - e**((e**x*x**2 - e**x + 4*x**3 - 3*x)/(x**2 - 1)))/e**3