Integrand size = 123, antiderivative size = 29 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=e^{-\frac {-1+e^{e^{-e^{-x} x+x^6}}}{x}+x}+x \]
Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x}+x \]
Integrate[(E^x*x^2 + E^((1 - E^E^((-x + E^x*x^6)/E^x) + x^2)/x)*(E^x*(-1 + x^2) + E^E^((-x + E^x*x^6)/E^x)*(E^x + E^((-x + E^x*x^6)/E^x)*(x - x^2 - 6*E^x*x^6))))/(E^x*x^2),x]
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.
\(\displaystyle \int \frac {e^{-x} \left (e^x x^2+e^{\frac {-e^{e^{e^{-x} \left (e^x x^6-x\right )}}+x^2+1}{x}} \left (e^x \left (x^2-1\right )+e^{e^{e^{-x} \left (e^x x^6-x\right )}} \left (e^{e^{-x} \left (e^x x^6-x\right )} \left (-6 e^x x^6-x^2+x\right )+e^x\right )\right )\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{-\frac {e^{e^{x^6-e^{-x} x}}}{x}} \left (e^{e^{x^6-e^{-x} x}+x+\frac {1}{x}}+e^{x+\frac {1}{x}} x^2+e^{\frac {e^{e^{x^6-e^{-x} x}}}{x}} x^2-e^{x+\frac {1}{x}}\right )}{x^2}-\frac {\left (6 e^x x^5+x-1\right ) \exp \left (x^6+e^{x^6-e^{-x} x}-\frac {e^{e^{x^6-e^{-x} x}}}{x}-e^{-x} x+\frac {1}{x}\right )}{x}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {e^{-\frac {e^{e^{x^6-e^{-x} x}}}{x}} \left (e^{e^{x^6-e^{-x} x}+x+\frac {1}{x}}+e^{x+\frac {1}{x}} x^2+e^{\frac {e^{e^{x^6-e^{-x} x}}}{x}} x^2-e^{x+\frac {1}{x}}\right )}{x^2}-\frac {\left (6 e^x x^5+x-1\right ) \exp \left (x^6+e^{x^6-e^{-x} x}-\frac {e^{e^{x^6-e^{-x} x}}}{x}-e^{-x} x+\frac {1}{x}\right )}{x}\right )dx\) |
Int[(E^x*x^2 + E^((1 - E^E^((-x + E^x*x^6)/E^x) + x^2)/x)*(E^x*(-1 + x^2) + E^E^((-x + E^x*x^6)/E^x)*(E^x + E^((-x + E^x*x^6)/E^x)*(x - x^2 - 6*E^x* x^6))))/(E^x*x^2),x]
3.14.18.3.1 Defintions of rubi rules used
Time = 51.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07
method | result | size |
risch | \(x +{\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{x \left (x^{5} {\mathrm e}^{x}-1\right ) {\mathrm e}^{-x}}}+x^{2}+1}{x}}\) | \(31\) |
parallelrisch | \(x +{\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{\left (x^{6} {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}}+x^{2}+1}{x}}\) | \(32\) |
int(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x ^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/exp(x)) )+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\left (\frac {x^{2} - e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 1}{x}\right )} \]
integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp( exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/e xp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm=\
Time = 9.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\frac {x^{2} - e^{e^{\left (x^{6} e^{x} - x\right ) e^{- x}}} + 1}{x}} \]
integrate(((((-6*x**6*exp(x)-x**2+x)*exp((x**6*exp(x)-x)/exp(x))+exp(x))*e xp(exp((x**6*exp(x)-x)/exp(x)))+(x**2-1)*exp(x))*exp((-exp(exp((x**6*exp(x )-x)/exp(x)))+x**2+1)/x)+exp(x)*x**2)/exp(x)/x**2,x)
Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\left (x - \frac {e^{\left (e^{\left (x^{6} - x e^{\left (-x\right )}\right )}\right )}}{x} + \frac {1}{x}\right )} \]
integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp( exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/e xp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm=\
\[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=\int { \frac {{\left (x^{2} e^{x} + {\left ({\left (x^{2} - 1\right )} e^{x} - {\left ({\left (6 \, x^{6} e^{x} + x^{2} - x\right )} e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )} - e^{x}\right )} e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (\frac {x^{2} - e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 1}{x}\right )}\right )} e^{\left (-x\right )}}{x^{2}} \,d x } \]
integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp( exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/e xp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm=\
integrate((x^2*e^x + ((x^2 - 1)*e^x - ((6*x^6*e^x + x^2 - x)*e^((x^6*e^x - x)*e^(-x)) - e^x)*e^(e^((x^6*e^x - x)*e^(-x))))*e^((x^2 - e^(e^((x^6*e^x - x)*e^(-x))) + 1)/x))*e^(-x)/x^2, x)
Time = 9.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x+{\mathrm {e}}^{-\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^6}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-x}}}}{x}}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^x \]