Integrand size = 60, antiderivative size = 27 \[ \int \frac {e^{-x} \left (9 e^x x+e^{e^{-x} \left (e^x \left (5-e^2\right )-x+e^x \log \left (x^2\right )\right )} \left (-2 e^x+x-x^2\right )\right )}{x} \, dx=3+9 x-e^{5-e^2-e^{-x} x} x^2 \] Output:
3+9*x-exp(5+ln(x^2)-x/exp(x)-exp(1)^2)
Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (9 e^x x+e^{e^{-x} \left (e^x \left (5-e^2\right )-x+e^x \log \left (x^2\right )\right )} \left (-2 e^x+x-x^2\right )\right )}{x} \, dx=9 x-e^{5-e^2-e^{-x} x} x^2 \] Input:
Integrate[(9*E^x*x + E^((E^x*(5 - E^2) - x + E^x*Log[x^2])/E^x)*(-2*E^x + x - x^2))/(E^x*x),x]
Output:
9*x - E^(5 - E^2 - x/E^x)*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.
\(\displaystyle \int \frac {e^{-x} \left (\left (-x^2+x-2 e^x\right ) \exp \left (e^{-x} \left (e^x \log \left (x^2\right )-x+\left (5-e^2\right ) e^x\right )\right )+9 e^x x\right )}{x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (e^{-e^{-x} x-x+5 \left (1-\frac {e^2}{5}\right )} x \left (-x^2+x-2 e^x\right )+9\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int e^{-e^{-x} \left (e^x x+x-5 e^x \left (1-\frac {e^2}{5}\right )\right )} x^3dx+\int e^{-e^{-x} \left (e^x x+x-5 e^x \left (1-\frac {e^2}{5}\right )\right )} x^2dx-2 \int e^{-e^{-x} \left (x-5 e^x \left (1-\frac {e^2}{5}\right )\right )} xdx+9 x\) |
Input:
Int[(9*E^x*x + E^((E^x*(5 - E^2) - x + E^x*Log[x^2])/E^x)*(-2*E^x + x - x^ 2))/(E^x*x),x]
Output:
$Aborted
Time = 0.49 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33
method | result | size |
parallelrisch | \(9 x -{\mathrm e}^{\left (-{\mathrm e}^{2} {\mathrm e}^{x}+{\mathrm e}^{x} \ln \left (x^{2}\right )+5 \,{\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\) | \(36\) |
default | \(\left (9 \,{\mathrm e}^{x} x -{\mathrm e}^{x} {\mathrm e}^{\left ({\mathrm e}^{x} \ln \left (x^{2}\right )+\left (5-{\mathrm e}^{2}\right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\) | \(44\) |
norman | \(\left (9 \,{\mathrm e}^{x} x -{\mathrm e}^{x} {\mathrm e}^{\left ({\mathrm e}^{x} \ln \left (x^{2}\right )+\left (5-{\mathrm e}^{2}\right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\) | \(44\) |
risch | \(9 x -{\mathrm e}^{\frac {\left (-i {\mathrm e}^{x} \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i {\mathrm e}^{x} \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-2 \,{\mathrm e}^{2+x}+4 \,{\mathrm e}^{x} \ln \left (x \right )+10 \,{\mathrm e}^{x}-2 x \right ) {\mathrm e}^{-x}}{2}}\) | \(89\) |
Input:
int(((-2*exp(x)-x^2+x)*exp((exp(x)*ln(x^2)+(-exp(1)^2+5)*exp(x)-x)/exp(x)) +9*exp(x)*x)/exp(x)/x,x,method=_RETURNVERBOSE)
Output:
9*x-exp((-exp(1)^2*exp(x)+exp(x)*ln(x^2)+5*exp(x)-x)/exp(x))
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-x} \left (9 e^x x+e^{e^{-x} \left (e^x \left (5-e^2\right )-x+e^x \log \left (x^2\right )\right )} \left (-2 e^x+x-x^2\right )\right )}{x} \, dx=9 \, x - e^{\left (-{\left ({\left (e^{2} - 5\right )} e^{x} - e^{x} \log \left (x^{2}\right ) + x\right )} e^{\left (-x\right )}\right )} \] Input:
integrate(((-2*exp(x)-x^2+x)*exp((exp(x)*log(x^2)+(-exp(1)^2+5)*exp(x)-x)/ exp(x))+9*exp(x)*x)/exp(x)/x,x, algorithm="fricas")
Output:
9*x - e^(-((e^2 - 5)*e^x - e^x*log(x^2) + x)*e^(-x))
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (9 e^x x+e^{e^{-x} \left (e^x \left (5-e^2\right )-x+e^x \log \left (x^2\right )\right )} \left (-2 e^x+x-x^2\right )\right )}{x} \, dx=9 x - e^{\left (- x + e^{x} \log {\left (x^{2} \right )} + \left (5 - e^{2}\right ) e^{x}\right ) e^{- x}} \] Input:
integrate(((-2*exp(x)-x**2+x)*exp((exp(x)*ln(x**2)+(-exp(1)**2+5)*exp(x)-x )/exp(x))+9*exp(x)*x)/exp(x)/x,x)
Output:
9*x - exp((-x + exp(x)*log(x**2) + (5 - exp(2))*exp(x))*exp(-x))
\[ \int \frac {e^{-x} \left (9 e^x x+e^{e^{-x} \left (e^x \left (5-e^2\right )-x+e^x \log \left (x^2\right )\right )} \left (-2 e^x+x-x^2\right )\right )}{x} \, dx=\int { -\frac {{\left ({\left (x^{2} - x + 2 \, e^{x}\right )} e^{\left (-{\left ({\left (e^{2} - 5\right )} e^{x} - e^{x} \log \left (x^{2}\right ) + x\right )} e^{\left (-x\right )}\right )} - 9 \, x e^{x}\right )} e^{\left (-x\right )}}{x} \,d x } \] Input:
integrate(((-2*exp(x)-x^2+x)*exp((exp(x)*log(x^2)+(-exp(1)^2+5)*exp(x)-x)/ exp(x))+9*exp(x)*x)/exp(x)/x,x, algorithm="maxima")
Output:
9*x - integrate((x^3*e^5 - x^2*e^5 + 2*x*e^(x + 5))*e^(-x*e^(-x) - x - e^2 ), x)
Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-x} \left (9 e^x x+e^{e^{-x} \left (e^x \left (5-e^2\right )-x+e^x \log \left (x^2\right )\right )} \left (-2 e^x+x-x^2\right )\right )}{x} \, dx=-{\left (x^{2} e^{\left (-x e^{\left (-x\right )} - x - e^{2} + 5\right )} - 9 \, x e^{\left (-x\right )}\right )} e^{x} \] Input:
integrate(((-2*exp(x)-x^2+x)*exp((exp(x)*log(x^2)+(-exp(1)^2+5)*exp(x)-x)/ exp(x))+9*exp(x)*x)/exp(x)/x,x, algorithm="giac")
Output:
-(x^2*e^(-x*e^(-x) - x - e^2 + 5) - 9*x*e^(-x))*e^x
Time = 2.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-x} \left (9 e^x x+e^{e^{-x} \left (e^x \left (5-e^2\right )-x+e^x \log \left (x^2\right )\right )} \left (-2 e^x+x-x^2\right )\right )}{x} \, dx=9\,x-x^2\,{\mathrm {e}}^{-{\mathrm {e}}^2}\,{\mathrm {e}}^5\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-x}} \] Input:
int(-(exp(-x)*(exp(-exp(-x)*(x + exp(x)*(exp(2) - 5) - log(x^2)*exp(x)))*( 2*exp(x) - x + x^2) - 9*x*exp(x)))/x,x)
Output:
9*x - x^2*exp(-exp(2))*exp(5)*exp(-x*exp(-x))
Time = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {e^{-x} \left (9 e^x x+e^{e^{-x} \left (e^x \left (5-e^2\right )-x+e^x \log \left (x^2\right )\right )} \left (-2 e^x+x-x^2\right )\right )}{x} \, dx=\frac {x \left (9 e^{\frac {e^{x} e^{2}+x}{e^{x}}}-e^{5} x \right )}{e^{\frac {e^{x} e^{2}+x}{e^{x}}}} \] Input:
int(((-2*exp(x)-x^2+x)*exp((exp(x)*log(x^2)+(-exp(1)^2+5)*exp(x)-x)/exp(x) )+9*exp(x)*x)/exp(x)/x,x)
Output:
(x*(9*e**((e**x*e**2 + x)/e**x) - e**5*x))/e**((e**x*e**2 + x)/e**x)