Integrand size = 69, antiderivative size = 26 \begin {dmath*} \int \frac {1}{36} e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}+e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}} x} \left (-36+e^{\frac {1}{36} \left (-108-36 e^x+x\right )} \left (-x+36 e^x x\right )\right ) \, dx=-4-e^{e^{1+e^{-3-e^x+\frac {x}{36}}} x} \end {dmath*}
Time = 1.73 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \begin {dmath*} \int \frac {1}{36} e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}+e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}} x} \left (-36+e^{\frac {1}{36} \left (-108-36 e^x+x\right )} \left (-x+36 e^x x\right )\right ) \, dx=-e^{e^{1+e^{-3-e^x+\frac {x}{36}}} x} \end {dmath*}
Integrate[(E^(1 + E^((-108 - 36*E^x + x)/36) + E^(1 + E^((-108 - 36*E^x + x)/36))*x)*(-36 + E^((-108 - 36*E^x + x)/36)*(-x + 36*E^x*x)))/36,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.
\(\displaystyle \int \frac {1}{36} \left (e^{\frac {1}{36} \left (x-36 e^x-108\right )} \left (36 e^x x-x\right )-36\right ) \exp \left (e^{e^{\frac {1}{36} \left (x-36 e^x-108\right )}+1} x+e^{\frac {1}{36} \left (x-36 e^x-108\right )}+1\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{36} \int -\exp \left (e^{1+e^{\frac {1}{36} \left (x-36 e^x-108\right )}} x+e^{\frac {1}{36} \left (x-36 e^x-108\right )}+1\right ) \left (e^{\frac {1}{36} \left (x-36 e^x-108\right )} \left (x-36 e^x x\right )+36\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{36} \int \exp \left (e^{1+e^{\frac {1}{36} \left (x-36 e^x-108\right )}} x+e^{\frac {1}{36} \left (x-36 e^x-108\right )}+1\right ) \left (e^{\frac {1}{36} \left (x-36 e^x-108\right )} \left (x-36 e^x x\right )+36\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{36} \int \left (\exp \left (e^{1+e^{\frac {1}{36} \left (x-36 e^x-108\right )}} x+e^{\frac {1}{36} \left (x-36 e^x-108\right )}+\frac {1}{36} \left (x-36 e^x-108\right )+1\right ) \left (x-36 e^x x\right )+36 \exp \left (e^{1+e^{\frac {1}{36} \left (x-36 e^x-108\right )}} x+e^{\frac {1}{36} \left (x-36 e^x-108\right )}+1\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{36} \left (-36 \int \exp \left (e^{1+e^{\frac {1}{36} \left (x-36 e^x-108\right )}} x+e^{\frac {1}{36} \left (x-36 e^x-108\right )}+1\right )dx-\int \exp \left (e^{1+e^{\frac {1}{36} \left (x-36 e^x-108\right )}} x+e^{\frac {1}{36} \left (x-36 e^x-108\right )}+\frac {1}{36} \left (x-36 e^x-108\right )+1\right ) xdx+36 \int \exp \left (e^{1+e^{\frac {1}{36} \left (x-36 e^x-108\right )}} x+x+e^{\frac {1}{36} \left (x-36 e^x-108\right )}+\frac {1}{36} \left (x-36 e^x-108\right )+1\right ) xdx\right )\) |
Int[(E^(1 + E^((-108 - 36*E^x + x)/36) + E^(1 + E^((-108 - 36*E^x + x)/36) )*x)*(-36 + E^((-108 - 36*E^x + x)/36)*(-x + 36*E^x*x)))/36,x]
3.7.75.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-{\mathrm e}^{x \,{\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x}+\frac {x}{36}-3}+1}}\) | \(19\) |
parallelrisch | \(-{\mathrm e}^{x \,{\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x}+\frac {x}{36}-3}+1}}\) | \(19\) |
int(1/36*((36*exp(x)*x-x)*exp(-exp(x)+1/36*x-3)-36)*exp(exp(-exp(x)+1/36*x -3)+1)*exp(x*exp(exp(-exp(x)+1/36*x-3)+1)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \begin {dmath*} \int \frac {1}{36} e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}+e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}} x} \left (-36+e^{\frac {1}{36} \left (-108-36 e^x+x\right )} \left (-x+36 e^x x\right )\right ) \, dx=-e^{\left (x e^{\left (e^{\left (\frac {1}{36} \, x - e^{x} - 3\right )} + 1\right )}\right )} \end {dmath*}
integrate(1/36*((36*exp(x)*x-x)*exp(-exp(x)+1/36*x-3)-36)*exp(exp(-exp(x)+ 1/36*x-3)+1)*exp(x*exp(exp(-exp(x)+1/36*x-3)+1)),x, algorithm=\
Time = 2.65 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \begin {dmath*} \int \frac {1}{36} e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}+e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}} x} \left (-36+e^{\frac {1}{36} \left (-108-36 e^x+x\right )} \left (-x+36 e^x x\right )\right ) \, dx=- e^{x e^{e^{\frac {x}{36} - e^{x} - 3} + 1}} \end {dmath*}
integrate(1/36*((36*exp(x)*x-x)*exp(-exp(x)+1/36*x-3)-36)*exp(exp(-exp(x)+ 1/36*x-3)+1)*exp(x*exp(exp(-exp(x)+1/36*x-3)+1)),x)
Time = 0.48 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \begin {dmath*} \int \frac {1}{36} e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}+e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}} x} \left (-36+e^{\frac {1}{36} \left (-108-36 e^x+x\right )} \left (-x+36 e^x x\right )\right ) \, dx=-e^{\left (x e^{\left (e^{\left (\frac {1}{36} \, x - e^{x} - 3\right )} + 1\right )}\right )} \end {dmath*}
integrate(1/36*((36*exp(x)*x-x)*exp(-exp(x)+1/36*x-3)-36)*exp(exp(-exp(x)+ 1/36*x-3)+1)*exp(x*exp(exp(-exp(x)+1/36*x-3)+1)),x, algorithm=\
\begin {dmath*} \int \frac {1}{36} e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}+e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}} x} \left (-36+e^{\frac {1}{36} \left (-108-36 e^x+x\right )} \left (-x+36 e^x x\right )\right ) \, dx=\int { \frac {1}{36} \, {\left ({\left (36 \, x e^{x} - x\right )} e^{\left (\frac {1}{36} \, x - e^{x} - 3\right )} - 36\right )} e^{\left (x e^{\left (e^{\left (\frac {1}{36} \, x - e^{x} - 3\right )} + 1\right )} + e^{\left (\frac {1}{36} \, x - e^{x} - 3\right )} + 1\right )} \,d x } \end {dmath*}
integrate(1/36*((36*exp(x)*x-x)*exp(-exp(x)+1/36*x-3)-36)*exp(exp(-exp(x)+ 1/36*x-3)+1)*exp(x*exp(exp(-exp(x)+1/36*x-3)+1)),x, algorithm=\
integrate(1/36*((36*x*e^x - x)*e^(1/36*x - e^x - 3) - 36)*e^(x*e^(e^(1/36* x - e^x - 3) + 1) + e^(1/36*x - e^x - 3) + 1), x)
Time = 15.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \begin {dmath*} \int \frac {1}{36} e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}+e^{1+e^{\frac {1}{36} \left (-108-36 e^x+x\right )}} x} \left (-36+e^{\frac {1}{36} \left (-108-36 e^x+x\right )} \left (-x+36 e^x x\right )\right ) \, dx=-{\mathrm {e}}^{x\,{\mathrm {e}}^{{\mathrm {e}}^{x/36}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{-{\mathrm {e}}^x}}\,\mathrm {e}} \end {dmath*}