Integrand size = 69, antiderivative size = 24 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{\frac {1}{4} \left (-\frac {17}{9}+e^x\right ) \left (-e^{x/4}+x\right )} \]
Time = 1.94 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{-\frac {1}{36} \left (-17+9 e^x\right ) \left (e^{x/4}-x\right )} \]
Integrate[(E^((17*E^(x/4) - 17*x + E^x*(-9*E^(x/4) + 9*x))/36)*(-68 + 17*E ^(x/4) + E^x*(36 - 45*E^(x/4) + 36*x)))/144,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.
\(\displaystyle \int \frac {1}{144} \left (e^x \left (36 x-45 e^{x/4}+36\right )+17 e^{x/4}-68\right ) \exp \left (\frac {1}{36} \left (-17 x+17 e^{x/4}+e^x \left (9 x-9 e^{x/4}\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{144} \int -\exp \left (\frac {1}{36} \left (-9 e^x \left (e^{x/4}-x\right )+17 e^{x/4}-17 x\right )\right ) \left (-9 e^x \left (4 x-5 e^{x/4}+4\right )-17 e^{x/4}+68\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{144} \int \exp \left (\frac {1}{36} \left (-9 e^x \left (e^{x/4}-x\right )+17 e^{x/4}-17 x\right )\right ) \left (-9 e^x \left (4 x-5 e^{x/4}+4\right )-17 e^{x/4}+68\right )dx\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle -\frac {1}{36} \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )} \left (-9 e^x \left (4 x-5 e^{x/4}+4\right )-17 e^{x/4}+68\right )d\frac {x}{4}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{36} \int \left (9 e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+x} \left (-4 x+5 e^{x/4}-4\right )+68 e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )}-17 e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+\frac {x}{4}}\right )d\frac {x}{4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{36} \left (-68 \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )}d\frac {x}{4}+17 \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+\frac {x}{4}}d\frac {x}{4}+36 \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+x}d\frac {x}{4}-45 \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+\frac {5 x}{4}}d\frac {x}{4}+144 \int \frac {1}{4} e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+x} xd\frac {x}{4}\right )\) |
Int[(E^((17*E^(x/4) - 17*x + E^x*(-9*E^(x/4) + 9*x))/36)*(-68 + 17*E^(x/4) + E^x*(36 - 45*E^(x/4) + 36*x)))/144,x]
3.15.27.3.1 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]]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
risch | \({\mathrm e}^{-\frac {{\mathrm e}^{\frac {5 x}{4}}}{4}+\frac {{\mathrm e}^{x} x}{4}+\frac {17 \,{\mathrm e}^{\frac {x}{4}}}{36}-\frac {17 x}{36}}\) | \(23\) |
parallelrisch | \({\mathrm e}^{\frac {\left (-9 \,{\mathrm e}^{\frac {x}{4}}+9 x \right ) {\mathrm e}^{x}}{36}+\frac {17 \,{\mathrm e}^{\frac {x}{4}}}{36}-\frac {17 x}{36}}\) | \(26\) |
norman | \({\mathrm e}^{\frac {\left (-9 \,{\mathrm e}^{\frac {x}{4}}+9 x \right ) {\mathrm e}^{x}}{36}+\frac {17 \,{\mathrm e}^{\frac {x}{4}}}{36}-\frac {17 x}{36}}\) | \(30\) |
int(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/36*(-9* exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{\left (\frac {1}{4} \, x e^{x} - \frac {17}{36} \, x - \frac {1}{4} \, e^{\left (\frac {5}{4} \, x\right )} + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )} \]
integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/3 6*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{- \frac {17 x}{36} + \left (\frac {x}{4} - \frac {e^{\frac {x}{4}}}{4}\right ) e^{x} + \frac {17 e^{\frac {x}{4}}}{36}} \]
integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/3 6*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x)
\[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=\int { \frac {1}{144} \, {\left (9 \, {\left (4 \, x - 5 \, e^{\left (\frac {1}{4} \, x\right )} + 4\right )} e^{x} + 17 \, e^{\left (\frac {1}{4} \, x\right )} - 68\right )} e^{\left (\frac {1}{4} \, {\left (x - e^{\left (\frac {1}{4} \, x\right )}\right )} e^{x} - \frac {17}{36} \, x + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )} \,d x } \]
integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/3 6*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x, algorithm=\
1/144*integrate((9*(4*x - 5*e^(1/4*x) + 4)*e^x + 17*e^(1/4*x) - 68)*e^(1/4 *(x - e^(1/4*x))*e^x - 17/36*x + 17/36*e^(1/4*x)), x)
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{\left (\frac {1}{4} \, x e^{x} - \frac {17}{36} \, x - \frac {1}{4} \, e^{\left (\frac {5}{4} \, x\right )} + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )} \]
integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/3 6*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x, algorithm=\
Time = 10.81 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx={\mathrm {e}}^{\frac {17\,{\mathrm {e}}^{x/4}}{36}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{4}}\,{\mathrm {e}}^{-\frac {17\,x}{36}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{x/4}\,{\mathrm {e}}^x}{4}} \]