Integrand size = 87, antiderivative size = 24 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=-5+e^{e^{e^x+\left (2+e^{e^{e^x}}\right )^2}}+3 x \]
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=e^{e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}}+3 x \]
Integrate[3 + E^(4 + 4*E^E^E^x + E^(2*E^E^x) + E^(4 + 4*E^E^E^x + E^(2*E^E ^x) + E^x) + E^x)*(E^x + 4*E^(E^E^x + E^x + x) + 2*E^(2*E^E^x + E^x + x)), 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.
\(\displaystyle \int \left (\left (e^x+4 e^{x+e^{e^x}+e^x}+2 e^{x+2 e^{e^x}+e^x}\right ) \exp \left (4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x+4}+e^x+4\right )+3\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \text {Subst}\left (\int e^{e^{\left (2+e^x\right )^2} x+4 e^x+e^{2 x}+4}dx,x,e^{e^x}\right )+4 \text {Subst}\left (\int e^{e^{\left (2+e^x\right )^2} x+x+4 e^x+e^{2 x}+4} xdx,x,e^{e^x}\right )+2 \text {Subst}\left (\int e^{e^{\left (2+e^x\right )^2} x+2 x+4 e^x+e^{2 x}+4} xdx,x,e^{e^x}\right )+3 x\) |
Int[3 + E^(4 + 4*E^E^E^x + E^(2*E^E^x) + E^(4 + 4*E^E^E^x + E^(2*E^E^x) + E^x) + E^x)*(E^x + 4*E^(E^E^x + E^x + x) + 2*E^(2*E^E^x + E^x + x)),x]
3.22.50.3.1 Defintions of rubi rules used
Time = 1.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
default | \(3 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{x}}}+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+{\mathrm e}^{x}+4}}\) | \(23\) |
risch | \(3 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{x}}}+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+{\mathrm e}^{x}+4}}\) | \(23\) |
parallelrisch | \(3 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{x}}}+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+{\mathrm e}^{x}+4}}\) | \(23\) |
int((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))^2+4*exp(x)*exp(exp(x))*exp(exp( exp(x)))+exp(x))*exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4)*exp(e xp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))+3,x,method=_RETURNVERB OSE)
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 10.08 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx={\left (3 \, x e^{\left ({\left ({\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (2 \, x + 2 \, e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{x}\right )}\right )} + e^{\left ({\left ({\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left ({\left ({\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (2 \, x + 2 \, e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{x}\right )} + 2 \, x + 2 \, e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (2 \, x + 2 \, e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{x}\right )}\right )}\right )} e^{\left (-{\left ({\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (2 \, x + 2 \, e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{x}\right )}\right )} \]
integrate((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))^2+4*exp(x)*exp(exp(x))*ex p(exp(exp(x)))+exp(x))*exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4) *exp(exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))+3,x, algorithm=\
(3*x*e^(((e^(3*x) + 4*e^(2*x))*e^(2*e^x) + e^(2*x + 2*e^x + 2*e^(e^x)) + 4 *e^(2*x + 2*e^x + e^(e^x)))*e^(-2*x - 2*e^x)) + e^(((e^(3*x) + 4*e^(2*x))* e^(2*e^x) + e^(((e^(3*x) + 4*e^(2*x))*e^(2*e^x) + e^(2*x + 2*e^x + 2*e^(e^ x)) + 4*e^(2*x + 2*e^x + e^(e^x)))*e^(-2*x - 2*e^x) + 2*x + 2*e^x) + e^(2* x + 2*e^x + 2*e^(e^x)) + 4*e^(2*x + 2*e^x + e^(e^x)))*e^(-2*x - 2*e^x)))*e ^(-((e^(3*x) + 4*e^(2*x))*e^(2*e^x) + e^(2*x + 2*e^x + 2*e^(e^x)) + 4*e^(2 *x + 2*e^x + e^(e^x)))*e^(-2*x - 2*e^x))
Time = 1.69 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=3 x + e^{e^{e^{x} + e^{2 e^{e^{x}}} + 4 e^{e^{e^{x}}} + 4}} \]
integrate((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))**2+4*exp(x)*exp(exp(x))*e xp(exp(exp(x)))+exp(x))*exp(exp(exp(exp(x)))**2+4*exp(exp(exp(x)))+exp(x)+ 4)*exp(exp(exp(exp(exp(x)))**2+4*exp(exp(exp(x)))+exp(x)+4))+3,x)
Time = 0.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=3 \, x + e^{\left (e^{\left (e^{x} + e^{\left (2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (e^{\left (e^{x}\right )}\right )} + 4\right )}\right )} \]
integrate((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))^2+4*exp(x)*exp(exp(x))*ex p(exp(exp(x)))+exp(x))*exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4) *exp(exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))+3,x, algorithm=\
\[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=\int { {\left (2 \, e^{\left (x + e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )} + e^{x}\right )} e^{\left (e^{x} + e^{\left (2 \, e^{\left (e^{x}\right )}\right )} + e^{\left (e^{x} + e^{\left (2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (e^{\left (e^{x}\right )}\right )} + 4\right )} + 4 \, e^{\left (e^{\left (e^{x}\right )}\right )} + 4\right )} + 3 \,d x } \]
integrate((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))^2+4*exp(x)*exp(exp(x))*ex p(exp(exp(x)))+exp(x))*exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4) *exp(exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))+3,x, algorithm=\
integrate((2*e^(x + e^x + 2*e^(e^x)) + 4*e^(x + e^x + e^(e^x)) + e^x)*e^(e ^x + e^(2*e^(e^x)) + e^(e^x + e^(2*e^(e^x)) + 4*e^(e^(e^x)) + 4) + 4*e^(e^ (e^x)) + 4) + 3, x)
Time = 11.81 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=3\,x+{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^x}}}\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}} \]