Integrand size = 79, antiderivative size = 30 \[ \int \frac {1}{3} e^{-\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )} \left (3+3 e^{\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )}+e^{e^5} \left (3+e^{7-e^3 x} (-3+x)-x\right )\right ) \, dx=1+e^{-\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )} (-3+x)+x \]
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {1}{3} e^{-\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )} \left (3+3 e^{\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )}+e^{e^5} \left (3+e^{7-e^3 x} (-3+x)-x\right )\right ) \, dx=e^{\frac {1}{3} e^{e^5} \left (-e^{4-e^3 x}-x\right )} (-3+x)+x \]
Integrate[(3 + 3*E^((E^E^5*(E^(4 - E^3*x) + x))/3) + E^E^5*(3 + E^(7 - E^3 *x)*(-3 + x) - x))/(3*E^((E^E^5*(E^(4 - E^3*x) + x))/3)),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.
\(\displaystyle \int \frac {1}{3} e^{-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )} \left (e^{e^5} \left (e^{7-e^3 x} (x-3)-x+3\right )+3 e^{\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )}+3\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int e^{-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )} \left (e^{e^5} \left (-e^{7-e^3 x} (3-x)-x+3\right )+3 e^{\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )}+3\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} \int \left (-e^{-e^3 x-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )+e^5} \left (-e^7+e^{e^3 x}\right ) (x-3)+3 e^{-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )}+3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-3 \int \exp \left (-e^3 x-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )+7 \left (1+\frac {e^5}{7}\right )\right )dx+\int \exp \left (-e^3 x-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )+7 \left (1+\frac {e^5}{7}\right )\right ) xdx+3 \int e^{-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )}dx+3 \int e^{e^5-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )}dx-\int e^{e^5-\frac {1}{3} e^{e^5} \left (x+e^{4-e^3 x}\right )} xdx+3 x\right )\) |
Int[(3 + 3*E^((E^E^5*(E^(4 - E^3*x) + x))/3) + E^E^5*(3 + E^(7 - E^3*x)*(- 3 + x) - x))/(3*E^((E^E^5*(E^(4 - E^3*x) + x))/3)),x]
3.30.68.3.1 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 = 0.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
method | result | size |
parts | \(x +\left (-3+x \right ) {\mathrm e}^{-\frac {\left ({\mathrm e}^{-x \,{\mathrm e}^{3}+4}+x \right ) {\mathrm e}^{{\mathrm e}^{5}}}{3}}\) | \(25\) |
risch | \(x +\frac {\left (3 x -9\right ) {\mathrm e}^{-\frac {\left ({\mathrm e}^{-x \,{\mathrm e}^{3}+4}+x \right ) {\mathrm e}^{{\mathrm e}^{5}}}{3}}}{3}\) | \(26\) |
norman | \(\left (-3+x +x \,{\mathrm e}^{\frac {\left ({\mathrm e}^{-x \,{\mathrm e}^{3}+4}+x \right ) {\mathrm e}^{{\mathrm e}^{5}}}{3}}\right ) {\mathrm e}^{-\frac {\left ({\mathrm e}^{-x \,{\mathrm e}^{3}+4}+x \right ) {\mathrm e}^{{\mathrm e}^{5}}}{3}}\) | \(41\) |
parallelrisch | \(-\frac {\left (27-9 x \,{\mathrm e}^{\frac {\left ({\mathrm e}^{-x \,{\mathrm e}^{3}+4}+x \right ) {\mathrm e}^{{\mathrm e}^{5}}}{3}}-9 x \right ) {\mathrm e}^{-\frac {\left ({\mathrm e}^{-x \,{\mathrm e}^{3}+4}+x \right ) {\mathrm e}^{{\mathrm e}^{5}}}{3}}}{9}\) | \(45\) |
int(1/3*(3*exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp(5)))+((-3+x)*exp(3)*exp(-x *exp(3)+4)+3-x)*exp(exp(5))+3)/exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp(5))),x ,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {1}{3} e^{-\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )} \left (3+3 e^{\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )}+e^{e^5} \left (3+e^{7-e^3 x} (-3+x)-x\right )\right ) \, dx={\left (x e^{\left (\frac {1}{3} \, {\left (x e^{3} + e^{\left (-x e^{3} + 7\right )}\right )} e^{\left (e^{5} - 3\right )}\right )} + x - 3\right )} e^{\left (-\frac {1}{3} \, {\left (x e^{3} + e^{\left (-x e^{3} + 7\right )}\right )} e^{\left (e^{5} - 3\right )}\right )} \]
integrate(1/3*(3*exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp(5)))+((-3+x)*exp(3)* exp(-x*exp(3)+4)+3-x)*exp(exp(5))+3)/exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp( 5))),x, algorithm=\
(x*e^(1/3*(x*e^3 + e^(-x*e^3 + 7))*e^(e^5 - 3)) + x - 3)*e^(-1/3*(x*e^3 + e^(-x*e^3 + 7))*e^(e^5 - 3))
Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {1}{3} e^{-\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )} \left (3+3 e^{\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )}+e^{e^5} \left (3+e^{7-e^3 x} (-3+x)-x\right )\right ) \, dx=x + \left (x - 3\right ) e^{- \left (\frac {x}{3} + \frac {e^{- x e^{3} + 4}}{3}\right ) e^{e^{5}}} \]
integrate(1/3*(3*exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp(5)))+((-3+x)*exp(3)* exp(-x*exp(3)+4)+3-x)*exp(exp(5))+3)/exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp( 5))),x)
Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{3} e^{-\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )} \left (3+3 e^{\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )}+e^{e^5} \left (3+e^{7-e^3 x} (-3+x)-x\right )\right ) \, dx={\left (x - 3\right )} e^{\left (-\frac {1}{3} \, x e^{\left (e^{5}\right )} - \frac {1}{3} \, e^{\left (-x e^{3} + e^{5} + 4\right )}\right )} + x \]
integrate(1/3*(3*exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp(5)))+((-3+x)*exp(3)* exp(-x*exp(3)+4)+3-x)*exp(exp(5))+3)/exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp( 5))),x, algorithm=\
\[ \int \frac {1}{3} e^{-\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )} \left (3+3 e^{\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )}+e^{e^5} \left (3+e^{7-e^3 x} (-3+x)-x\right )\right ) \, dx=\int { \frac {1}{3} \, {\left ({\left ({\left (x - 3\right )} e^{\left (-x e^{3} + 7\right )} - x + 3\right )} e^{\left (e^{5}\right )} + 3 \, e^{\left (\frac {1}{3} \, {\left (x + e^{\left (-x e^{3} + 4\right )}\right )} e^{\left (e^{5}\right )}\right )} + 3\right )} e^{\left (-\frac {1}{3} \, {\left (x + e^{\left (-x e^{3} + 4\right )}\right )} e^{\left (e^{5}\right )}\right )} \,d x } \]
integrate(1/3*(3*exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp(5)))+((-3+x)*exp(3)* exp(-x*exp(3)+4)+3-x)*exp(exp(5))+3)/exp(1/3*(exp(-x*exp(3)+4)+x)*exp(exp( 5))),x, algorithm=\
integrate(1/3*(((x - 3)*e^(-x*e^3 + 7) - x + 3)*e^(e^5) + 3*e^(1/3*(x + e^ (-x*e^3 + 4))*e^(e^5)) + 3)*e^(-1/3*(x + e^(-x*e^3 + 4))*e^(e^5)), x)
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {1}{3} e^{-\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )} \left (3+3 e^{\frac {1}{3} e^{e^5} \left (e^{4-e^3 x}+x\right )}+e^{e^5} \left (3+e^{7-e^3 x} (-3+x)-x\right )\right ) \, dx=x-3\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^5}}{3}-\frac {{\mathrm {e}}^4\,{\mathrm {e}}^{-x\,{\mathrm {e}}^3}\,{\mathrm {e}}^{{\mathrm {e}}^5}}{3}}+x\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^5}}{3}-\frac {{\mathrm {e}}^4\,{\mathrm {e}}^{-x\,{\mathrm {e}}^3}\,{\mathrm {e}}^{{\mathrm {e}}^5}}{3}} \]
int(exp(-(exp(exp(5))*(x + exp(4 - x*exp(3))))/3)*(exp((exp(exp(5))*(x + e xp(4 - x*exp(3))))/3) + (exp(exp(5))*(exp(4 - x*exp(3))*exp(3)*(x - 3) - x + 3))/3 + 1),x)