\(\int \frac {e^2 (e^{2 x}-e^{-12+6 x-3 e^x x}) (-2 e^{2 x}+e^{-12+6 x-3 e^x x} (6+e^x (-3-3 x)))}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx\) [6845]

Optimal result

Integrand size = 81, antiderivative size = 29 \[ \int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx=e^2 \left (e^{2 x}-e^{3 \left (2-e^x-\frac {4}{x}\right ) x}\right ) \]

[Out]

exp(ln(exp(x)^2-exp(x*(6-12/x-3*exp(x))))+2)

Rubi [F]

\[ \int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx=\int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx \]

[In]

Int[(E^2*(E^(2*x) - E^(-12 + 6*x - 3*E^x*x))*(-2*E^(2*x) + E^(-12 + 6*x - 3*E^x*x)*(6 + E^x*(-3 - 3*x))))/(-E^
(2*x) + E^(-12 + 6*x - 3*E^x*x)),x]

[Out]

E^(2 + 2*x) + 3*E^2*Defer[Int][E^(-12 + 7*x - 3*E^x*x), x] - 6*E^2*Defer[Int][E^(-3*(4 + (-2 + E^x)*x)), x] +
3*E^2*Defer[Int][E^(-12 + 7*x - 3*E^x*x)*x, x]

Rubi steps \begin{align*} \text {integral}& = e^2 \int \frac {\left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx \\ & = e^2 \int \left (2 e^{2 x}-6 e^{-3 \left (4+\left (-2+e^x\right ) x\right )}+3 e^{-12+7 x-3 e^x x} (1+x)\right ) \, dx \\ & = \left (2 e^2\right ) \int e^{2 x} \, dx+\left (3 e^2\right ) \int e^{-12+7 x-3 e^x x} (1+x) \, dx-\left (6 e^2\right ) \int e^{-3 \left (4+\left (-2+e^x\right ) x\right )} \, dx \\ & = e^{2+2 x}+\left (3 e^2\right ) \int \left (e^{-12+7 x-3 e^x x}+e^{-12+7 x-3 e^x x} x\right ) \, dx-\left (6 e^2\right ) \int e^{-3 \left (4+\left (-2+e^x\right ) x\right )} \, dx \\ & = e^{2+2 x}+\left (3 e^2\right ) \int e^{-12+7 x-3 e^x x} \, dx+\left (3 e^2\right ) \int e^{-12+7 x-3 e^x x} x \, dx-\left (6 e^2\right ) \int e^{-3 \left (4+\left (-2+e^x\right ) x\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx=e^{2+2 x}-e^{-10+6 x-3 e^x x} \]

[In]

Integrate[(E^2*(E^(2*x) - E^(-12 + 6*x - 3*E^x*x))*(-2*E^(2*x) + E^(-12 + 6*x - 3*E^x*x)*(6 + E^x*(-3 - 3*x)))
)/(-E^(2*x) + E^(-12 + 6*x - 3*E^x*x)),x]

[Out]

E^(2 + 2*x) - E^(-10 + 6*x - 3*E^x*x)

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72

method	result	size
risch	\(-{\mathrm e}^{-10-3 \,{\mathrm e}^{x} x +6 x}+{\mathrm e}^{2+2 x}\)	\(21\)
parallelrisch	\(\frac {2 \,{\mathrm e}^{\ln \left (-{\mathrm e}^{-3 \,{\mathrm e}^{x} x +6 x -12}+{\mathrm e}^{2 x}\right )+2} {\mathrm e}^{4 x}-4 \,{\mathrm e}^{\ln \left (-{\mathrm e}^{-3 \,{\mathrm e}^{x} x +6 x -12}+{\mathrm e}^{2 x}\right )+2} {\mathrm e}^{2 x} {\mathrm e}^{-3 \,{\mathrm e}^{x} x +6 x -12}+2 \,{\mathrm e}^{\ln \left (-{\mathrm e}^{-3 \,{\mathrm e}^{x} x +6 x -12}+{\mathrm e}^{2 x}\right )+2} {\mathrm e}^{-6 \,{\mathrm e}^{x} x +12 x -24}}{2 \left (-{\mathrm e}^{-3 \,{\mathrm e}^{x} x +6 x -12}+{\mathrm e}^{2 x}\right )^{2}}\)	\(128\)

[In]

int((((-3*x-3)*exp(x)+6)*exp(-3*exp(x)*x+6*x-12)-2*exp(x)^2)*exp(ln(-exp(-3*exp(x)*x+6*x-12)+exp(x)^2)+2)/(exp
(-3*exp(x)*x+6*x-12)-exp(x)^2),x,method=_RETURNVERBOSE)

[Out]

-exp(-10-3*exp(x)*x+6*x)+exp(2+2*x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx=-e^{\left (-3 \, x e^{x} + 6 \, x - 10\right )} + e^{\left (2 \, x + 2\right )} \]

[In]

integrate((((-3*x-3)*exp(x)+6)*exp(-3*exp(x)*x+6*x-12)-2*exp(x)^2)*exp(log(-exp(-3*exp(x)*x+6*x-12)+exp(x)^2)+
2)/(exp(-3*exp(x)*x+6*x-12)-exp(x)^2),x, algorithm="fricas")

[Out]

-e^(-3*x*e^x + 6*x - 10) + e^(2*x + 2)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx=e^{2} e^{2 x} - e^{2} e^{- 3 x e^{x} + 6 x - 12} \]

[In]

integrate((((-3*x-3)*exp(x)+6)*exp(-3*exp(x)*x+6*x-12)-2*exp(x)**2)*exp(ln(-exp(-3*exp(x)*x+6*x-12)+exp(x)**2)
+2)/(exp(-3*exp(x)*x+6*x-12)-exp(x)**2),x)

[Out]

exp(2)*exp(2*x) - exp(2)*exp(-3*x*exp(x) + 6*x - 12)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx=-{\left (e^{\left (-3 \, x e^{x} + 6 \, x\right )} - e^{\left (2 \, x + 12\right )}\right )} e^{\left (-10\right )} \]

[In]

integrate((((-3*x-3)*exp(x)+6)*exp(-3*exp(x)*x+6*x-12)-2*exp(x)^2)*exp(log(-exp(-3*exp(x)*x+6*x-12)+exp(x)^2)+
2)/(exp(-3*exp(x)*x+6*x-12)-exp(x)^2),x, algorithm="maxima")

[Out]

-(e^(-3*x*e^x + 6*x) - e^(2*x + 12))*e^(-10)

Giac [F(-1)]

Timed out. \[ \int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx=\text {Timed out} \]

[In]

integrate((((-3*x-3)*exp(x)+6)*exp(-3*exp(x)*x+6*x-12)-2*exp(x)^2)*exp(log(-exp(-3*exp(x)*x+6*x-12)+exp(x)^2)+
2)/(exp(-3*exp(x)*x+6*x-12)-exp(x)^2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 11.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {e^2 \left (e^{2 x}-e^{-12+6 x-3 e^x x}\right ) \left (-2 e^{2 x}+e^{-12+6 x-3 e^x x} \left (6+e^x (-3-3 x)\right )\right )}{-e^{2 x}+e^{-12+6 x-3 e^x x}} \, dx={\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2-{\mathrm {e}}^{-3\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{-10} \]

[In]

int((exp(log(exp(2*x) - exp(6*x - 3*x*exp(x) - 12)) + 2)*(2*exp(2*x) + exp(6*x - 3*x*exp(x) - 12)*(exp(x)*(3*x
+ 3) - 6)))/(exp(2*x) - exp(6*x - 3*x*exp(x) - 12)),x)

[Out]

exp(2*x)*exp(2) - exp(-3*x*exp(x))*exp(6*x)*exp(-10)