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

Optimal. Leaf size=29 \[ e^2 \left (e^{2 x}-e^{3 \left (2-e^x-\frac {4}{x}\right ) x}\right ) \]

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Rubi [F]  time = 0.47, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 23, normalized size = 0.79 \begin {gather*} e^{2+2 x}-e^{-10+6 x-3 e^x x} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.50, size = 20, normalized size = 0.69 \begin {gather*} -e^{\left (-3 \, x e^{x} + 6 \, x - 10\right )} + e^{\left (2 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.05, size = 21, normalized size = 0.72

method	result	size
risch	\(-{\mathrm e}^{-10-3 \,{\mathrm e}^{x} x +6 x}+{\mathrm e}^{2 x +2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[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*x+2)

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maxima [A]  time = 0.43, size = 23, normalized size = 0.79 \begin {gather*} -{\left (e^{\left (-3 \, x e^{x} + 6 \, x\right )} - e^{\left (2 \, x + 12\right )}\right )} e^{\left (-10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.19, size = 22, normalized size = 0.76 \begin {gather*} {\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2-{\mathrm {e}}^{-3\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{-10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.37, size = 24, normalized size = 0.83 \begin {gather*} e^{2} e^{2 x} - e^{2} e^{- 3 x e^{x} + 6 x - 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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