3.61.6 \(\int e^{13+e^x-2 x-x^2} (e^{2 x}+e^{-13-e^x+2 x+x^2}+e^x (-1-2 x)) \, dx\)

Optimal. Leaf size=17 \[ e^{13+e^x-x-x^2}+x \]

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Rubi [F]  time = 0.38, 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 e^{13+e^x-2 x-x^2} \left (e^{2 x}+e^{-13-e^x+2 x+x^2}+e^x (-1-2 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(13 + E^x - 2*x - x^2)*(E^(2*x) + E^(-13 - E^x + 2*x + x^2) + E^x*(-1 - 2*x)),x]

[Out]

x + Defer[Int][E^(13 + E^x - x^2), x] - Defer[Int][E^(13 + E^x - x - x^2), x] - 2*Defer[Int][E^(13 + E^x - x -
x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+e^{13+e^x-x^2}-e^{13+e^x-x-x^2} (1+2 x)\right ) \, dx\\ &=x+\int e^{13+e^x-x^2} \, dx-\int e^{13+e^x-x-x^2} (1+2 x) \, dx\\ &=x+\int e^{13+e^x-x^2} \, dx-\int \left (e^{13+e^x-x-x^2}+2 e^{13+e^x-x-x^2} x\right ) \, dx\\ &=x-2 \int e^{13+e^x-x-x^2} x \, dx+\int e^{13+e^x-x^2} \, dx-\int e^{13+e^x-x-x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 17, normalized size = 1.00 \begin {gather*} e^{13+e^x-x-x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(13 + E^x - 2*x - x^2)*(E^(2*x) + E^(-13 - E^x + 2*x + x^2) + E^x*(-1 - 2*x)),x]

[Out]

E^(13 + E^x - x - x^2) + x

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fricas [A]  time = 0.68, size = 15, normalized size = 0.88 \begin {gather*} x + e^{\left (-x^{2} - x + e^{x} + 13\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-exp(x)+x^2+2*x-13)+exp(x)^2+(-2*x-1)*exp(x))/exp(-exp(x)+x^2+2*x-13),x, algorithm="fricas")

[Out]

x + e^(-x^2 - x + e^x + 13)

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giac [A]  time = 0.15, size = 20, normalized size = 1.18 \begin {gather*} {\left (x e^{x} + e^{\left (-x^{2} + e^{x} + 13\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-exp(x)+x^2+2*x-13)+exp(x)^2+(-2*x-1)*exp(x))/exp(-exp(x)+x^2+2*x-13),x, algorithm="giac")

[Out]

(x*e^x + e^(-x^2 + e^x + 13))*e^(-x)

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maple [A]  time = 0.05, size = 16, normalized size = 0.94

method	result	size
risch	\(x +{\mathrm e}^{-x +{\mathrm e}^{x}-x^{2}+13}\)	\(16\)
norman	\(\left (x \,{\mathrm e}^{-{\mathrm e}^{x}+x^{2}+2 x -13}+{\mathrm e}^{x}\right ) {\mathrm e}^{{\mathrm e}^{x}-x^{2}-2 x +13}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-exp(x)+x^2+2*x-13)+exp(x)^2+(-2*x-1)*exp(x))/exp(-exp(x)+x^2+2*x-13),x,method=_RETURNVERBOSE)

[Out]

x+exp(-x+exp(x)-x^2+13)

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maxima [A]  time = 0.41, size = 15, normalized size = 0.88 \begin {gather*} x + e^{\left (-x^{2} - x + e^{x} + 13\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-exp(x)+x^2+2*x-13)+exp(x)^2+(-2*x-1)*exp(x))/exp(-exp(x)+x^2+2*x-13),x, algorithm="maxima")

[Out]

x + e^(-x^2 - x + e^x + 13)

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mupad [B]  time = 0.13, size = 18, normalized size = 1.06 \begin {gather*} x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{13}\,{\mathrm {e}}^{-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(x) - 2*x - x^2 + 13)*(exp(2*x) + exp(2*x - exp(x) + x^2 - 13) - exp(x)*(2*x + 1)),x)

[Out]

x + exp(-x)*exp(exp(x))*exp(13)*exp(-x^2)

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sympy [A]  time = 0.18, size = 17, normalized size = 1.00 \begin {gather*} x + e^{x} e^{- x^{2} - 2 x + e^{x} + 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-exp(x)+x**2+2*x-13)+exp(x)**2+(-2*x-1)*exp(x))/exp(-exp(x)+x**2+2*x-13),x)

[Out]

x + exp(x)*exp(-x**2 - 2*x + exp(x) + 13)

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