3.44.41 \(\int e^{\frac {1}{2} (-e^x+x)} (-6 x^2-x^3+e^x x^3) \, dx\)

Optimal. Leaf size=20 \[ -4-2 e^{\frac {1}{2} \left (-e^x+x\right )} x^3 \]

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Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2288} \begin {gather*} -\frac {2 e^{\frac {1}{2} \left (x-e^x\right )} \left (x^3-e^x x^3\right )}{1-e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*E^((-E^x + x)/2)*(x^3 - E^x*x^3))/(1 - E^x)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {2 e^{\frac {1}{2} \left (-e^x+x\right )} \left (x^3-e^x x^3\right )}{1-e^x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.90 \begin {gather*} -2 e^{\frac {1}{2} \left (-e^x+x\right )} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*E^((-E^x + x)/2)*x^3

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fricas [A]  time = 0.69, size = 14, normalized size = 0.70 \begin {gather*} -2 \, x^{3} e^{\left (\frac {1}{2} \, x - \frac {1}{2} \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x^3-x^3-6*x^2)*exp(1/2*x-1/2*exp(x)),x, algorithm="fricas")

[Out]

-2*x^3*e^(1/2*x - 1/2*e^x)

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giac [A]  time = 0.20, size = 14, normalized size = 0.70 \begin {gather*} -2 \, x^{3} e^{\left (\frac {1}{2} \, x - \frac {1}{2} \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x^3-x^3-6*x^2)*exp(1/2*x-1/2*exp(x)),x, algorithm="giac")

[Out]

-2*x^3*e^(1/2*x - 1/2*e^x)

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maple [A]  time = 0.04, size = 15, normalized size = 0.75

method	result	size
norman	\(-2 x^{3} {\mathrm e}^{\frac {x}{2}-\frac {{\mathrm e}^{x}}{2}}\)	\(15\)
risch	\(-2 x^{3} {\mathrm e}^{\frac {x}{2}-\frac {{\mathrm e}^{x}}{2}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*x^3-x^3-6*x^2)*exp(1/2*x-1/2*exp(x)),x,method=_RETURNVERBOSE)

[Out]

-2*x^3*exp(1/2*x-1/2*exp(x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} e^{x} - x^{3} - 6 \, x^{2}\right )} e^{\left (\frac {1}{2} \, x - \frac {1}{2} \, e^{x}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x^3-x^3-6*x^2)*exp(1/2*x-1/2*exp(x)),x, algorithm="maxima")

[Out]

integrate((x^3*e^x - x^3 - 6*x^2)*e^(1/2*x - 1/2*e^x), x)

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mupad [B]  time = 3.19, size = 14, normalized size = 0.70 \begin {gather*} -2\,x^3\,{\mathrm {e}}^{x/2}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x^3*exp(x/2)*exp(-exp(x)/2)

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sympy [A]  time = 0.15, size = 15, normalized size = 0.75 \begin {gather*} - 2 x^{3} e^{\frac {x}{2} - \frac {e^{x}}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x**3-x**3-6*x**2)*exp(1/2*x-1/2*exp(x)),x)

[Out]

-2*x**3*exp(x/2 - exp(x)/2)

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