3.25.47 \(\int \frac {e^{2+4 x-x^2} (32+128 x-64 x^2)}{-3+32 e^{2+4 x-x^2} x} \, dx\)

Optimal. Leaf size=20 \[ -5+\log \left (-3+32 e^{2+4 x-x^2} x\right ) \]

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Rubi [A]  time = 0.20, antiderivative size = 27, normalized size of antiderivative = 1.35, number of steps used = 1, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6684} \begin {gather*} \log \left (e^{-x^2} \left (3 e^{x^2}-32 e^{4 x+2} x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2 + 4*x - x^2)*(32 + 128*x - 64*x^2))/(-3 + 32*E^(2 + 4*x - x^2)*x),x]

[Out]

Log[(3*E^x^2 - 32*E^(2 + 4*x)*x)/E^x^2]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (e^{-x^2} \left (3 e^{x^2}-32 e^{2+4 x} x\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 33, normalized size = 1.65 \begin {gather*} 32 \left (-\frac {x^2}{32}+\frac {1}{32} \log \left (3 e^{x^2}-32 e^{2+4 x} x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2 + 4*x - x^2)*(32 + 128*x - 64*x^2))/(-3 + 32*E^(2 + 4*x - x^2)*x),x]

[Out]

32*(-1/32*x^2 + Log[3*E^x^2 - 32*E^(2 + 4*x)*x]/32)

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fricas [A]  time = 0.95, size = 24, normalized size = 1.20 \begin {gather*} \log \relax (x) + \log \left (\frac {32 \, x e^{\left (-x^{2} + 4 \, x + 2\right )} - 3}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2+128*x+32)*exp(-x^2+4*x+2)/(32*x*exp(-x^2+4*x+2)-3),x, algorithm="fricas")

[Out]

log(x) + log((32*x*e^(-x^2 + 4*x + 2) - 3)/x)

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giac [A]  time = 0.26, size = 17, normalized size = 0.85 \begin {gather*} \log \left (32 \, x e^{\left (-x^{2} + 4 \, x + 2\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2+128*x+32)*exp(-x^2+4*x+2)/(32*x*exp(-x^2+4*x+2)-3),x, algorithm="giac")

[Out]

log(32*x*e^(-x^2 + 4*x + 2) - 3)

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maple [A]  time = 0.04, size = 18, normalized size = 0.90

method	result	size
norman	\(\ln \left (32 x \,{\mathrm e}^{-x^{2}+4 x +2}-3\right )\)	\(18\)
risch	\(\ln \relax (x )-2+\ln \left ({\mathrm e}^{-x^{2}+4 x +2}-\frac {3}{32 x}\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-64*x^2+128*x+32)*exp(-x^2+4*x+2)/(32*x*exp(-x^2+4*x+2)-3),x,method=_RETURNVERBOSE)

[Out]

ln(32*x*exp(-x^2+4*x+2)-3)

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maxima [A]  time = 0.65, size = 21, normalized size = 1.05 \begin {gather*} -x^{2} + \log \left (-\frac {32}{3} \, x e^{\left (4 \, x + 2\right )} + e^{\left (x^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2+128*x+32)*exp(-x^2+4*x+2)/(32*x*exp(-x^2+4*x+2)-3),x, algorithm="maxima")

[Out]

-x^2 + log(-32/3*x*e^(4*x + 2) + e^(x^2))

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mupad [B]  time = 0.20, size = 18, normalized size = 0.90 \begin {gather*} \ln \left (32\,x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{-x^2}-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x - x^2 + 2)*(128*x - 64*x^2 + 32))/(32*x*exp(4*x - x^2 + 2) - 3),x)

[Out]

log(32*x*exp(4*x)*exp(2)*exp(-x^2) - 3)

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sympy [A]  time = 0.20, size = 19, normalized size = 0.95 \begin {gather*} \log {\relax (x )} + \log {\left (e^{- x^{2} + 4 x + 2} - \frac {3}{32 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x**2+128*x+32)*exp(-x**2+4*x+2)/(32*x*exp(-x**2+4*x+2)-3),x)

[Out]

log(x) + log(exp(-x**2 + 4*x + 2) - 3/(32*x))

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