3.10.49 \(\int \frac {e^{-e^{x^2}+x} (4 e^{e^{x^2}-x}+4 x^2 \log (3)-8 e^{x^2} x^3 \log (3))}{x^2 \log (3)} \, dx\)

Optimal. Leaf size=31 \[ \frac {2 x+4 e^{-e^{x^2}+x} x-\frac {4+x}{\log (3)}}{x} \]

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Rubi [F]  time = 1.25, 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^{-e^{x^2}+x} \left (4 e^{e^{x^2}-x}+4 x^2 \log (3)-8 e^{x^2} x^3 \log (3)\right )}{x^2 \log (3)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-E^x^2 + x)*(4*E^(E^x^2 - x) + 4*x^2*Log[3] - 8*E^x^2*x^3*Log[3]))/(x^2*Log[3]),x]

[Out]

-4/(x*Log[3]) + 4*Defer[Int][E^(-E^x^2 + x), x] - 8*Defer[Int][E^(-E^x^2 + x + x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{-e^{x^2}+x} \left (4 e^{e^{x^2}-x}+4 x^2 \log (3)-8 e^{x^2} x^3 \log (3)\right )}{x^2} \, dx}{\log (3)}\\ &=\frac {\int \left (-8 e^{-e^{x^2}+x+x^2} x \log (3)+\frac {4 e^{-e^{x^2}} \left (e^{e^{x^2}}+e^x x^2 \log (3)\right )}{x^2}\right ) \, dx}{\log (3)}\\ &=-\left (8 \int e^{-e^{x^2}+x+x^2} x \, dx\right )+\frac {4 \int \frac {e^{-e^{x^2}} \left (e^{e^{x^2}}+e^x x^2 \log (3)\right )}{x^2} \, dx}{\log (3)}\\ &=-\left (8 \int e^{-e^{x^2}+x+x^2} x \, dx\right )+\frac {4 \int \left (\frac {1}{x^2}+e^{-e^{x^2}+x} \log (3)\right ) \, dx}{\log (3)}\\ &=-\frac {4}{x \log (3)}+4 \int e^{-e^{x^2}+x} \, dx-8 \int e^{-e^{x^2}+x+x^2} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 23, normalized size = 0.74 \begin {gather*} 4 \left (e^{-e^{x^2}+x}-\frac {1}{x \log (3)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^x^2 + x)*(4*E^(E^x^2 - x) + 4*x^2*Log[3] - 8*E^x^2*x^3*Log[3]))/(x^2*Log[3]),x]

[Out]

4*(E^(-E^x^2 + x) - 1/(x*Log[3]))

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fricas [A]  time = 0.88, size = 34, normalized size = 1.10 \begin {gather*} \frac {4 \, {\left (x \log \relax (3) - e^{\left (-x + e^{\left (x^{2}\right )}\right )}\right )} e^{\left (x - e^{\left (x^{2}\right )}\right )}}{x \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(exp(x^2)-x)-8*x^3*log(3)*exp(x^2)+4*x^2*log(3))/x^2/log(3)/exp(exp(x^2)-x),x, algorithm="fric
as")

[Out]

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

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giac [A]  time = 0.43, size = 24, normalized size = 0.77 \begin {gather*} \frac {4 \, {\left (x e^{\left (x - e^{\left (x^{2}\right )}\right )} \log \relax (3) - 1\right )}}{x \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(exp(x^2)-x)-8*x^3*log(3)*exp(x^2)+4*x^2*log(3))/x^2/log(3)/exp(exp(x^2)-x),x, algorithm="giac
")

[Out]

4*(x*e^(x - e^(x^2))*log(3) - 1)/(x*log(3))

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maple [A]  time = 0.07, size = 22, normalized size = 0.71

method	result	size
risch	\(-\frac {4}{x \ln \relax (3)}+4 \,{\mathrm e}^{-{\mathrm e}^{x^{2}}+x}\)	\(22\)
norman	\(\frac {\left (4 x -\frac {4 \,{\mathrm e}^{{\mathrm e}^{x^{2}}-x}}{\ln \relax (3)}\right ) {\mathrm e}^{-{\mathrm e}^{x^{2}}+x}}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(exp(x^2)-x)-8*x^3*ln(3)*exp(x^2)+4*x^2*ln(3))/x^2/ln(3)/exp(exp(x^2)-x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.85, size = 24, normalized size = 0.77 \begin {gather*} \frac {4 \, {\left (e^{\left (x - e^{\left (x^{2}\right )}\right )} \log \relax (3) - \frac {1}{x}\right )}}{\log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(exp(x^2)-x)-8*x^3*log(3)*exp(x^2)+4*x^2*log(3))/x^2/log(3)/exp(exp(x^2)-x),x, algorithm="maxi
ma")

[Out]

4*(e^(x - e^(x^2))*log(3) - 1/x)/log(3)

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mupad [B]  time = 0.86, size = 21, normalized size = 0.68 \begin {gather*} 4\,{\mathrm {e}}^{x-{\mathrm {e}}^{x^2}}-\frac {4}{x\,\ln \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - exp(x^2))*(4*exp(exp(x^2) - x) + 4*x^2*log(3) - 8*x^3*exp(x^2)*log(3)))/(x^2*log(3)),x)

[Out]

4*exp(x - exp(x^2)) - 4/(x*log(3))

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sympy [A]  time = 0.31, size = 15, normalized size = 0.48 \begin {gather*} 4 e^{x - e^{x^{2}}} - \frac {4}{x \log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(exp(x**2)-x)-8*x**3*ln(3)*exp(x**2)+4*x**2*ln(3))/x**2/ln(3)/exp(exp(x**2)-x),x)

[Out]

4*exp(x - exp(x**2)) - 4/(x*log(3))

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