3.87.64 \(\int (e^x x)^{4 x^2} (\frac {2 \log (3)}{9})^{4 x^2} (4 x+4 x^2+8 x \log (\frac {2}{9} e^x x \log (3))) \, dx\)

Optimal. Leaf size=24 \[ \left (e^x x\right )^{4 x^2} \left (\frac {2 \log (3)}{9}\right )^{4 x^2} \]

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Rubi [F]  time = 180.01, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*x)^(4*x^2)*((2*Log[3])/9)^(4*x^2)*(4*x + 4*x^2 + 8*x*Log[(2*E^x*x*Log[3])/9]),x]

[Out]

$Aborted

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 2^{2+4 x^2} x \left (e^x x\right )^{4 x^2} \left (\frac {\log (3)}{9}\right )^{4 x^2} \left (1+x+2 \log \left (\frac {2}{9} e^x x \log (3)\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.48, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e^x x\right )^{4 x^2} \left (\frac {2 \log (3)}{9}\right )^{4 x^2} \left (4 x+4 x^2+8 x \log \left (\frac {2}{9} e^x x \log (3)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^x*x)^(4*x^2)*((2*Log[3])/9)^(4*x^2)*(4*x + 4*x^2 + 8*x*Log[(2*E^x*x*Log[3])/9]),x]

[Out]

Integrate[(E^x*x)^(4*x^2)*((2*Log[3])/9)^(4*x^2)*(4*x + 4*x^2 + 8*x*Log[(2*E^x*x*Log[3])/9]), x]

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fricas [A]  time = 0.51, size = 13, normalized size = 0.54 \begin {gather*} \left (\frac {2}{9} \, x e^{x} \log \relax (3)\right )^{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(2/9*x*log(3)*exp(x))+4*x^2+4*x)*exp(4*x^2*log(2/9*x*log(3)*exp(x))),x, algorithm="fricas")

[Out]

(2/9*x*e^x*log(3))^(4*x^2)

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giac [A]  time = 0.22, size = 25, normalized size = 1.04 \begin {gather*} e^{\left (4 \, x^{3} - 8 \, x^{2} \log \relax (3) + 4 \, x^{2} \log \left (2 \, x \log \relax (3)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(2/9*x*log(3)*exp(x))+4*x^2+4*x)*exp(4*x^2*log(2/9*x*log(3)*exp(x))),x, algorithm="giac")

[Out]

e^(4*x^3 - 8*x^2*log(3) + 4*x^2*log(2*x*log(3)))

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maple [C]  time = 0.20, size = 102, normalized size = 4.25

method	result	size
risch	\({\mathrm e}^{2 x^{2} \left (-i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{3}+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+2 \ln \relax (x )+2 \ln \left (\ln \relax (3)\right )+2 \ln \relax (2)-4 \ln \relax (3)+2 \ln \left ({\mathrm e}^{x}\right )\right )}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x*ln(2/9*x*ln(3)*exp(x))+4*x^2+4*x)*exp(4*x^2*ln(2/9*x*ln(3)*exp(x))),x,method=_RETURNVERBOSE)

[Out]

exp(2*x^2*(-I*Pi*csgn(I*x*exp(x))^3+I*Pi*csgn(I*x*exp(x))^2*csgn(I*x)+I*Pi*csgn(I*x*exp(x))^2*csgn(I*exp(x))-I
*Pi*csgn(I*x*exp(x))*csgn(I*x)*csgn(I*exp(x))+2*ln(x)+2*ln(ln(3))+2*ln(2)-4*ln(3)+2*ln(exp(x))))

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maxima [B]  time = 0.49, size = 36, normalized size = 1.50 \begin {gather*} e^{\left (4 \, x^{3} - 8 \, x^{2} \log \relax (3) + 4 \, x^{2} \log \relax (2) + 4 \, x^{2} \log \relax (x) + 4 \, x^{2} \log \left (\log \relax (3)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*log(2/9*x*log(3)*exp(x))+4*x^2+4*x)*exp(4*x^2*log(2/9*x*log(3)*exp(x))),x, algorithm="maxima")

[Out]

e^(4*x^3 - 8*x^2*log(3) + 4*x^2*log(2) + 4*x^2*log(x) + 4*x^2*log(log(3)))

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mupad [B]  time = 5.52, size = 24, normalized size = 1.00 \begin {gather*} x^{4\,x^2}\,{\mathrm {e}}^{4\,x^3}\,{\left (\frac {16\,{\ln \relax (3)}^4}{6561}\right )}^{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*x^2*log((2*x*exp(x)*log(3))/9))*(4*x + 8*x*log((2*x*exp(x)*log(3))/9) + 4*x^2),x)

[Out]

x^(4*x^2)*exp(4*x^3)*((16*log(3)^4)/6561)^(x^2)

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sympy [A]  time = 117.13, size = 19, normalized size = 0.79 \begin {gather*} e^{4 x^{2} \log {\left (\frac {2 x e^{x} \log {\relax (3 )}}{9} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x*ln(2/9*x*ln(3)*exp(x))+4*x**2+4*x)*exp(4*x**2*ln(2/9*x*ln(3)*exp(x))),x)

[Out]

exp(4*x**2*log(2*x*exp(x)*log(3)/9))

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