3.66.12 \(\int \frac {e^{\frac {1}{4} e^{\frac {x^2+2 x \log (4)+\log ^2(4)}{\log ^2(\frac {2 x^2}{5})}}+\frac {x^2+2 x \log (4)+\log ^2(4)}{\log ^2(\frac {2 x^2}{5})}} (-6 x^2-12 x \log (4)-6 \log ^2(4)+(3 x^2+3 x \log (4)) \log (\frac {2 x^2}{5}))}{2 x \log ^3(\frac {2 x^2}{5})} \, dx\)

Optimal. Leaf size=27 \[ 3 e^{\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}} \]

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Rubi [F]  time = 17.38, 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 \frac {\exp \left (\frac {1}{4} e^{\frac {x^2+2 x \log (4)+\log ^2(4)}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {x^2+2 x \log (4)+\log ^2(4)}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) \left (-6 x^2-12 x \log (4)-6 \log ^2(4)+\left (3 x^2+3 x \log (4)\right ) \log \left (\frac {2 x^2}{5}\right )\right )}{2 x \log ^3\left (\frac {2 x^2}{5}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((x^2 + 2*x*Log[4] + Log[4]^2)/Log[(2*x^2)/5]^2)/4 + (x^2 + 2*x*Log[4] + Log[4]^2)/Log[(2*x^2)/5]^2)
*(-6*x^2 - 12*x*Log[4] - 6*Log[4]^2 + (3*x^2 + 3*x*Log[4])*Log[(2*x^2)/5]))/(2*x*Log[(2*x^2)/5]^3),x]

[Out]

-6*Log[4]*Defer[Int][E^(E^((x + Log[4])^2/Log[(2*x^2)/5]^2)/4 + (x + Log[4])^2/Log[(2*x^2)/5]^2)/Log[(2*x^2)/5
]^3, x] - 3*Log[4]^2*Defer[Int][E^(E^((x + Log[4])^2/Log[(2*x^2)/5]^2)/4 + (x + Log[4])^2/Log[(2*x^2)/5]^2)/(x
*Log[(2*x^2)/5]^3), x] - 3*Defer[Int][(E^(E^((x + Log[4])^2/Log[(2*x^2)/5]^2)/4 + (x + Log[4])^2/Log[(2*x^2)/5
]^2)*x)/Log[(2*x^2)/5]^3, x] + (3*Log[4]*Defer[Int][E^(E^((x + Log[4])^2/Log[(2*x^2)/5]^2)/4 + (x + Log[4])^2/
Log[(2*x^2)/5]^2)/Log[(2*x^2)/5]^2, x])/2 + (3*Defer[Int][(E^(E^((x + Log[4])^2/Log[(2*x^2)/5]^2)/4 + (x + Log
[4])^2/Log[(2*x^2)/5]^2)*x)/Log[(2*x^2)/5]^2, x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {\exp \left (\frac {1}{4} e^{\frac {x^2+2 x \log (4)+\log ^2(4)}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {x^2+2 x \log (4)+\log ^2(4)}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) \left (-6 x^2-12 x \log (4)-6 \log ^2(4)+\left (3 x^2+3 x \log (4)\right ) \log \left (\frac {2 x^2}{5}\right )\right )}{x \log ^3\left (\frac {2 x^2}{5}\right )} \, dx\\ &=\frac {1}{2} \int \frac {3 \exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) (x+\log (4)) \left (-2 (x+\log (4))+x \log \left (\frac {2 x^2}{5}\right )\right )}{x \log ^3\left (\frac {2 x^2}{5}\right )} \, dx\\ &=\frac {3}{2} \int \frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) (x+\log (4)) \left (-2 (x+\log (4))+x \log \left (\frac {2 x^2}{5}\right )\right )}{x \log ^3\left (\frac {2 x^2}{5}\right )} \, dx\\ &=\frac {3}{2} \int \left (-\frac {2 \exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) (x+\log (4))^2}{x \log ^3\left (\frac {2 x^2}{5}\right )}+\frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) (x+\log (4))}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) \, dx\\ &=\frac {3}{2} \int \frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) (x+\log (4))}{\log ^2\left (\frac {2 x^2}{5}\right )} \, dx-3 \int \frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) (x+\log (4))^2}{x \log ^3\left (\frac {2 x^2}{5}\right )} \, dx\\ &=\frac {3}{2} \int \left (\frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) x}{\log ^2\left (\frac {2 x^2}{5}\right )}+\frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) \log (4)}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) \, dx-3 \int \left (\frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) x}{\log ^3\left (\frac {2 x^2}{5}\right )}+\frac {2 \exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) \log (4)}{\log ^3\left (\frac {2 x^2}{5}\right )}+\frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) \log ^2(4)}{x \log ^3\left (\frac {2 x^2}{5}\right )}\right ) \, dx\\ &=\frac {3}{2} \int \frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) x}{\log ^2\left (\frac {2 x^2}{5}\right )} \, dx-3 \int \frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right ) x}{\log ^3\left (\frac {2 x^2}{5}\right )} \, dx+\frac {1}{2} (3 \log (4)) \int \frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right )}{\log ^2\left (\frac {2 x^2}{5}\right )} \, dx-(6 \log (4)) \int \frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right )}{\log ^3\left (\frac {2 x^2}{5}\right )} \, dx-\left (3 \log ^2(4)\right ) \int \frac {\exp \left (\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}+\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}\right )}{x \log ^3\left (\frac {2 x^2}{5}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 4.43, size = 27, normalized size = 1.00 \begin {gather*} 3 e^{\frac {1}{4} e^{\frac {(x+\log (4))^2}{\log ^2\left (\frac {2 x^2}{5}\right )}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((x^2 + 2*x*Log[4] + Log[4]^2)/Log[(2*x^2)/5]^2)/4 + (x^2 + 2*x*Log[4] + Log[4]^2)/Log[(2*x^2)
/5]^2)*(-6*x^2 - 12*x*Log[4] - 6*Log[4]^2 + (3*x^2 + 3*x*Log[4])*Log[(2*x^2)/5]))/(2*x*Log[(2*x^2)/5]^3),x]

[Out]

3*E^(E^((x + Log[4])^2/Log[(2*x^2)/5]^2)/4)

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fricas [B]  time = 0.71, size = 90, normalized size = 3.33 \begin {gather*} 3 \, e^{\left (\frac {e^{\left (\frac {x^{2} + 4 \, x \log \relax (2) + 4 \, \log \relax (2)^{2}}{\log \left (\frac {2}{5} \, x^{2}\right )^{2}}\right )} \log \left (\frac {2}{5} \, x^{2}\right )^{2} + 4 \, x^{2} + 16 \, x \log \relax (2) + 16 \, \log \relax (2)^{2}}{4 \, \log \left (\frac {2}{5} \, x^{2}\right )^{2}} - \frac {x^{2} + 4 \, x \log \relax (2) + 4 \, \log \relax (2)^{2}}{\log \left (\frac {2}{5} \, x^{2}\right )^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((6*x*log(2)+3*x^2)*log(2/5*x^2)-24*log(2)^2-24*x*log(2)-6*x^2)*exp((4*log(2)^2+4*x*log(2)+x^2)/
log(2/5*x^2)^2)*exp(1/4*exp((4*log(2)^2+4*x*log(2)+x^2)/log(2/5*x^2)^2))/x/log(2/5*x^2)^3,x, algorithm="fricas
")

[Out]

3*e^(1/4*(e^((x^2 + 4*x*log(2) + 4*log(2)^2)/log(2/5*x^2)^2)*log(2/5*x^2)^2 + 4*x^2 + 16*x*log(2) + 16*log(2)^
2)/log(2/5*x^2)^2 - (x^2 + 4*x*log(2) + 4*log(2)^2)/log(2/5*x^2)^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, {\left (2 \, x^{2} + 8 \, x \log \relax (2) + 8 \, \log \relax (2)^{2} - {\left (x^{2} + 2 \, x \log \relax (2)\right )} \log \left (\frac {2}{5} \, x^{2}\right )\right )} e^{\left (\frac {x^{2} + 4 \, x \log \relax (2) + 4 \, \log \relax (2)^{2}}{\log \left (\frac {2}{5} \, x^{2}\right )^{2}} + \frac {1}{4} \, e^{\left (\frac {x^{2} + 4 \, x \log \relax (2) + 4 \, \log \relax (2)^{2}}{\log \left (\frac {2}{5} \, x^{2}\right )^{2}}\right )}\right )}}{2 \, x \log \left (\frac {2}{5} \, x^{2}\right )^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((6*x*log(2)+3*x^2)*log(2/5*x^2)-24*log(2)^2-24*x*log(2)-6*x^2)*exp((4*log(2)^2+4*x*log(2)+x^2)/
log(2/5*x^2)^2)*exp(1/4*exp((4*log(2)^2+4*x*log(2)+x^2)/log(2/5*x^2)^2))/x/log(2/5*x^2)^3,x, algorithm="giac")

[Out]

integrate(-3/2*(2*x^2 + 8*x*log(2) + 8*log(2)^2 - (x^2 + 2*x*log(2))*log(2/5*x^2))*e^((x^2 + 4*x*log(2) + 4*lo
g(2)^2)/log(2/5*x^2)^2 + 1/4*e^((x^2 + 4*x*log(2) + 4*log(2)^2)/log(2/5*x^2)^2))/(x*log(2/5*x^2)^3), x)

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maple [A]  time = 0.41, size = 24, normalized size = 0.89

method	result	size
risch	\(3 \,{\mathrm e}^{\frac {{\mathrm e}^{\frac {\left (x +2 \ln \relax (2)\right )^{2}}{\ln \left (\frac {2 x^{2}}{5}\right )^{2}}}}{4}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((6*x*ln(2)+3*x^2)*ln(2/5*x^2)-24*ln(2)^2-24*x*ln(2)-6*x^2)*exp((4*ln(2)^2+4*x*ln(2)+x^2)/ln(2/5*x^2)^
2)*exp(1/4*exp((4*ln(2)^2+4*x*ln(2)+x^2)/ln(2/5*x^2)^2))/x/ln(2/5*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

3*exp(1/4*exp((x+2*ln(2))^2/ln(2/5*x^2)^2))

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maxima [B]  time = 0.82, size = 124, normalized size = 4.59 \begin {gather*} 3 \, e^{\left (\frac {1}{4} \, e^{\left (\frac {x^{2}}{\log \relax (5)^{2} - 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2} - 4 \, {\left (\log \relax (5) - \log \relax (2)\right )} \log \relax (x) + 4 \, \log \relax (x)^{2}} + \frac {4 \, x \log \relax (2)}{\log \relax (5)^{2} - 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2} - 4 \, {\left (\log \relax (5) - \log \relax (2)\right )} \log \relax (x) + 4 \, \log \relax (x)^{2}} + \frac {4 \, \log \relax (2)^{2}}{\log \relax (5)^{2} - 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2} - 4 \, {\left (\log \relax (5) - \log \relax (2)\right )} \log \relax (x) + 4 \, \log \relax (x)^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((6*x*log(2)+3*x^2)*log(2/5*x^2)-24*log(2)^2-24*x*log(2)-6*x^2)*exp((4*log(2)^2+4*x*log(2)+x^2)/
log(2/5*x^2)^2)*exp(1/4*exp((4*log(2)^2+4*x*log(2)+x^2)/log(2/5*x^2)^2))/x/log(2/5*x^2)^3,x, algorithm="maxima
")

[Out]

3*e^(1/4*e^(x^2/(log(5)^2 - 2*log(5)*log(2) + log(2)^2 - 4*(log(5) - log(2))*log(x) + 4*log(x)^2) + 4*x*log(2)
/(log(5)^2 - 2*log(5)*log(2) + log(2)^2 - 4*(log(5) - log(2))*log(x) + 4*log(x)^2) + 4*log(2)^2/(log(5)^2 - 2*
log(5)*log(2) + log(2)^2 - 4*(log(5) - log(2))*log(x) + 4*log(x)^2)))

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mupad [B]  time = 4.80, size = 139, normalized size = 5.15 \begin {gather*} 3\,{\mathrm {e}}^{\frac {2^{\frac {4\,x}{2\,\ln \left (x^2\right )\,\ln \relax (2)-2\,\ln \left (x^2\right )\,\ln \relax (5)-2\,\ln \relax (2)\,\ln \relax (5)+{\ln \left (x^2\right )}^2+{\ln \relax (2)}^2+{\ln \relax (5)}^2}}\,{\mathrm {e}}^{\frac {4\,{\ln \relax (2)}^2}{2\,\ln \left (x^2\right )\,\ln \relax (2)-2\,\ln \left (x^2\right )\,\ln \relax (5)-2\,\ln \relax (2)\,\ln \relax (5)+{\ln \left (x^2\right )}^2+{\ln \relax (2)}^2+{\ln \relax (5)}^2}}\,{\mathrm {e}}^{\frac {x^2}{2\,\ln \left (x^2\right )\,\ln \relax (2)-2\,\ln \left (x^2\right )\,\ln \relax (5)-2\,\ln \relax (2)\,\ln \relax (5)+{\ln \left (x^2\right )}^2+{\ln \relax (2)}^2+{\ln \relax (5)}^2}}}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp((4*x*log(2) + 4*log(2)^2 + x^2)/log((2*x^2)/5)^2)/4)*exp((4*x*log(2) + 4*log(2)^2 + x^2)/log((2*
x^2)/5)^2)*(24*x*log(2) + 24*log(2)^2 + 6*x^2 - log((2*x^2)/5)*(6*x*log(2) + 3*x^2)))/(2*x*log((2*x^2)/5)^3),x
)

[Out]

3*exp((2^((4*x)/(2*log(x^2)*log(2) - 2*log(x^2)*log(5) - 2*log(2)*log(5) + log(x^2)^2 + log(2)^2 + log(5)^2))*
exp((4*log(2)^2)/(2*log(x^2)*log(2) - 2*log(x^2)*log(5) - 2*log(2)*log(5) + log(x^2)^2 + log(2)^2 + log(5)^2))
*exp(x^2/(2*log(x^2)*log(2) - 2*log(x^2)*log(5) - 2*log(2)*log(5) + log(x^2)^2 + log(2)^2 + log(5)^2)))/4)

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sympy [A]  time = 1.68, size = 32, normalized size = 1.19 \begin {gather*} 3 e^{\frac {e^{\frac {x^{2} + 4 x \log {\relax (2 )} + 4 \log {\relax (2 )}^{2}}{\log {\left (\frac {2 x^{2}}{5} \right )}^{2}}}}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((6*x*ln(2)+3*x**2)*ln(2/5*x**2)-24*ln(2)**2-24*x*ln(2)-6*x**2)*exp((4*ln(2)**2+4*x*ln(2)+x**2)/
ln(2/5*x**2)**2)*exp(1/4*exp((4*ln(2)**2+4*x*ln(2)+x**2)/ln(2/5*x**2)**2))/x/ln(2/5*x**2)**3,x)

[Out]

3*exp(exp((x**2 + 4*x*log(2) + 4*log(2)**2)/log(2*x**2/5)**2)/4)

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