3.29.53 \(\int \frac {e^{\frac {x^2}{e^{\frac {2}{3} (648+e^2 (8-3 x)-243 x)}+2 e^{\frac {1}{3} (648+e^2 (8-3 x)-243 x)} \log (x)+\log ^2(x)}} (-2 x+e^{\frac {1}{3} (648+e^2 (8-3 x)-243 x)} (2 x+162 x^2+2 e^2 x^2)+2 x \log (x))}{e^{648+e^2 (8-3 x)-243 x}+3 e^{\frac {2}{3} (648+e^2 (8-3 x)-243 x)} \log (x)+3 e^{\frac {1}{3} (648+e^2 (8-3 x)-243 x)} \log ^2(x)+\log ^3(x)} \, dx\)

Optimal. Leaf size=28 \[ 1+e^{\frac {x^2}{\left (e^{\left (81+e^2\right ) \left (\frac {8}{3}-x\right )}+\log (x)\right )^2}} \]

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Rubi [F]  time = 27.46, 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 {x^2}{e^{\frac {2}{3} \left (648+e^2 (8-3 x)-243 x\right )}+2 e^{\frac {1}{3} \left (648+e^2 (8-3 x)-243 x\right )} \log (x)+\log ^2(x)}\right ) \left (-2 x+e^{\frac {1}{3} \left (648+e^2 (8-3 x)-243 x\right )} \left (2 x+162 x^2+2 e^2 x^2\right )+2 x \log (x)\right )}{e^{648+e^2 (8-3 x)-243 x}+3 e^{\frac {2}{3} \left (648+e^2 (8-3 x)-243 x\right )} \log (x)+3 e^{\frac {1}{3} \left (648+e^2 (8-3 x)-243 x\right )} \log ^2(x)+\log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(x^2/(E^((2*(648 + E^2*(8 - 3*x) - 243*x))/3) + 2*E^((648 + E^2*(8 - 3*x) - 243*x)/3)*Log[x] + Log[x]^2
))*(-2*x + E^((648 + E^2*(8 - 3*x) - 243*x)/3)*(2*x + 162*x^2 + 2*E^2*x^2) + 2*x*Log[x]))/(E^(648 + E^2*(8 - 3
*x) - 243*x) + 3*E^((2*(648 + E^2*(8 - 3*x) - 243*x))/3)*Log[x] + 3*E^((648 + E^2*(8 - 3*x) - 243*x)/3)*Log[x]
^2 + Log[x]^3),x]

[Out]

2*(81 + E^2)*Defer[Int][(E^((8*(81 + E^2))/3 + 162*(1 - E^2/162)*x + (E^(162*x)*x^2)/(E^(216 + E^2*(8/3 - x))
+ E^(81*x)*Log[x])^2)*x^2)/(E^(216 + E^2*(8/3 - x)) + E^(81*x)*Log[x])^3, x] + 2*Defer[Int][(E^((8*(81 + E^2))
/3 + 162*(1 - E^2/162)*x + (E^(162*x)*x^2)/(E^(216 + E^2*(8/3 - x)) + E^(81*x)*Log[x])^2)*x)/(Log[x]*(E^(216 +
E^2*(8/3 - x)) + E^(81*x)*Log[x])^3), x] + 2*Defer[Int][(E^(162*x + (E^(162*x)*x^2)/(E^(216 + E^2*(8/3 - x))
+ E^(81*x)*Log[x])^2)*x)/(E^(216 + E^2*(8/3 - x)) + E^(81*x)*Log[x])^2, x] - 2*Defer[Int][(E^(162*x + (E^(162*
x)*x^2)/(E^(216 + E^2*(8/3 - x)) + E^(81*x)*Log[x])^2)*x)/(Log[x]*(E^(216 + E^2*(8/3 - x)) + E^(81*x)*Log[x])^
2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x \left (-e^{81 x}+e^{218+e^2 \left (\frac {8}{3}-x\right )} x+e^{216+e^2 \left (\frac {8}{3}-x\right )} (1+81 x)+e^{81 x} \log (x)\right )}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3} \, dx\\ &=2 \int \frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x \left (-e^{81 x}+e^{218+e^2 \left (\frac {8}{3}-x\right )} x+e^{216+e^2 \left (\frac {8}{3}-x\right )} (1+81 x)+e^{81 x} \log (x)\right )}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3} \, dx\\ &=2 \int \left (\frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x (-1+\log (x))}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}+\frac {\exp \left (\frac {8}{3} \left (81+e^2\right )+162 x-e^2 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x \left (1+81 \left (1+\frac {e^2}{81}\right ) x \log (x)\right )}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3}\right ) \, dx\\ &=2 \int \frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x (-1+\log (x))}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2} \, dx+2 \int \frac {\exp \left (\frac {8}{3} \left (81+e^2\right )+162 x-e^2 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x \left (1+81 \left (1+\frac {e^2}{81}\right ) x \log (x)\right )}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3} \, dx\\ &=2 \int \frac {\exp \left (\frac {8}{3} \left (81+e^2\right )+162 \left (1-\frac {e^2}{162}\right ) x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x \left (1+81 \left (1+\frac {e^2}{81}\right ) x \log (x)\right )}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3} \, dx+2 \int \left (\frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}-\frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) \, dx\\ &=2 \int \frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2} \, dx-2 \int \frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2} \, dx+2 \int \left (\frac {\exp \left (\frac {8}{3} \left (81+e^2\right )+162 \left (1-\frac {e^2}{162}\right ) x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) \left (81+e^2\right ) x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3}+\frac {\exp \left (\frac {8}{3} \left (81+e^2\right )+162 \left (1-\frac {e^2}{162}\right ) x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {8}{3} \left (81+e^2\right )+162 \left (1-\frac {e^2}{162}\right ) x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3} \, dx+2 \int \frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2} \, dx-2 \int \frac {\exp \left (162 x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x}{\log (x) \left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2} \, dx+\left (2 \left (81+e^2\right )\right ) \int \frac {\exp \left (\frac {8}{3} \left (81+e^2\right )+162 \left (1-\frac {e^2}{162}\right ) x+\frac {e^{162 x} x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^2}\right ) x^2}{\left (e^{216+e^2 \left (\frac {8}{3}-x\right )}+e^{81 x} \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 39, normalized size = 1.39 \begin {gather*} e^{\frac {e^{162 x} x^2}{\left (e^{216+\frac {8 e^2}{3}-e^2 x}+e^{81 x} \log (x)\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2/(E^((2*(648 + E^2*(8 - 3*x) - 243*x))/3) + 2*E^((648 + E^2*(8 - 3*x) - 243*x)/3)*Log[x] + Lo
g[x]^2))*(-2*x + E^((648 + E^2*(8 - 3*x) - 243*x)/3)*(2*x + 162*x^2 + 2*E^2*x^2) + 2*x*Log[x]))/(E^(648 + E^2*
(8 - 3*x) - 243*x) + 3*E^((2*(648 + E^2*(8 - 3*x) - 243*x))/3)*Log[x] + 3*E^((648 + E^2*(8 - 3*x) - 243*x)/3)*
Log[x]^2 + Log[x]^3),x]

[Out]

E^((E^(162*x)*x^2)/(E^(216 + (8*E^2)/3 - E^2*x) + E^(81*x)*Log[x])^2)

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fricas [A]  time = 0.73, size = 46, normalized size = 1.64 \begin {gather*} e^{\left (\frac {x^{2}}{2 \, e^{\left (-\frac {1}{3} \, {\left (3 \, x - 8\right )} e^{2} - 81 \, x + 216\right )} \log \relax (x) + \log \relax (x)^{2} + e^{\left (-\frac {2}{3} \, {\left (3 \, x - 8\right )} e^{2} - 162 \, x + 432\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)+(2*x^2*exp(2)+162*x^2+2*x)*exp(1/3*(-3*x+8)*exp(2)-81*x+216)-2*x)*exp(x^2/(log(x)^2+2*ex
p(1/3*(-3*x+8)*exp(2)-81*x+216)*log(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)^2))/(log(x)^3+3*exp(1/3*(-3*x+8)*exp(
2)-81*x+216)*log(x)^2+3*exp(1/3*(-3*x+8)*exp(2)-81*x+216)^2*log(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)^3),x, alg
orithm="fricas")

[Out]

e^(x^2/(2*e^(-1/3*(3*x - 8)*e^2 - 81*x + 216)*log(x) + log(x)^2 + e^(-2/3*(3*x - 8)*e^2 - 162*x + 432)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left ({\left (x^{2} e^{2} + 81 \, x^{2} + x\right )} e^{\left (-\frac {1}{3} \, {\left (3 \, x - 8\right )} e^{2} - 81 \, x + 216\right )} + x \log \relax (x) - x\right )} e^{\left (\frac {x^{2}}{2 \, e^{\left (-\frac {1}{3} \, {\left (3 \, x - 8\right )} e^{2} - 81 \, x + 216\right )} \log \relax (x) + \log \relax (x)^{2} + e^{\left (-\frac {2}{3} \, {\left (3 \, x - 8\right )} e^{2} - 162 \, x + 432\right )}}\right )}}{3 \, e^{\left (-\frac {1}{3} \, {\left (3 \, x - 8\right )} e^{2} - 81 \, x + 216\right )} \log \relax (x)^{2} + \log \relax (x)^{3} + 3 \, e^{\left (-\frac {2}{3} \, {\left (3 \, x - 8\right )} e^{2} - 162 \, x + 432\right )} \log \relax (x) + e^{\left (-{\left (3 \, x - 8\right )} e^{2} - 243 \, x + 648\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)+(2*x^2*exp(2)+162*x^2+2*x)*exp(1/3*(-3*x+8)*exp(2)-81*x+216)-2*x)*exp(x^2/(log(x)^2+2*ex
p(1/3*(-3*x+8)*exp(2)-81*x+216)*log(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)^2))/(log(x)^3+3*exp(1/3*(-3*x+8)*exp(
2)-81*x+216)*log(x)^2+3*exp(1/3*(-3*x+8)*exp(2)-81*x+216)^2*log(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)^3),x, alg
orithm="giac")

[Out]

integrate(2*((x^2*e^2 + 81*x^2 + x)*e^(-1/3*(3*x - 8)*e^2 - 81*x + 216) + x*log(x) - x)*e^(x^2/(2*e^(-1/3*(3*x
- 8)*e^2 - 81*x + 216)*log(x) + log(x)^2 + e^(-2/3*(3*x - 8)*e^2 - 162*x + 432)))/(3*e^(-1/3*(3*x - 8)*e^2 -
81*x + 216)*log(x)^2 + log(x)^3 + 3*e^(-2/3*(3*x - 8)*e^2 - 162*x + 432)*log(x) + e^(-(3*x - 8)*e^2 - 243*x +
648)), x)

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maple [A]  time = 0.05, size = 41, normalized size = 1.46

method	result	size
risch	\({\mathrm e}^{\frac {x^{2}}{\ln \relax (x )^{2}+2 \,{\mathrm e}^{-\frac {\left (3 x -8\right ) \left (81+{\mathrm e}^{2}\right )}{3}} \ln \relax (x )+{\mathrm e}^{-\frac {2 \left (3 x -8\right ) \left (81+{\mathrm e}^{2}\right )}{3}}}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*ln(x)+(2*x^2*exp(2)+162*x^2+2*x)*exp(1/3*(-3*x+8)*exp(2)-81*x+216)-2*x)*exp(x^2/(ln(x)^2+2*exp(1/3*(-
3*x+8)*exp(2)-81*x+216)*ln(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)^2))/(ln(x)^3+3*exp(1/3*(-3*x+8)*exp(2)-81*x+21
6)*ln(x)^2+3*exp(1/3*(-3*x+8)*exp(2)-81*x+216)^2*ln(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)^3),x,method=_RETURNVE
RBOSE)

[Out]

exp(x^2/(ln(x)^2+2*exp(-1/3*(3*x-8)*(81+exp(2)))*ln(x)+exp(-2/3*(3*x-8)*(81+exp(2)))))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)+(2*x^2*exp(2)+162*x^2+2*x)*exp(1/3*(-3*x+8)*exp(2)-81*x+216)-2*x)*exp(x^2/(log(x)^2+2*ex
p(1/3*(-3*x+8)*exp(2)-81*x+216)*log(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)^2))/(log(x)^3+3*exp(1/3*(-3*x+8)*exp(
2)-81*x+216)*log(x)^2+3*exp(1/3*(-3*x+8)*exp(2)-81*x+216)^2*log(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)^3),x, alg
orithm="maxima")

[Out]

Timed out

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mupad [B]  time = 2.16, size = 51, normalized size = 1.82 \begin {gather*} {\mathrm {e}}^{\frac {x^2}{{\ln \relax (x)}^2+2\,{\mathrm {e}}^{\frac {8\,{\mathrm {e}}^2}{3}}\,{\mathrm {e}}^{-81\,x}\,{\mathrm {e}}^{216}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^2}\,\ln \relax (x)+{\mathrm {e}}^{\frac {16\,{\mathrm {e}}^2}{3}}\,{\mathrm {e}}^{-162\,x}\,{\mathrm {e}}^{432}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2/(exp(432 - (2*exp(2)*(3*x - 8))/3 - 162*x) + log(x)^2 + 2*exp(216 - (exp(2)*(3*x - 8))/3 - 81*x)*
log(x)))*(exp(216 - (exp(2)*(3*x - 8))/3 - 81*x)*(2*x + 2*x^2*exp(2) + 162*x^2) - 2*x + 2*x*log(x)))/(exp(648
- exp(2)*(3*x - 8) - 243*x) + 3*exp(216 - (exp(2)*(3*x - 8))/3 - 81*x)*log(x)^2 + log(x)^3 + 3*exp(432 - (2*ex
p(2)*(3*x - 8))/3 - 162*x)*log(x)),x)

[Out]

exp(x^2/(log(x)^2 + exp((16*exp(2))/3)*exp(-162*x)*exp(432)*exp(-2*x*exp(2)) + 2*exp((8*exp(2))/3)*exp(-81*x)*
exp(216)*exp(-x*exp(2))*log(x)))

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sympy [B]  time = 1.34, size = 46, normalized size = 1.64 \begin {gather*} e^{\frac {x^{2}}{e^{- 162 x + 2 \left (\frac {8}{3} - x\right ) e^{2} + 432} + 2 e^{- 81 x + \left (\frac {8}{3} - x\right ) e^{2} + 216} \log {\relax (x )} + \log {\relax (x )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*ln(x)+(2*x**2*exp(2)+162*x**2+2*x)*exp(1/3*(-3*x+8)*exp(2)-81*x+216)-2*x)*exp(x**2/(ln(x)**2+2*
exp(1/3*(-3*x+8)*exp(2)-81*x+216)*ln(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)**2))/(ln(x)**3+3*exp(1/3*(-3*x+8)*ex
p(2)-81*x+216)*ln(x)**2+3*exp(1/3*(-3*x+8)*exp(2)-81*x+216)**2*ln(x)+exp(1/3*(-3*x+8)*exp(2)-81*x+216)**3),x)

[Out]

exp(x**2/(exp(-162*x + 2*(8/3 - x)*exp(2) + 432) + 2*exp(-81*x + (8/3 - x)*exp(2) + 216)*log(x) + log(x)**2))

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