3.24.11 \(\int \frac {1}{4} e^{\frac {1}{4} (20 x^2+4 x^3+16 x^4-16 x^3 \log ^2(3)+4 x^2 \log ^4(3)+(-5 x-x^2-4 x^3+4 x^2 \log ^2(3)-x \log ^4(3)) \log (x))} (-5+39 x+8 x^2+64 x^3+(4 x-48 x^2) \log ^2(3)+(-1+8 x) \log ^4(3)+(-5-2 x-12 x^2+8 x \log ^2(3)-\log ^4(3)) \log (x)) \, dx\)

Optimal. Leaf size=32 \[ e^{\frac {1}{4} \left (5+x+\left (2 x-\log ^2(3)\right )^2\right ) \left (4 x^2-x \log (x)\right )} \]

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Rubi [F]  time = 5.45, 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 {1}{4} \exp \left (\frac {1}{4} \left (20 x^2+4 x^3+16 x^4-16 x^3 \log ^2(3)+4 x^2 \log ^4(3)+\left (-5 x-x^2-4 x^3+4 x^2 \log ^2(3)-x \log ^4(3)\right ) \log (x)\right )\right ) \left (-5+39 x+8 x^2+64 x^3+\left (4 x-48 x^2\right ) \log ^2(3)+(-1+8 x) \log ^4(3)+\left (-5-2 x-12 x^2+8 x \log ^2(3)-\log ^4(3)\right ) \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((20*x^2 + 4*x^3 + 16*x^4 - 16*x^3*Log[3]^2 + 4*x^2*Log[3]^4 + (-5*x - x^2 - 4*x^3 + 4*x^2*Log[3]^2 - x
*Log[3]^4)*Log[x])/4)*(-5 + 39*x + 8*x^2 + 64*x^3 + (4*x - 48*x^2)*Log[3]^2 + (-1 + 8*x)*Log[3]^4 + (-5 - 2*x
- 12*x^2 + 8*x*Log[3]^2 - Log[3]^4)*Log[x]))/4,x]

[Out]

(-5*Defer[Int][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x]))/4), x])/4 - (Log[3]^4*Defer[I
nt][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x]))/4), x])/4 + (39*Defer[Int][E^((x*(5 + 4*
x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x]))/4)*x, x])/4 + Log[3]^2*Defer[Int][E^((x*(5 + 4*x^2 + Log[
3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x]))/4)*x, x] + 2*Log[3]^4*Defer[Int][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1
- 4*Log[3]^2))*(4*x - Log[x]))/4)*x, x] + 2*Defer[Int][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x
- Log[x]))/4)*x^2, x] - 12*Log[3]^2*Defer[Int][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x
]))/4)*x^2, x] + 16*Defer[Int][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x]))/4)*x^3, x] -
((5 + Log[3]^4)*Defer[Int][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x]))/4)*Log[x], x])/4
- ((1 - 4*Log[3]^2)*Defer[Int][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x]))/4)*x*Log[x],
x])/2 - 3*Defer[Int][E^((x*(5 + 4*x^2 + Log[3]^4 + x*(1 - 4*Log[3]^2))*(4*x - Log[x]))/4)*x^2*Log[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \exp \left (\frac {1}{4} \left (20 x^2+4 x^3+16 x^4-16 x^3 \log ^2(3)+4 x^2 \log ^4(3)+\left (-5 x-x^2-4 x^3+4 x^2 \log ^2(3)-x \log ^4(3)\right ) \log (x)\right )\right ) \left (-5+39 x+8 x^2+64 x^3+\left (4 x-48 x^2\right ) \log ^2(3)+(-1+8 x) \log ^4(3)+\left (-5-2 x-12 x^2+8 x \log ^2(3)-\log ^4(3)\right ) \log (x)\right ) \, dx\\ &=\frac {1}{4} \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \left (-5+39 x+8 x^2+64 x^3+\left (4 x-48 x^2\right ) \log ^2(3)+(-1+8 x) \log ^4(3)+\left (-5-2 x-12 x^2+8 x \log ^2(3)-\log ^4(3)\right ) \log (x)\right ) \, dx\\ &=\frac {1}{4} \int \left (-5 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right )+39 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x+8 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^2+64 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^3-4 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x (-1+12 x) \log ^2(3)+\exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) (-1+8 x) \log ^4(3)+\exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \left (-5-12 x^2-\log ^4(3)-2 x \left (1-4 \log ^2(3)\right )\right ) \log (x)\right ) \, dx\\ &=\frac {1}{4} \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \left (-5-12 x^2-\log ^4(3)-2 x \left (1-4 \log ^2(3)\right )\right ) \log (x) \, dx-\frac {5}{4} \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \, dx+2 \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^2 \, dx+\frac {39}{4} \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x \, dx+16 \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^3 \, dx-\log ^2(3) \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x (-1+12 x) \, dx+\frac {1}{4} \log ^4(3) \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) (-1+8 x) \, dx\\ &=\frac {1}{4} \int \left (-12 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^2 \log (x)+2 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x \left (-1+4 \log ^2(3)\right ) \log (x)-5 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \left (1+\frac {\log ^4(3)}{5}\right ) \log (x)\right ) \, dx-\frac {5}{4} \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \, dx+2 \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^2 \, dx+\frac {39}{4} \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x \, dx+16 \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^3 \, dx-\log ^2(3) \int \left (-\exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x+12 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^2\right ) \, dx+\frac {1}{4} \log ^4(3) \int \left (-\exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right )+8 \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x\right ) \, dx\\ &=-\left (\frac {5}{4} \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \, dx\right )+2 \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^2 \, dx-3 \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^2 \log (x) \, dx+\frac {39}{4} \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x \, dx+16 \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^3 \, dx+\log ^2(3) \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x \, dx-\left (12 \log ^2(3)\right ) \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x^2 \, dx-\frac {1}{4} \log ^4(3) \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \, dx+\left (2 \log ^4(3)\right ) \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x \, dx+\frac {1}{2} \left (-1+4 \log ^2(3)\right ) \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) x \log (x) \, dx+\frac {1}{4} \left (-5-\log ^4(3)\right ) \int \exp \left (\frac {1}{4} x \left (5+4 x^2+\log ^4(3)+x \left (1-4 \log ^2(3)\right )\right ) (4 x-\log (x))\right ) \log (x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 52, normalized size = 1.62 \begin {gather*} e^{x^2 \left (5+x+4 x^2-4 x \log ^2(3)+\log ^4(3)\right )} x^{-\frac {1}{4} x \left (5+x+4 x^2-4 x \log ^2(3)+\log ^4(3)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((20*x^2 + 4*x^3 + 16*x^4 - 16*x^3*Log[3]^2 + 4*x^2*Log[3]^4 + (-5*x - x^2 - 4*x^3 + 4*x^2*Log[3]
^2 - x*Log[3]^4)*Log[x])/4)*(-5 + 39*x + 8*x^2 + 64*x^3 + (4*x - 48*x^2)*Log[3]^2 + (-1 + 8*x)*Log[3]^4 + (-5
- 2*x - 12*x^2 + 8*x*Log[3]^2 - Log[3]^4)*Log[x]))/4,x]

[Out]

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

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fricas [B]  time = 0.69, size = 63, normalized size = 1.97 \begin {gather*} e^{\left (x^{2} \log \relax (3)^{4} - 4 \, x^{3} \log \relax (3)^{2} + 4 \, x^{4} + x^{3} + 5 \, x^{2} - \frac {1}{4} \, {\left (x \log \relax (3)^{4} - 4 \, x^{2} \log \relax (3)^{2} + 4 \, x^{3} + x^{2} + 5 \, x\right )} \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-log(3)^4+8*x*log(3)^2-12*x^2-2*x-5)*log(x)+(8*x-1)*log(3)^4+(-48*x^2+4*x)*log(3)^2+64*x^3+8*x
^2+39*x-5)*exp(1/4*(-x*log(3)^4+4*x^2*log(3)^2-4*x^3-x^2-5*x)*log(x)+x^2*log(3)^4-4*x^3*log(3)^2+4*x^4+x^3+5*x
^2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.37, size = 70, normalized size = 2.19 \begin {gather*} e^{\left (x^{2} \log \relax (3)^{4} - \frac {1}{4} \, x \log \relax (3)^{4} \log \relax (x) - 4 \, x^{3} \log \relax (3)^{2} + x^{2} \log \relax (3)^{2} \log \relax (x) + 4 \, x^{4} - x^{3} \log \relax (x) + x^{3} - \frac {1}{4} \, x^{2} \log \relax (x) + 5 \, x^{2} - \frac {5}{4} \, x \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-log(3)^4+8*x*log(3)^2-12*x^2-2*x-5)*log(x)+(8*x-1)*log(3)^4+(-48*x^2+4*x)*log(3)^2+64*x^3+8*x
^2+39*x-5)*exp(1/4*(-x*log(3)^4+4*x^2*log(3)^2-4*x^3-x^2-5*x)*log(x)+x^2*log(3)^4-4*x^3*log(3)^2+4*x^4+x^3+5*x
^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.11, size = 50, normalized size = 1.56

method	result	size
risch	\(x^{-\frac {x \left (\ln \relax (3)^{4}-4 x \ln \relax (3)^{2}+4 x^{2}+x +5\right )}{4}} {\mathrm e}^{x^{2} \left (\ln \relax (3)^{4}-4 x \ln \relax (3)^{2}+4 x^{2}+x +5\right )}\)	\(50\)
norman	\({\mathrm e}^{\frac {\left (-x \ln \relax (3)^{4}+4 x^{2} \ln \relax (3)^{2}-4 x^{3}-x^{2}-5 x \right ) \ln \relax (x )}{4}+x^{2} \ln \relax (3)^{4}-4 x^{3} \ln \relax (3)^{2}+4 x^{4}+x^{3}+5 x^{2}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((-ln(3)^4+8*x*ln(3)^2-12*x^2-2*x-5)*ln(x)+(8*x-1)*ln(3)^4+(-48*x^2+4*x)*ln(3)^2+64*x^3+8*x^2+39*x-5)*
exp(1/4*(-x*ln(3)^4+4*x^2*ln(3)^2-4*x^3-x^2-5*x)*ln(x)+x^2*ln(3)^4-4*x^3*ln(3)^2+4*x^4+x^3+5*x^2),x,method=_RE
TURNVERBOSE)

[Out]

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

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maxima [B]  time = 1.17, size = 70, normalized size = 2.19 \begin {gather*} e^{\left (x^{2} \log \relax (3)^{4} - \frac {1}{4} \, x \log \relax (3)^{4} \log \relax (x) - 4 \, x^{3} \log \relax (3)^{2} + x^{2} \log \relax (3)^{2} \log \relax (x) + 4 \, x^{4} - x^{3} \log \relax (x) + x^{3} - \frac {1}{4} \, x^{2} \log \relax (x) + 5 \, x^{2} - \frac {5}{4} \, x \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-log(3)^4+8*x*log(3)^2-12*x^2-2*x-5)*log(x)+(8*x-1)*log(3)^4+(-48*x^2+4*x)*log(3)^2+64*x^3+8*x
^2+39*x-5)*exp(1/4*(-x*log(3)^4+4*x^2*log(3)^2-4*x^3-x^2-5*x)*log(x)+x^2*log(3)^4-4*x^3*log(3)^2+4*x^4+x^3+5*x
^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.78, size = 73, normalized size = 2.28 \begin {gather*} x^{x^2\,{\ln \relax (3)}^2-\frac {x^2}{4}-x^3}\,{\mathrm {e}}^{-\frac {5\,x\,\ln \relax (x)}{4}}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{x^2\,{\ln \relax (3)}^4}\,{\mathrm {e}}^{-4\,x^3\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{5\,x^2}\,{\mathrm {e}}^{4\,x^4}\,{\mathrm {e}}^{-\frac {x\,{\ln \relax (3)}^4\,\ln \relax (x)}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2*log(3)^4 - 4*x^3*log(3)^2 - (log(x)*(5*x - 4*x^2*log(3)^2 + x*log(3)^4 + x^2 + 4*x^3))/4 + 5*x^2
+ x^3 + 4*x^4)*(39*x - log(x)*(2*x - 8*x*log(3)^2 + log(3)^4 + 12*x^2 + 5) + log(3)^4*(8*x - 1) + log(3)^2*(4*
x - 48*x^2) + 8*x^2 + 64*x^3 - 5))/4,x)

[Out]

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

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sympy [B]  time = 0.56, size = 66, normalized size = 2.06 \begin {gather*} e^{4 x^{4} - 4 x^{3} \log {\relax (3 )}^{2} + x^{3} + x^{2} \log {\relax (3 )}^{4} + 5 x^{2} + \left (- x^{3} - \frac {x^{2}}{4} + x^{2} \log {\relax (3 )}^{2} - \frac {5 x}{4} - \frac {x \log {\relax (3 )}^{4}}{4}\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-ln(3)**4+8*x*ln(3)**2-12*x**2-2*x-5)*ln(x)+(8*x-1)*ln(3)**4+(-48*x**2+4*x)*ln(3)**2+64*x**3+8
*x**2+39*x-5)*exp(1/4*(-x*ln(3)**4+4*x**2*ln(3)**2-4*x**3-x**2-5*x)*ln(x)+x**2*ln(3)**4-4*x**3*ln(3)**2+4*x**4
+x**3+5*x**2),x)

[Out]

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

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