3.100.88 \(\int e^{-x-3 x^3+3 x \log (x)+2 x \log (3 x)-x \log ^2(3 x)} (4-9 x^2+3 \log (x)-\log ^2(3 x)) \, dx\)

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

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Rubi [A]  time = 0.38, antiderivative size = 31, normalized size of antiderivative = 1.11, number of steps used = 1, number of rules used = 1, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6706} \begin {gather*} 3^{2 x} x^{5 x} e^{-3 x^3-x-x \log ^2(3 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=3^{2 x} e^{-x-3 x^3-x \log ^2(3 x)} x^{5 x}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 3.30, size = 0, normalized size = 0.00 \begin {gather*} \int e^{-x-3 x^3+3 x \log (x)+2 x \log (3 x)-x \log ^2(3 x)} \left (4-9 x^2+3 \log (x)-\log ^2(3 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.23, size = 42, normalized size = 1.50 \begin {gather*} e^{\left (-3 \, x^{3} - x \log \relax (3)^{2} - x \log \relax (x)^{2} + 2 \, x \log \relax (3) - {\left (2 \, x \log \relax (3) - 5 \, x\right )} \log \relax (x) - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3*x)^2+3*log(x)-9*x^2+4)*exp(-x*log(3*x)^2+2*x*log(3*x)+3*x*log(x)-3*x^3-x),x, algorithm="fric
as")

[Out]

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

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giac [A]  time = 0.28, size = 31, normalized size = 1.11 \begin {gather*} e^{\left (-3 \, x^{3} - x \log \left (3 \, x\right )^{2} + 2 \, x \log \left (3 \, x\right ) + 3 \, x \log \relax (x) - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 32, normalized size = 1.14

method	result	size
derivativedivides	\({\mathrm e}^{-x \ln \left (3 x \right )^{2}+2 x \ln \left (3 x \right )+3 x \ln \relax (x )-3 x^{3}-x}\)	\(32\)
default	\({\mathrm e}^{-x \ln \left (3 x \right )^{2}+2 x \ln \left (3 x \right )+3 x \ln \relax (x )-3 x^{3}-x}\)	\(32\)
risch	\(x^{3 x} x^{-2 x \ln \relax (3)} x^{2 x} 9^{x} {\mathrm e}^{-x \left (\ln \relax (x )^{2}+\ln \relax (3)^{2}+3 x^{2}+1\right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 31, normalized size = 1.11 \begin {gather*} e^{\left (-3 \, x^{3} - x \log \left (3 \, x\right )^{2} + 2 \, x \log \left (3 \, x\right ) + 3 \, x \log \relax (x) - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3*x)^2+3*log(x)-9*x^2+4)*exp(-x*log(3*x)^2+2*x*log(3*x)+3*x*log(x)-3*x^3-x),x, algorithm="maxi
ma")

[Out]

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

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mupad [B]  time = 8.80, size = 41, normalized size = 1.46 \begin {gather*} 9^x\,x^{5\,x-2\,x\,\ln \relax (3)}\,{\mathrm {e}}^{-x\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-x\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{-3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9^x*x^(5*x - 2*x*log(3))*exp(-x*log(3)^2)*exp(-x)*exp(-x*log(x)^2)*exp(-3*x^3)

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sympy [A]  time = 0.40, size = 34, normalized size = 1.21 \begin {gather*} e^{- 3 x^{3} - x \left (\log {\relax (x )} + \log {\relax (3 )}\right )^{2} + 2 x \left (\log {\relax (x )} + \log {\relax (3 )}\right ) + 3 x \log {\relax (x )} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-3*x**3 - x*(log(x) + log(3))**2 + 2*x*(log(x) + log(3)) + 3*x*log(x) - x)

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