3.89.50 \(\int \frac {e^{x-16 e^{2+2 e^3} x^2-32 e^{2+2 e^3} x \log (3 x)-16 e^{2+2 e^3} \log ^2(3 x)} (x+e^{2+2 e^3} (-32 x-32 x^2)+e^{2+2 e^3} (-32-32 x) \log (3 x))}{x} \, dx\)

Optimal. Leaf size=25 \[ 16+e^{x-16 e^{2+2 e^3} (x+\log (3 x))^2} \]

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Rubi [B]  time = 1.76, antiderivative size = 64, normalized size of antiderivative = 2.56, number of steps used = 1, number of rules used = 1, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6706} \begin {gather*} 3^{-32 e^{2+2 e^3} x} x^{-32 e^{2+2 e^3} x} \exp \left (-16 e^{2+2 e^3} x^2+x-16 e^{2+2 e^3} \log ^2(3 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(x - 16*E^(2 + 2*E^3)*x^2 - 32*E^(2 + 2*E^3)*x*Log[3*x] - 16*E^(2 + 2*E^3)*Log[3*x]^2)*(x + E^(2 + 2*E^
3)*(-32*x - 32*x^2) + E^(2 + 2*E^3)*(-32 - 32*x)*Log[3*x]))/x,x]

[Out]

E^(x - 16*E^(2 + 2*E^3)*x^2 - 16*E^(2 + 2*E^3)*Log[3*x]^2)/(3^(32*E^(2 + 2*E^3)*x)*x^(32*E^(2 + 2*E^3)*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^{-32 e^{2+2 e^3} x} \exp \left (x-16 e^{2+2 e^3} x^2-16 e^{2+2 e^3} \log ^2(3 x)\right ) x^{-32 e^{2+2 e^3} x}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.16, size = 51, normalized size = 2.04 \begin {gather*} e^{x-16 e^{2+2 e^3} x^2-32 e^{2+2 e^3} x \log (3 x)-16 e^{2+2 e^3} \log ^2(3 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x - 16*E^(2 + 2*E^3)*x^2 - 32*E^(2 + 2*E^3)*x*Log[3*x] - 16*E^(2 + 2*E^3)*Log[3*x]^2)*(x + E^(2
+ 2*E^3)*(-32*x - 32*x^2) + E^(2 + 2*E^3)*(-32 - 32*x)*Log[3*x]))/x,x]

[Out]

E^(x - 16*E^(2 + 2*E^3)*x^2 - 32*E^(2 + 2*E^3)*x*Log[3*x] - 16*E^(2 + 2*E^3)*Log[3*x]^2)

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fricas [A]  time = 0.52, size = 44, normalized size = 1.76 \begin {gather*} e^{\left (-16 \, x^{2} e^{\left (2 \, e^{3} + 2\right )} - 32 \, x e^{\left (2 \, e^{3} + 2\right )} \log \left (3 \, x\right ) - 16 \, e^{\left (2 \, e^{3} + 2\right )} \log \left (3 \, x\right )^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x-32)*exp(exp(3)+1)^2*log(3*x)+(-32*x^2-32*x)*exp(exp(3)+1)^2+x)*exp(-16*exp(exp(3)+1)^2*log(3
*x)^2-32*x*exp(exp(3)+1)^2*log(3*x)-16*x^2*exp(exp(3)+1)^2+x)/x,x, algorithm="fricas")

[Out]

e^(-16*x^2*e^(2*e^3 + 2) - 32*x*e^(2*e^3 + 2)*log(3*x) - 16*e^(2*e^3 + 2)*log(3*x)^2 + x)

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giac [A]  time = 0.37, size = 44, normalized size = 1.76 \begin {gather*} e^{\left (-16 \, x^{2} e^{\left (2 \, e^{3} + 2\right )} - 32 \, x e^{\left (2 \, e^{3} + 2\right )} \log \left (3 \, x\right ) - 16 \, e^{\left (2 \, e^{3} + 2\right )} \log \left (3 \, x\right )^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x-32)*exp(exp(3)+1)^2*log(3*x)+(-32*x^2-32*x)*exp(exp(3)+1)^2+x)*exp(-16*exp(exp(3)+1)^2*log(3
*x)^2-32*x*exp(exp(3)+1)^2*log(3*x)-16*x^2*exp(exp(3)+1)^2+x)/x,x, algorithm="giac")

[Out]

e^(-16*x^2*e^(2*e^3 + 2) - 32*x*e^(2*e^3 + 2)*log(3*x) - 16*e^(2*e^3 + 2)*log(3*x)^2 + x)

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maple [A]  time = 0.11, size = 45, normalized size = 1.80

method	result	size
norman	\({\mathrm e}^{-16 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2} \ln \left (3 x \right )^{2}-32 x \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2} \ln \left (3 x \right )-16 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{3}+2}+x}\)	\(45\)
risch	\(\left (3 x \right )^{-32 x \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2}} {\mathrm e}^{-16 \,{\mathrm e}^{2 \,{\mathrm e}^{3}+2} \ln \left (3 x \right )^{2}-16 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{3}+2}+x}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-32*x-32)*exp(exp(3)+1)^2*ln(3*x)+(-32*x^2-32*x)*exp(exp(3)+1)^2+x)*exp(-16*exp(exp(3)+1)^2*ln(3*x)^2-32
*x*exp(exp(3)+1)^2*ln(3*x)-16*x^2*exp(exp(3)+1)^2+x)/x,x,method=_RETURNVERBOSE)

[Out]

exp(-16*exp(exp(3)+1)^2*ln(3*x)^2-32*x*exp(exp(3)+1)^2*ln(3*x)-16*x^2*exp(exp(3)+1)^2+x)

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maxima [B]  time = 0.71, size = 78, normalized size = 3.12 \begin {gather*} e^{\left (-16 \, x^{2} e^{\left (2 \, e^{3} + 2\right )} - 32 \, x e^{\left (2 \, e^{3} + 2\right )} \log \relax (3) - 16 \, e^{\left (2 \, e^{3} + 2\right )} \log \relax (3)^{2} - 32 \, x e^{\left (2 \, e^{3} + 2\right )} \log \relax (x) - 32 \, e^{\left (2 \, e^{3} + 2\right )} \log \relax (3) \log \relax (x) - 16 \, e^{\left (2 \, e^{3} + 2\right )} \log \relax (x)^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x-32)*exp(exp(3)+1)^2*log(3*x)+(-32*x^2-32*x)*exp(exp(3)+1)^2+x)*exp(-16*exp(exp(3)+1)^2*log(3
*x)^2-32*x*exp(exp(3)+1)^2*log(3*x)-16*x^2*exp(exp(3)+1)^2+x)/x,x, algorithm="maxima")

[Out]

e^(-16*x^2*e^(2*e^3 + 2) - 32*x*e^(2*e^3 + 2)*log(3) - 16*e^(2*e^3 + 2)*log(3)^2 - 32*x*e^(2*e^3 + 2)*log(x) -
32*e^(2*e^3 + 2)*log(3)*log(x) - 16*e^(2*e^3 + 2)*log(x)^2 + x)

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mupad [B]  time = 5.45, size = 87, normalized size = 3.48 \begin {gather*} \frac {{\mathrm {e}}^{-16\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^2\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{-16\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^2}\,{\mathrm {e}}^{-16\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^2\,{\ln \relax (x)}^2}\,{\mathrm {e}}^x}{3^{32\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^2}\,x^{32\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^2\,\ln \relax (3)}\,x^{32\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x - 16*log(3*x)^2*exp(2*exp(3) + 2) - 16*x^2*exp(2*exp(3) + 2) - 32*x*log(3*x)*exp(2*exp(3) + 2))*(e
xp(2*exp(3) + 2)*(32*x + 32*x^2) - x + log(3*x)*exp(2*exp(3) + 2)*(32*x + 32)))/x,x)

[Out]

(exp(-16*exp(2*exp(3))*exp(2)*log(3)^2)*exp(-16*x^2*exp(2*exp(3))*exp(2))*exp(-16*exp(2*exp(3))*exp(2)*log(x)^
2)*exp(x))/(3^(32*x*exp(2*exp(3))*exp(2))*x^(32*exp(2*exp(3))*exp(2)*log(3))*x^(32*x*exp(2*exp(3))*exp(2)))

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sympy [B]  time = 0.44, size = 49, normalized size = 1.96 \begin {gather*} e^{- 16 x^{2} e^{2 + 2 e^{3}} - 32 x e^{2 + 2 e^{3}} \log {\left (3 x \right )} + x - 16 e^{2 + 2 e^{3}} \log {\left (3 x \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x-32)*exp(exp(3)+1)**2*ln(3*x)+(-32*x**2-32*x)*exp(exp(3)+1)**2+x)*exp(-16*exp(exp(3)+1)**2*ln
(3*x)**2-32*x*exp(exp(3)+1)**2*ln(3*x)-16*x**2*exp(exp(3)+1)**2+x)/x,x)

[Out]

exp(-16*x**2*exp(2 + 2*exp(3)) - 32*x*exp(2 + 2*exp(3))*log(3*x) + x - 16*exp(2 + 2*exp(3))*log(3*x)**2)

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