3.27.82 \(\int e^{-e^{2+e^4}-x+4 x \log (x)-8 e^2 x \log ^2(x)} (3+(4-16 e^2) \log (x)-8 e^2 \log ^2(x)) \, dx\)

Optimal. Leaf size=29 \[ e^{-e^{2+e^4}-x+4 \log (x) \left (x-2 e^2 x \log (x)\right )} \]

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Rubi [A]  time = 0.16, antiderivative size = 31, normalized size of antiderivative = 1.07, 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*} x^{4 x} e^{-x-8 e^2 x \log ^2(x)-e^{2+e^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-E^(2 + E^4) - x + 4*x*Log[x] - 8*E^2*x*Log[x]^2)*(3 + (4 - 16*E^2)*Log[x] - 8*E^2*Log[x]^2),x]

[Out]

E^(-E^(2 + E^4) - x - 8*E^2*x*Log[x]^2)*x^(4*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} &=e^{-e^{2+e^4}-x-8 e^2 x \log ^2(x)} x^{4 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 30, normalized size = 1.03 \begin {gather*} e^{-e^{2+e^4}-x+4 x \log (x)-8 e^2 x \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-E^(2 + E^4) - x + 4*x*Log[x] - 8*E^2*x*Log[x]^2)*(3 + (4 - 16*E^2)*Log[x] - 8*E^2*Log[x]^2),x]

[Out]

E^(-E^(2 + E^4) - x + 4*x*Log[x] - 8*E^2*x*Log[x]^2)

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fricas [A]  time = 0.60, size = 26, normalized size = 0.90 \begin {gather*} e^{\left (-8 \, x e^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x) - x - e^{\left (e^{4} + 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(2)*log(x)^2+(-16*exp(2)+4)*log(x)+3)*exp(-8*x*exp(2)*log(x)^2+4*x*log(x)-exp(2+exp(4))-x),x,
algorithm="fricas")

[Out]

e^(-8*x*e^2*log(x)^2 + 4*x*log(x) - x - e^(e^4 + 2))

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giac [A]  time = 0.98, size = 26, normalized size = 0.90 \begin {gather*} e^{\left (-8 \, x e^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x) - x - e^{\left (e^{4} + 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(2)*log(x)^2+(-16*exp(2)+4)*log(x)+3)*exp(-8*x*exp(2)*log(x)^2+4*x*log(x)-exp(2+exp(4))-x),x,
algorithm="giac")

[Out]

e^(-8*x*e^2*log(x)^2 + 4*x*log(x) - x - e^(e^4 + 2))

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maple [A]  time = 0.04, size = 27, normalized size = 0.93

method	result	size
norman	\({\mathrm e}^{-8 x \,{\mathrm e}^{2} \ln \relax (x )^{2}+4 x \ln \relax (x )-{\mathrm e}^{2+{\mathrm e}^{4}}-x}\)	\(27\)
risch	\(x^{4 x} {\mathrm e}^{-8 x \,{\mathrm e}^{2} \ln \relax (x )^{2}-{\mathrm e}^{2+{\mathrm e}^{4}}-x}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*exp(2)*ln(x)^2+(-16*exp(2)+4)*ln(x)+3)*exp(-8*x*exp(2)*ln(x)^2+4*x*ln(x)-exp(2+exp(4))-x),x,method=_RE
TURNVERBOSE)

[Out]

exp(-8*x*exp(2)*ln(x)^2+4*x*ln(x)-exp(2+exp(4))-x)

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maxima [A]  time = 0.72, size = 26, normalized size = 0.90 \begin {gather*} e^{\left (-8 \, x e^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x) - x - e^{\left (e^{4} + 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(2)*log(x)^2+(-16*exp(2)+4)*log(x)+3)*exp(-8*x*exp(2)*log(x)^2+4*x*log(x)-exp(2+exp(4))-x),x,
algorithm="maxima")

[Out]

e^(-8*x*e^2*log(x)^2 + 4*x*log(x) - x - e^(e^4 + 2))

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mupad [B]  time = 1.59, size = 28, normalized size = 0.97 \begin {gather*} x^{4\,x}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-8\,x\,{\mathrm {e}}^2\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{-{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(4*x*log(x) - exp(exp(4) + 2) - x - 8*x*exp(2)*log(x)^2)*(8*exp(2)*log(x)^2 + log(x)*(16*exp(2) - 4) -
3),x)

[Out]

x^(4*x)*exp(-x)*exp(-8*x*exp(2)*log(x)^2)*exp(-exp(2)*exp(exp(4)))

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sympy [A]  time = 0.33, size = 27, normalized size = 0.93 \begin {gather*} e^{- 8 x e^{2} \log {\relax (x )}^{2} + 4 x \log {\relax (x )} - x - e^{2 + e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(2)*ln(x)**2+(-16*exp(2)+4)*ln(x)+3)*exp(-8*x*exp(2)*ln(x)**2+4*x*ln(x)-exp(2+exp(4))-x),x)

[Out]

exp(-8*x*exp(2)*log(x)**2 + 4*x*log(x) - x - exp(2 + exp(4)))

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