3.6.90 \(\int e^{-\frac {1}{125} e^{14+e^x}+x \log (2) \log (3)} (-\frac {1}{125} e^{14+e^x+x}+\log (2) \log (3)) \, dx\)

Optimal. Leaf size=26 \[ e^{-e^{-1+e^x+3 (5-\log (5))}+x \log (2) \log (3)} \]

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Rubi [A]  time = 0.15, antiderivative size = 20, normalized size of antiderivative = 0.77, number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6706} \begin {gather*} e^{-\frac {1}{125} e^{e^x+14}} 2^{x \log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-1/125*E^(14 + E^x) + x*Log[2]*Log[3])*(-1/125*E^(14 + E^x + x) + Log[2]*Log[3]),x]

[Out]

2^(x*Log[3])/E^(E^(14 + E^x)/125)

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} &=2^{x \log (3)} e^{-\frac {1}{125} e^{14+e^x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 20, normalized size = 0.77 \begin {gather*} e^{-\frac {1}{125} e^{14+e^x}+x \log (2) \log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-1/125*E^(14 + E^x) + x*Log[2]*Log[3])*(-1/125*E^(14 + E^x + x) + Log[2]*Log[3]),x]

[Out]

E^(-1/125*E^(14 + E^x) + x*Log[2]*Log[3])

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fricas [A]  time = 0.64, size = 27, normalized size = 1.04 \begin {gather*} e^{\left ({\left (x e^{x} \log \relax (3) \log \relax (2) - e^{\left (x + e^{x} - 3 \, \log \relax (5) + 14\right )}\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*exp(exp(x)-3*log(5)+14)+log(2)*log(3))*exp(-exp(exp(x)-3*log(5)+14)+x*log(2)*log(3)),x, alg
orithm="fricas")

[Out]

e^((x*e^x*log(3)*log(2) - e^(x + e^x - 3*log(5) + 14))*e^(-x))

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giac [A]  time = 0.32, size = 19, normalized size = 0.73 \begin {gather*} e^{\left (x \log \relax (3) \log \relax (2) - e^{\left (e^{x} - 3 \, \log \relax (5) + 14\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*exp(exp(x)-3*log(5)+14)+log(2)*log(3))*exp(-exp(exp(x)-3*log(5)+14)+x*log(2)*log(3)),x, alg
orithm="giac")

[Out]

e^(x*log(3)*log(2) - e^(e^x - 3*log(5) + 14))

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maple [A]  time = 0.08, size = 16, normalized size = 0.62

method	result	size
risch	\(2^{x \ln \relax (3)} {\mathrm e}^{-\frac {{\mathrm e}^{{\mathrm e}^{x}+14}}{125}}\)	\(16\)
derivativedivides	\({\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x}-3 \ln \relax (5)+14}+x \ln \relax (2) \ln \relax (3)}\)	\(20\)
default	\({\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x}-3 \ln \relax (5)+14}+x \ln \relax (2) \ln \relax (3)}\)	\(20\)
norman	\({\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x}-3 \ln \relax (5)+14}+x \ln \relax (2) \ln \relax (3)}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x)*exp(exp(x)-3*ln(5)+14)+ln(2)*ln(3))*exp(-exp(exp(x)-3*ln(5)+14)+x*ln(2)*ln(3)),x,method=_RETURNVE
RBOSE)

[Out]

2^(x*ln(3))*exp(-1/125*exp(exp(x)+14))

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maxima [A]  time = 0.42, size = 15, normalized size = 0.58 \begin {gather*} e^{\left (x \log \relax (3) \log \relax (2) - \frac {1}{125} \, e^{\left (e^{x} + 14\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*exp(exp(x)-3*log(5)+14)+log(2)*log(3))*exp(-exp(exp(x)-3*log(5)+14)+x*log(2)*log(3)),x, alg
orithm="maxima")

[Out]

e^(x*log(3)*log(2) - 1/125*e^(e^x + 14))

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mupad [B]  time = 0.13, size = 15, normalized size = 0.58 \begin {gather*} 2^{x\,\ln \relax (3)}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{14}}{125}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x*log(2)*log(3) - exp(exp(x) - 3*log(5) + 14))*(log(2)*log(3) - exp(exp(x) - 3*log(5) + 14)*exp(x)),x)

[Out]

2^(x*log(3))*exp(-(exp(exp(x))*exp(14))/125)

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sympy [A]  time = 0.24, size = 17, normalized size = 0.65 \begin {gather*} e^{x \log {\relax (2 )} \log {\relax (3 )} - \frac {e^{e^{x} + 14}}{125}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*exp(exp(x)-3*ln(5)+14)+ln(2)*ln(3))*exp(-exp(exp(x)-3*ln(5)+14)+x*ln(2)*ln(3)),x)

[Out]

exp(x*log(2)*log(3) - exp(exp(x) + 14)/125)

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