3.2.75 \(\int \frac {e^{-e^3+4 e^4-4 x} (-e^{e^3-4 e^4+4 x}-60 e^{3+x} x)}{x} \, dx\)

Optimal. Leaf size=26 \[ 20 e^{3-e^3+x-4 \left (-e^4+x\right )}-\log (x) \]

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Rubi [A]  time = 0.28, antiderivative size = 24, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {6688, 2194} \begin {gather*} 20 e^{-3 x+4 e^4-e^3+3}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-E^3 + 4*E^4 - 4*x)*(-E^(E^3 - 4*E^4 + 4*x) - 60*E^(3 + x)*x))/x,x]

[Out]

20*E^(3 - E^3 + 4*E^4 - 3*x) - Log[x]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-60 e^{3-e^3+4 e^4-3 x}-\frac {1}{x}\right ) \, dx\\ &=-\log (x)-60 \int e^{3-e^3+4 e^4-3 x} \, dx\\ &=20 e^{3-e^3+4 e^4-3 x}-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 24, normalized size = 0.92 \begin {gather*} 20 e^{3-e^3+4 e^4-3 x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^3 + 4*E^4 - 4*x)*(-E^(E^3 - 4*E^4 + 4*x) - 60*E^(3 + x)*x))/x,x]

[Out]

20*E^(3 - E^3 + 4*E^4 - 3*x) - Log[x]

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fricas [A]  time = 0.55, size = 43, normalized size = 1.65 \begin {gather*} -{\left (e^{\left (3 \, x + 6 \, \log \relax (2) + 9\right )} \log \relax (x) - 5 \, e^{\left (4 \, e^{4} - e^{3} + 8 \, \log \relax (2) + 12\right )}\right )} e^{\left (-3 \, x - 6 \, \log \relax (2) - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-15*x*exp(x)*exp(2*log(2)+3)-exp(-4*exp(4)+exp(3)+4*x))/x/exp(-4*exp(4)+exp(3)+4*x),x, algorithm="f
ricas")

[Out]

-(e^(3*x + 6*log(2) + 9)*log(x) - 5*e^(4*e^4 - e^3 + 8*log(2) + 12))*e^(-3*x - 6*log(2) - 9)

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giac [A]  time = 0.35, size = 21, normalized size = 0.81 \begin {gather*} 20 \, e^{\left (-3 \, x + 4 \, e^{4} - e^{3} + 3\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-15*x*exp(x)*exp(2*log(2)+3)-exp(-4*exp(4)+exp(3)+4*x))/x/exp(-4*exp(4)+exp(3)+4*x),x, algorithm="g
iac")

[Out]

20*e^(-3*x + 4*e^4 - e^3 + 3) - log(x)

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maple [A]  time = 0.07, size = 22, normalized size = 0.85

method	result	size
risch	\(-\ln \relax (x )+20 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-{\mathrm e}^{3}+3-3 x}\)	\(22\)
default	\(-\ln \relax (x )+20 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} {\mathrm e}^{-{\mathrm e}^{3}} {\mathrm e}^{3} {\mathrm e}^{-3 x}\)	\(24\)
norman	\(-\ln \relax (x )+20 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} {\mathrm e}^{-{\mathrm e}^{3}} {\mathrm e}^{3} {\mathrm e}^{-3 x}\)	\(24\)
meijerg	\(15 \,{\mathrm e}^{2 \ln \relax (2)+3+4 \,{\mathrm e}^{4}-{\mathrm e}^{3}-4 x} \left (1-{\mathrm e}^{x}\right )-{\mathrm e}^{-4 x} \left (-\ln \left (-4 x \right )-\expIntegralEi \left (1, -4 x \right )+\ln \relax (x )+2 \ln \relax (2)+i \pi \right )\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-15*x*exp(x)*exp(2*ln(2)+3)-exp(-4*exp(4)+exp(3)+4*x))/x/exp(-4*exp(4)+exp(3)+4*x),x,method=_RETURNVERBOS
E)

[Out]

-ln(x)+20*exp(4*exp(4)-exp(3)+3-3*x)

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maxima [A]  time = 0.69, size = 21, normalized size = 0.81 \begin {gather*} 20 \, e^{\left (-3 \, x + 4 \, e^{4} - e^{3} + 3\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-15*x*exp(x)*exp(2*log(2)+3)-exp(-4*exp(4)+exp(3)+4*x))/x/exp(-4*exp(4)+exp(3)+4*x),x, algorithm="m
axima")

[Out]

20*e^(-3*x + 4*e^4 - e^3 + 3) - log(x)

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mupad [B]  time = 0.24, size = 23, normalized size = 0.88 \begin {gather*} 20\,{\mathrm {e}}^{-{\mathrm {e}}^3}\,{\mathrm {e}}^{4\,{\mathrm {e}}^4}\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^3-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4*exp(4) - exp(3) - 4*x)*(exp(4*x + exp(3) - 4*exp(4)) + 15*x*exp(2*log(2) + 3)*exp(x)))/x,x)

[Out]

20*exp(-exp(3))*exp(4*exp(4))*exp(-3*x)*exp(3) - log(x)

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sympy [A]  time = 0.18, size = 24, normalized size = 0.92 \begin {gather*} - \log {\relax (x )} + \frac {20 e^{3} e^{- 3 x} e^{4 e^{4}}}{e^{e^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-15*x*exp(x)*exp(2*ln(2)+3)-exp(-4*exp(4)+exp(3)+4*x))/x/exp(-4*exp(4)+exp(3)+4*x),x)

[Out]

-log(x) + 20*exp(3)*exp(-3*x)*exp(-exp(3))*exp(4*exp(4))

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