3.36.84 \(\int \frac {e^{\frac {1}{16} (-48-e^{-12+64 e^x}+16 \log (x))} (-1+4 e^{-12+64 e^x+x} x)}{x} \, dx\)

Optimal. Leaf size=22 \[ 5-e^{-3-\frac {1}{16} e^{-12+64 e^x}} x \]

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Rubi [A]  time = 0.14, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2274, 2288} \begin {gather*} -e^{\frac {1}{16} \left (-e^{64 e^x-12}-48\right )} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-48 - E^(-12 + 64*E^x) + 16*Log[x])/16)*(-1 + 4*E^(-12 + 64*E^x + x)*x))/x,x]

[Out]

-(E^((-48 - E^(-12 + 64*E^x))/16)*x)

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{\frac {1}{16} \left (-48-e^{-12+64 e^x}\right )} \left (-1+4 e^{-12+64 e^x+x} x\right ) \, dx\\ &=-e^{\frac {1}{16} \left (-48-e^{-12+64 e^x}\right )} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 20, normalized size = 0.91 \begin {gather*} -e^{-3-\frac {1}{16} e^{-12+64 e^x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-48 - E^(-12 + 64*E^x) + 16*Log[x])/16)*(-1 + 4*E^(-12 + 64*E^x + x)*x))/x,x]

[Out]

-(E^(-3 - E^(-12 + 64*E^x)/16)*x)

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fricas [A]  time = 0.82, size = 30, normalized size = 1.36 \begin {gather*} -e^{\left (\frac {1}{16} \, {\left (16 \, e^{x} \log \relax (x) - e^{\left (x + 64 \, e^{x} - 12\right )} - 48 \, e^{x}\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(x)*exp(16*exp(x)-3)^4-1)*exp(-1/16*exp(16*exp(x)-3)^4+log(x)-3)/x,x, algorithm="fricas")

[Out]

-e^(1/16*(16*e^x*log(x) - e^(x + 64*e^x - 12) - 48*e^x)*e^(-x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x e^{\left (x + 64 \, e^{x} - 12\right )} - 1\right )} e^{\left (-\frac {1}{16} \, e^{\left (64 \, e^{x} - 12\right )} + \log \relax (x) - 3\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(x)*exp(16*exp(x)-3)^4-1)*exp(-1/16*exp(16*exp(x)-3)^4+log(x)-3)/x,x, algorithm="giac")

[Out]

integrate((4*x*e^(x + 64*e^x - 12) - 1)*e^(-1/16*e^(64*e^x - 12) + log(x) - 3)/x, x)

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maple [A]  time = 0.10, size = 16, normalized size = 0.73

method	result	size
risch	\(-x \,{\mathrm e}^{-\frac {{\mathrm e}^{64 \,{\mathrm e}^{x}-12}}{16}-3}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*exp(x)*exp(16*exp(x)-3)^4-1)*exp(-1/16*exp(16*exp(x)-3)^4+ln(x)-3)/x,x,method=_RETURNVERBOSE)

[Out]

-x*exp(-1/16*exp(64*exp(x)-12)-3)

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maxima [A]  time = 1.36, size = 15, normalized size = 0.68 \begin {gather*} -x e^{\left (-\frac {1}{16} \, e^{\left (64 \, e^{x} - 12\right )} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(x)*exp(16*exp(x)-3)^4-1)*exp(-1/16*exp(16*exp(x)-3)^4+log(x)-3)/x,x, algorithm="maxima")

[Out]

-x*e^(-1/16*e^(64*e^x - 12) - 3)

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mupad [B]  time = 2.20, size = 15, normalized size = 0.68 \begin {gather*} -x\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-12}\,{\mathrm {e}}^{64\,{\mathrm {e}}^x}}{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x) - exp(64*exp(x) - 12)/16 - 3)*(4*x*exp(64*exp(x) - 12)*exp(x) - 1))/x,x)

[Out]

-x*exp(-3)*exp(-(exp(-12)*exp(64*exp(x)))/16)

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sympy [A]  time = 2.36, size = 17, normalized size = 0.77 \begin {gather*} - x e^{- \frac {e^{64 e^{x} - 12}}{16} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(x)*exp(16*exp(x)-3)**4-1)*exp(-1/16*exp(16*exp(x)-3)**4+ln(x)-3)/x,x)

[Out]

-x*exp(-exp(64*exp(x) - 12)/16 - 3)

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