3.76.45 \(\int -\frac {4 e^{8+e^{5-4 e^3 \log (16 x)}-4 e^3 \log (16 x)}}{x} \, dx\)

Optimal. Leaf size=15 \[ e^{e^{5-4 e^3 \log (16 x)}} \]

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Rubi [A]  time = 0.19, antiderivative size = 20, normalized size of antiderivative = 1.33, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 2274, 6706} \begin {gather*} e^{16^{-4 e^3} e^5 x^{-4 e^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*E^(8 + E^(5 - 4*E^3*Log[16*x]) - 4*E^3*Log[16*x]))/x,x]

[Out]

E^(E^5/(16^(4*E^3)*x^(4*E^3)))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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} &=-\left (4 \int \frac {e^{8+e^{5-4 e^3 \log (16 x)}-4 e^3 \log (16 x)}}{x} \, dx\right )\\ &=-\left (4 \int 16^{-4 e^3} e^{8+e^{5-4 e^3 \log (16 x)}} x^{-1-4 e^3} \, dx\right )\\ &=-\left (4^{1-8 e^3} \int e^{8+e^{5-4 e^3 \log (16 x)}} x^{-1-4 e^3} \, dx\right )\\ &=e^{16^{-4 e^3} e^5 x^{-4 e^3}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 15, normalized size = 1.00 \begin {gather*} e^{e^{5-4 e^3 \log (16 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*E^(8 + E^(5 - 4*E^3*Log[16*x]) - 4*E^3*Log[16*x]))/x,x]

[Out]

E^E^(5 - 4*E^3*Log[16*x])

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fricas [A]  time = 0.59, size = 12, normalized size = 0.80 \begin {gather*} e^{\left (e^{\left (-4 \, e^{3} \log \left (16 \, x\right ) + 5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(3)*exp(-4*exp(3)*log(16*x)+5)*exp(exp(-4*exp(3)*log(16*x)+5))/x,x, algorithm="fricas")

[Out]

e^(e^(-4*e^3*log(16*x) + 5))

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giac [A]  time = 0.22, size = 12, normalized size = 0.80 \begin {gather*} e^{\left (e^{\left (-4 \, e^{3} \log \left (16 \, x\right ) + 5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(3)*exp(-4*exp(3)*log(16*x)+5)*exp(exp(-4*exp(3)*log(16*x)+5))/x,x, algorithm="giac")

[Out]

e^(e^(-4*e^3*log(16*x) + 5))

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maple [A]  time = 0.05, size = 13, normalized size = 0.87

method	result	size
derivativedivides	\({\mathrm e}^{{\mathrm e}^{-4 \,{\mathrm e}^{3} \ln \left (16 x \right )+5}}\)	\(13\)
default	\({\mathrm e}^{{\mathrm e}^{-4 \,{\mathrm e}^{3} \ln \left (16 x \right )+5}}\)	\(13\)
norman	\({\mathrm e}^{{\mathrm e}^{-4 \,{\mathrm e}^{3} \ln \left (16 x \right )+5}}\)	\(13\)
risch	\({\mathrm e}^{\left (16 x \right )^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{5}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4*exp(3)*exp(-4*exp(3)*ln(16*x)+5)*exp(exp(-4*exp(3)*ln(16*x)+5))/x,x,method=_RETURNVERBOSE)

[Out]

exp(exp(-4*exp(3)*ln(16*x)+5))

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maxima [A]  time = 0.36, size = 12, normalized size = 0.80 \begin {gather*} e^{\left (e^{\left (-4 \, e^{3} \log \left (16 \, x\right ) + 5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(3)*exp(-4*exp(3)*log(16*x)+5)*exp(exp(-4*exp(3)*log(16*x)+5))/x,x, algorithm="maxima")

[Out]

e^(e^(-4*e^3*log(16*x) + 5))

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mupad [B]  time = 5.27, size = 20, normalized size = 1.33 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^5}{2^{16\,{\mathrm {e}}^3}\,x^{4\,{\mathrm {e}}^3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*exp(exp(5 - 4*log(16*x)*exp(3)))*exp(3)*exp(5 - 4*log(16*x)*exp(3)))/x,x)

[Out]

exp(exp(5)/(2^(16*exp(3))*x^(4*exp(3))))

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sympy [A]  time = 2.51, size = 15, normalized size = 1.00 \begin {gather*} e^{\frac {e^{5}}{65536^{e^{3}} x^{4 e^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*exp(3)*exp(-4*exp(3)*ln(16*x)+5)*exp(exp(-4*exp(3)*ln(16*x)+5))/x,x)

[Out]

exp(65536**(-exp(3))*x**(-4*exp(3))*exp(5))

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