3.13.59 \(\int \frac {e^{-\frac {x}{\log (\frac {48 x}{4+3 x})}} (1036+(-1036-777 x) \log (\frac {48 x}{4+3 x}))}{(4+3 x) \log ^2(\frac {48 x}{4+3 x})} \, dx\)

Optimal. Leaf size=20 \[ 259 e^{-\frac {x}{\log \left (\frac {16 x}{\frac {4}{3}+x}\right )}} \]

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Rubi [A]  time = 0.53, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6706} \begin {gather*} 259 e^{-\frac {x}{\log \left (\frac {48 x}{3 x+4}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1036 + (-1036 - 777*x)*Log[(48*x)/(4 + 3*x)])/(E^(x/Log[(48*x)/(4 + 3*x)])*(4 + 3*x)*Log[(48*x)/(4 + 3*x)
]^2),x]

[Out]

259/E^(x/Log[(48*x)/(4 + 3*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} &=259 e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 20, normalized size = 1.00 \begin {gather*} 259 e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1036 + (-1036 - 777*x)*Log[(48*x)/(4 + 3*x)])/(E^(x/Log[(48*x)/(4 + 3*x)])*(4 + 3*x)*Log[(48*x)/(4
+ 3*x)]^2),x]

[Out]

259/E^(x/Log[(48*x)/(4 + 3*x)])

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fricas [A]  time = 0.65, size = 19, normalized size = 0.95 \begin {gather*} 259 \, e^{\left (-\frac {x}{\log \left (\frac {48 \, x}{3 \, x + 4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-777*x-1036)*log(48*x/(4+3*x))+1036)/(4+3*x)/log(48*x/(4+3*x))^2/exp(x/log(48*x/(4+3*x))),x, algor
ithm="fricas")

[Out]

259*e^(-x/log(48*x/(3*x + 4)))

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giac [A]  time = 0.89, size = 19, normalized size = 0.95 \begin {gather*} 259 \, e^{\left (-\frac {x}{\log \left (\frac {48 \, x}{3 \, x + 4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-777*x-1036)*log(48*x/(4+3*x))+1036)/(4+3*x)/log(48*x/(4+3*x))^2/exp(x/log(48*x/(4+3*x))),x, algor
ithm="giac")

[Out]

259*e^(-x/log(48*x/(3*x + 4)))

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maple [A]  time = 0.20, size = 20, normalized size = 1.00

method	result	size
risch	\(259 \,{\mathrm e}^{-\frac {x}{\ln \left (\frac {48 x}{4+3 x}\right )}}\)	\(20\)
norman	\(259 \,{\mathrm e}^{-\frac {x}{\ln \left (\frac {48 x}{4+3 x}\right )}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-777*x-1036)*ln(48*x/(4+3*x))+1036)/(4+3*x)/ln(48*x/(4+3*x))^2/exp(x/ln(48*x/(4+3*x))),x,method=_RETURNV
ERBOSE)

[Out]

259*exp(-x/ln(48*x/(4+3*x)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -259 \, \int \frac {{\left ({\left (3 \, x + 4\right )} \log \left (\frac {48 \, x}{3 \, x + 4}\right ) - 4\right )} e^{\left (-\frac {x}{\log \left (\frac {48 \, x}{3 \, x + 4}\right )}\right )}}{{\left (3 \, x + 4\right )} \log \left (\frac {48 \, x}{3 \, x + 4}\right )^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-777*x-1036)*log(48*x/(4+3*x))+1036)/(4+3*x)/log(48*x/(4+3*x))^2/exp(x/log(48*x/(4+3*x))),x, algor
ithm="maxima")

[Out]

-259*integrate(((3*x + 4)*log(48*x/(3*x + 4)) - 4)*e^(-x/log(48*x/(3*x + 4)))/((3*x + 4)*log(48*x/(3*x + 4))^2
), x)

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mupad [B]  time = 1.19, size = 19, normalized size = 0.95 \begin {gather*} 259\,{\mathrm {e}}^{-\frac {x}{\ln \left (\frac {48\,x}{3\,x+4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x/log((48*x)/(3*x + 4)))*(log((48*x)/(3*x + 4))*(777*x + 1036) - 1036))/(log((48*x)/(3*x + 4))^2*(3
*x + 4)),x)

[Out]

259*exp(-x/log((48*x)/(3*x + 4)))

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sympy [A]  time = 0.46, size = 14, normalized size = 0.70 \begin {gather*} 259 e^{- \frac {x}{\log {\left (\frac {48 x}{3 x + 4} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-777*x-1036)*ln(48*x/(4+3*x))+1036)/(4+3*x)/ln(48*x/(4+3*x))**2/exp(x/ln(48*x/(4+3*x))),x)

[Out]

259*exp(-x/log(48*x/(3*x + 4)))

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