3.71 \(\int \frac{\log (-\frac{g (d+e x)}{e f-d g})}{f+g x} \, dx\)

Optimal. Leaf size=24 \[ -\frac{\text{PolyLog}\left (2,\frac{e (f+g x)}{e f-d g}\right )}{g} \]

[Out]

-(PolyLog[2, (e*(f + g*x))/(e*f - d*g)]/g)

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Rubi [A]  time = 0.026475, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2393, 2391} \[ -\frac{\text{PolyLog}\left (2,\frac{e (f+g x)}{e f-d g}\right )}{g} \]

Antiderivative was successfully verified.

[In]

Int[Log[-((g*(d + e*x))/(e*f - d*g))]/(f + g*x),x]

[Out]

-(PolyLog[2, (e*(f + g*x))/(e*f - d*g)]/g)

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\log \left (-\frac{g (d+e x)}{e f-d g}\right )}{f+g x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{e x}{e f-d g}\right )}{x} \, dx,x,f+g x\right )}{g}\\ &=-\frac{\text{Li}_2\left (\frac{e (f+g x)}{e f-d g}\right )}{g}\\ \end{align*}

Mathematica [A]  time = 0.00671, size = 24, normalized size = 1. \[ -\frac{\text{PolyLog}\left (2,\frac{e (f+g x)}{e f-d g}\right )}{g} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[-((g*(d + e*x))/(e*f - d*g))]/(f + g*x),x]

[Out]

-(PolyLog[2, (e*(f + g*x))/(e*f - d*g)]/g)

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Maple [A]  time = 0.061, size = 35, normalized size = 1.5 \begin{align*} -{\frac{1}{g}{\it dilog} \left ({\frac{egx}{dg-fe}}+{\frac{dg}{dg-fe}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(-g*(e*x+d)/(-d*g+e*f))/(g*x+f),x)

[Out]

-1/g*dilog(e*g/(d*g-e*f)*x+d*g/(d*g-e*f))

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Maxima [B]  time = 1.22021, size = 138, normalized size = 5.75 \begin{align*} -\frac{\log \left (e x + d\right ) \log \left (g x + f\right )}{g} + \frac{\log \left (g x + f\right ) \log \left (-\frac{{\left (e x + d\right )} g}{e f - d g}\right )}{g} + \frac{\log \left (e x + d\right ) \log \left (\frac{e g x + d g}{e f - d g} + 1\right ) +{\rm Li}_2\left (-\frac{e g x + d g}{e f - d g}\right )}{g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-g*(e*x+d)/(-d*g+e*f))/(g*x+f),x, algorithm="maxima")

[Out]

-log(e*x + d)*log(g*x + f)/g + log(g*x + f)*log(-(e*x + d)*g/(e*f - d*g))/g + (log(e*x + d)*log((e*g*x + d*g)/
(e*f - d*g) + 1) + dilog(-(e*g*x + d*g)/(e*f - d*g)))/g

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Fricas [A]  time = 2.07007, size = 55, normalized size = 2.29 \begin{align*} -\frac{{\rm Li}_2\left (\frac{e g x + d g}{e f - d g} + 1\right )}{g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-g*(e*x+d)/(-d*g+e*f))/(g*x+f),x, algorithm="fricas")

[Out]

-dilog((e*g*x + d*g)/(e*f - d*g) + 1)/g

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (- \frac{d g}{- d g + e f} - \frac{e g x}{- d g + e f} \right )}}{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(-g*(e*x+d)/(-d*g+e*f))/(g*x+f),x)

[Out]

Integral(log(-d*g/(-d*g + e*f) - e*g*x/(-d*g + e*f))/(f + g*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (-\frac{{\left (e x + d\right )} g}{e f - d g}\right )}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(-g*(e*x+d)/(-d*g+e*f))/(g*x+f),x, algorithm="giac")

[Out]

integrate(log(-(e*x + d)*g/(e*f - d*g))/(g*x + f), x)