3.1.71 \(\int \frac {\log (-\frac {g (d+e x)}{e f-d g})}{f+g x} \, dx\) [71]

Optimal. Leaf size=24 \[ -\frac {\text {Li}_2\left (\frac {e (f+g x)}{e f-d g}\right )}{g} \]

[Out]

-polylog(2,e*(g*x+f)/(-d*g+e*f))/g

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {2440, 2438} \begin {gather*} -\frac {\text {PolyLog}\left (2,\frac {e (f+g x)}{e f-d g}\right )}{g} \end {gather*}

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 2438

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]

Rule 2440

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} -\frac {\text {Li}_2\left (\frac {e (f+g x)}{e f-d g}\right )}{g} \end {gather*}

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 = 0.37, size = 35, normalized size = 1.46

method	result	size
derivativedivides	\(-\frac {\dilog \left (\frac {e g x}{d g -e f}+\frac {d g}{d g -e f}\right )}{g}\)	\(35\)
default	\(-\frac {\dilog \left (\frac {e g x}{d g -e f}+\frac {d g}{d g -e f}\right )}{g}\)	\(35\)
risch	\(-\frac {\dilog \left (\frac {e g x}{d g -e f}+\frac {d g}{d g -e f}\right )}{g}\)	\(35\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (26) = 52\).
time = 0.29, size = 109, normalized size = 4.54 \begin {gather*} -\frac {\log \left (g x + f\right ) \log \left (x e + d\right )}{g} + \frac {\log \left (g x + f\right ) \log \left (\frac {{\left (x e + d\right )} g}{d g - f e}\right )}{g} + \frac {\log \left (x e + d\right ) \log \left (-\frac {g x e + d g}{d g - f e} + 1\right ) + {\rm Li}_2\left (\frac {g x e + d g}{d g - f e}\right )}{g} \end {gather*}

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

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Fricas [A]
time = 0.35, size = 30, normalized size = 1.25 \begin {gather*} -\frac {{\rm Li}_2\left (-\frac {g x e + d g}{d g - f e} + 1\right )}{g} \end {gather*}

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(-(g*x*e + d*g)/(d*g - f*e) + 1)/g

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [B]
time = 0.39, size = 23, normalized size = 0.96 \begin {gather*} -\frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {g\,\left (d+e\,x\right )}{d\,g-e\,f}\right )}{g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-dilog((g*(d + e*x))/(d*g - e*f))/g

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