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

Optimal result

Integrand size = 27, antiderivative size = 24 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {e (f+g x)}{e f-d g}\right )}{g} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2440, 2438} \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {e (f+g x)}{e f-d g}\right )}{g} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {e (f+g x)}{e f-d g}\right )}{g} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(24)=48\).

Time = 1.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25

method	result	size
derivativedivides	\(\frac {\left (-d g +e f \right ) \operatorname {dilog}\left (-\frac {g e x}{-d g +e f}-\frac {d g}{-d g +e f}\right )}{g \left (d g -e f \right )}\)	\(54\)
default	\(\frac {\left (-d g +e f \right ) \operatorname {dilog}\left (-\frac {g e x}{-d g +e f}-\frac {d g}{-d g +e f}\right )}{g \left (d g -e f \right )}\)	\(54\)
risch	\(-\frac {\operatorname {dilog}\left (-\frac {g e x}{-d g +e f}-\frac {d g}{-d g +e f}\right ) d}{d g -e f}+\frac {\operatorname {dilog}\left (-\frac {g e x}{-d g +e f}-\frac {d g}{-d g +e f}\right ) e f}{g \left (d g -e f \right )}\)	\(93\)
parts	\(\frac {\ln \left (-\frac {g \left (e x +d \right )}{-d g +e f}\right ) \ln \left (g x +f \right )}{g}-\frac {e \left (\frac {\operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{e}+\frac {\ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{e}\right )}{g}\)	\(106\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {{\rm Li}_2\left (\frac {e g x + d g}{e f - d g} + 1\right )}{g} \]

[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

Sympy [F]

\[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=\int \frac {\log {\left (- \frac {d g}{- d g + e f} - \frac {e g x}{- d g + e f} \right )}}{f + g x}\, dx \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (23) = 46\).

Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.25 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {\log \left (e x + d\right ) \log \left (g x + f\right )}{g} + \frac {\log \left (g x + f\right ) \log \left (-\frac {{\left (e x + d\right )} g}{e f - d g}\right )}{g} + \frac {\log \left (e x + d\right ) \log \left (\frac {e g x + d g}{e f - d g} + 1\right ) + {\rm Li}_2\left (-\frac {e g x + d g}{e f - d g}\right )}{g} \]

[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

Giac [F]

\[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=\int { \frac {\log \left (-\frac {{\left (e x + d\right )} g}{e f - d g}\right )}{g x + f} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 1.51 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {\log \left (-\frac {g (d+e x)}{e f-d g}\right )}{f+g x} \, dx=-\frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {g\,\left (d+e\,x\right )}{d\,g-e\,f}\right )}{g} \]

[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