\(\int \frac {\log (\frac {a-c g+b x-d g x}{a+b x})}{(a+b x) (c+d x)} \, dx\) [258]

Optimal result

Integrand size = 38, antiderivative size = 27 \[ \int \frac {\log \left (\frac {a-c g+b x-d g x}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {g (c+d x)}{a+b x}\right )}{b c-a d} \]

[Out]

polylog(2,g*(d*x+c)/(b*x+a))/(-a*d+b*c)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2571, 2565, 2352} \[ \int \frac {\log \left (\frac {a-c g+b x-d g x}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {g (c+d x)}{a+b x}\right )}{b c-a d} \]

[In]

Int[Log[(a - c*g + b*x - d*g*x)/(a + b*x)]/((a + b*x)*(c + d*x)),x]

[Out]

PolyLog[2, (g*(c + d*x))/(a + b*x)]/(b*c - a*d)

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2565

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(q + 1)*(i/d)^q, Subst[Int[(b*f - a*g - (d*f - c
*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e
, f, g, h, i, A, B, n}, x] && NeQ[b*c - a*d, 0] && IntegersQ[m, q] && IGtQ[p, 0] && EqQ[d*h - c*i, 0]

Rule 2571

Int[((A_.) + Log[(e_.)*((u_)/(v_))^(n_.)]*(B_.))^(p_.)*(w_)^(m_.)*(y_)^(q_.), x_Symbol] :> Int[ExpandToSum[w,
x]^m*ExpandToSum[y, x]^q*(A + B*Log[e*(ExpandToSum[u, x]/ExpandToSum[v, x])^n])^p, x] /; FreeQ[{e, A, B, m, n,
p, q}, x] && LinearQ[{u, v, w, y}, x] &&  !LinearMatchQ[{u, v, w, y}, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (\frac {a-c g+(b-d g) x}{a+b x}\right )}{(a+b x) (c+d x)} \, dx \\ & = \text {Subst}\left (\int \frac {\log (x)}{-d (a-c g)+c (b-d g)+(-b c+a d) x} \, dx,x,\frac {a-c g+(b-d g) x}{a+b x}\right ) \\ & = \frac {\text {Li}_2\left (\frac {g (c+d x)}{a+b x}\right )}{b c-a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(375\) vs. \(2(27)=54\).

Time = 0.01 (sec) , antiderivative size = 375, normalized size of antiderivative = 13.89 \[ \int \frac {\log \left (\frac {a-c g+b x-d g x}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {-\log ^2\left (\frac {a}{b}+x\right )+2 \log \left (\frac {a}{b}+x\right ) \log (a+b x)-2 \log \left (\frac {a-c g}{b-d g}+x\right ) \log (a+b x)+2 \log \left (\frac {a-c g}{b-d g}+x\right ) \log \left (\frac {(b-d g) (a+b x)}{(b c-a d) g}\right )-2 \log \left (\frac {a}{b}+x\right ) \log (c+d x)+2 \log \left (\frac {a-c g}{b-d g}+x\right ) \log (c+d x)+2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )-2 \log \left (\frac {a-c g}{b-d g}+x\right ) \log \left (\frac {(b-d g) (c+d x)}{b c-a d}\right )+2 \log (a+b x) \log \left (\frac {a-c g+b x-d g x}{a+b x}\right )-2 \log (c+d x) \log \left (\frac {a-c g+b x-d g x}{a+b x}\right )+2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+2 \operatorname {PolyLog}\left (2,-\frac {b (a-c g+b x-d g x)}{(b c-a d) g}\right )-2 \operatorname {PolyLog}\left (2,-\frac {d (-a+c g-b x+d g x)}{-b c+a d}\right )}{2 b c-2 a d} \]

[In]

Integrate[Log[(a - c*g + b*x - d*g*x)/(a + b*x)]/((a + b*x)*(c + d*x)),x]

[Out]

(-Log[a/b + x]^2 + 2*Log[a/b + x]*Log[a + b*x] - 2*Log[(a - c*g)/(b - d*g) + x]*Log[a + b*x] + 2*Log[(a - c*g)
/(b - d*g) + x]*Log[((b - d*g)*(a + b*x))/((b*c - a*d)*g)] - 2*Log[a/b + x]*Log[c + d*x] + 2*Log[(a - c*g)/(b
- d*g) + x]*Log[c + d*x] + 2*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] - 2*Log[(a - c*g)/(b - d*g) + x]*Log[
((b - d*g)*(c + d*x))/(b*c - a*d)] + 2*Log[a + b*x]*Log[(a - c*g + b*x - d*g*x)/(a + b*x)] - 2*Log[c + d*x]*Lo
g[(a - c*g + b*x - d*g*x)/(a + b*x)] + 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 2*PolyLog[2, -((b*(a - c*g
+ b*x - d*g*x))/((b*c - a*d)*g))] - 2*PolyLog[2, -((d*(-a + c*g - b*x + d*g*x))/(-(b*c) + a*d))])/(2*b*c - 2*
a*d)

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70

method	result	size
derivativedivides	\(-\frac {\operatorname {dilog}\left (\frac {a d g -b c g}{b \left (b x +a \right )}+\frac {-d g +b}{b}\right )}{a d -c b}\)	\(46\)
default	\(-\frac {\operatorname {dilog}\left (\frac {a d g -b c g}{b \left (b x +a \right )}+\frac {-d g +b}{b}\right )}{a d -c b}\)	\(46\)
risch	\(-\frac {\operatorname {dilog}\left (\frac {a d g -b c g}{b \left (b x +a \right )}+\frac {-d g +b}{b}\right )}{a d -c b}\)	\(46\)
parts	\(\frac {\ln \left (\frac {-d g x +b x -c g +a}{b x +a}\right ) \ln \left (d x +c \right )}{a d -c b}-\frac {\ln \left (\frac {-d g x +b x -c g +a}{b x +a}\right ) \ln \left (b x +a \right )}{a d -c b}-\frac {g \left (\frac {b \ln \left (b x +a \right )^{2}}{2 g \left (a d -c b \right )}-\frac {b \left (-d g +b \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (-d g +b \right ) \left (b x +a \right )+a d g -b c g}{a d g -b c g}\right )}{-d g +b}+\frac {\ln \left (b x +a \right ) \ln \left (\frac {\left (-d g +b \right ) \left (b x +a \right )+a d g -b c g}{a d g -b c g}\right )}{-d g +b}\right )}{g \left (a d -c b \right )}\right )}{b}+\frac {g \left (-\frac {d \left (-d g +b \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (-d g +b \right ) \left (d x +c \right )+a d -c b}{a d -c b}\right )}{-d g +b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {\left (-d g +b \right ) \left (d x +c \right )+a d -c b}{a d -c b}\right )}{-d g +b}\right )}{\left (a d -c b \right ) g}+\frac {d b \left (\frac {\operatorname {dilog}\left (\frac {a d -c b +b \left (d x +c \right )}{a d -c b}\right )}{b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {a d -c b +b \left (d x +c \right )}{a d -c b}\right )}{b}\right )}{\left (a d -c b \right ) g}\right )}{d}\)	\(435\)

[In]

int(ln((-d*g*x+b*x-c*g+a)/(b*x+a))/(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)

[Out]

-1/(a*d-b*c)*dilog((a*d*g-b*c*g)/b/(b*x+a)+(-d*g+b)/b)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {\log \left (\frac {a-c g+b x-d g x}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {{\rm Li}_2\left (\frac {c g + {\left (d g - b\right )} x - a}{b x + a} + 1\right )}{b c - a d} \]

[In]

integrate(log((-d*g*x+b*x-c*g+a)/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

dilog((c*g + (d*g - b)*x - a)/(b*x + a) + 1)/(b*c - a*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (\frac {a-c g+b x-d g x}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(ln((-d*g*x+b*x-c*g+a)/(b*x+a))/(b*x+a)/(d*x+c),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 343, normalized size of antiderivative = 12.70 \[ \int \frac {\log \left (\frac {a-c g+b x-d g x}{a+b x}\right )}{(a+b x) (c+d x)} \, dx={\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (-\frac {d g x + c g - b x - a}{b x + a}\right ) + \frac {\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right )}{2 \, {\left (b c - a d\right )}} - \frac {\log \left (b x + a\right ) \log \left (\frac {{\left (d g - b\right )} a + {\left (b d g - b^{2}\right )} x}{b c g - a d g} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (d g - b\right )} a + {\left (b d g - b^{2}\right )} x}{b c g - a d g}\right )}{b c - a d} + \frac {\log \left (d x + c\right ) \log \left (\frac {c d g - b c + {\left (d^{2} g - b d\right )} x}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {c d g - b c + {\left (d^{2} g - b d\right )} x}{b c - a d}\right )}{b c - a d} + \frac {\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )}{b c - a d} \]

[In]

integrate(log((-d*g*x+b*x-c*g+a)/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d))*log(-(d*g*x + c*g - b*x - a)/(b*x + a)) + 1/2*(log(b*x +
a)^2 - 2*log(b*x + a)*log(d*x + c))/(b*c - a*d) - (log(b*x + a)*log(((d*g - b)*a + (b*d*g - b^2)*x)/(b*c*g -
a*d*g) + 1) + dilog(-((d*g - b)*a + (b*d*g - b^2)*x)/(b*c*g - a*d*g)))/(b*c - a*d) + (log(d*x + c)*log((c*d*g
- b*c + (d^2*g - b*d)*x)/(b*c - a*d) + 1) + dilog(-(c*d*g - b*c + (d^2*g - b*d)*x)/(b*c - a*d)))/(b*c - a*d) +
(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))/(b*c - a*d)

Giac [F]

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

[In]

integrate(log((-d*g*x+b*x-c*g+a)/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

integrate(log(-(d*g*x + c*g - b*x - a)/(b*x + a))/((b*x + a)*(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (\frac {a-c g+b x-d g x}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\ln \left (\frac {a-c\,g+b\,x-d\,g\,x}{a+b\,x}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

int(log((a - c*g + b*x - d*g*x)/(a + b*x))/((a + b*x)*(c + d*x)),x)

[Out]

int(log((a - c*g + b*x - d*g*x)/(a + b*x))/((a + b*x)*(c + d*x)), x)