3.3.56 \(\int \frac {\log (\frac {a-c g+(b-d g) x}{a+b x})}{(a+b x) (c+d x)} \, dx\) [256]

Optimal. Leaf size=27 \[ \frac {\text {Li}_2\left (\frac {g (c+d x)}{a+b x}\right )}{b c-a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2565, 2352} \begin {gather*} \frac {\text {PolyLog}\left (2,\frac {g (c+d x)}{a+b x}\right )}{b c-a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[(a - c*g + (b - 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(320\) vs. \(2(27)=54\).
time = 0.18, size = 320, normalized size = 11.85 \begin {gather*} \frac {\log ^2\left (\frac {(b c-a d) g}{(b-d g) (a+b x)}\right )-2 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+2 \log (c+d x) \log \left (-\frac {d (a-c g+b x-d g x)}{b c-a d}\right )+2 \log \left (\frac {(b c-a d) g}{(b-d g) (a+b x)}\right ) \log \left (-\frac {b (a-c g+b x-d g x)}{(b c-a d) g}\right )-2 \log \left (\frac {(b c-a d) g}{(b-d g) (a+b x)}\right ) \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 \text {Li}_2\left (\frac {(b-d g) (a+b x)}{(b c-a d) g}\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+2 \text {Li}_2\left (\frac {(b-d g) (c+d x)}{b c-a d}\right )}{2 b c-2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.86, size = 45, normalized size = 1.67

method	result	size
derivativedivides	\(-\frac {\dilog \left (\frac {-d g +b}{b}+\frac {g \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{a d -c b}\)	\(45\)
default	\(-\frac {\dilog \left (\frac {-d g +b}{b}+\frac {g \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{a d -c b}\)	\(45\)
risch	\(-\frac {\dilog \left (\frac {-d g +b}{b}+\frac {g \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{a d -c b}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (26) = 52\).
time = 0.30, size = 344, normalized size = 12.74 \begin {gather*} {\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 {c g + {\left (d g - b\right )} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 38, normalized size = 1.41 \begin {gather*} \frac {{\rm Li}_2\left (\frac {c g + {\left (d g - b\right )} x - a}{b x + a} + 1\right )}{b c - a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((a-c*g+(-d*g+b)*x)/(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
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \left (\frac {a-c\,g+x\,\left (b-d\,g\right )}{a+b\,x}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________