3.255 \(\int \frac{\log (\frac{c+d x}{a+b x})}{(a+b x) ((a-c) h+(b-d) h x)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\text{PolyLog}\left (2,1-\frac{c+d x}{a+b x}\right )}{h (b c-a d)} \]

[Out]

-(PolyLog[2, 1 - (c + d*x)/(a + b*x)]/((b*c - a*d)*h))

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Rubi [A]  time = 0.0950319, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2502, 2315} \[ -\frac{\text{PolyLog}\left (2,1-\frac{c+d x}{a+b x}\right )}{h (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(PolyLog[2, 1 - (c + d*x)/(a + b*x)]/((b*c - a*d)*h))

Rule 2502

Int[Log[((e_.)*((c_.) + (d_.)*(x_)))/((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{g = Coeff[Simplify[1/(u*(a
+ b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Dist[(b - d*e)/(h*(b*c - a*d)), Subst[Int[Log[
e*x]/(1 - e*x), x], x, (c + d*x)/(a + b*x)], x] /; EqQ[g*(b - d*e) - h*(a - c*e), 0]] /; FreeQ[{a, b, c, d, e}
, x] && NeQ[b*c - a*d, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]

Rule 2315

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

Rubi steps

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

Mathematica [B]  time = 0.173132, size = 298, normalized size = 9.03 \[ \frac{2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+2 \text{PolyLog}\left (2,-\frac{b (a+b x-c-d x)}{b c-a d}\right )-2 \text{PolyLog}\left (2,-\frac{d (-a-b x+c+d x)}{a d-b c}\right )-\log ^2\left (\frac{a d-b c}{d (a+b x)}\right )-2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log \left (\frac{a d-b c}{d (a+b x)}\right )+2 \log \left (\frac{c+d x}{a+b x}\right ) \log \left (\frac{a d-b c}{d (a+b x)}\right )+2 \log \left (\frac{(b-d) (a+b x)}{b c-a d}\right ) \log (a+b x-c-d x)-2 \log (a+b x-c-d x) \log \left (\frac{(b-d) (c+d x)}{b c-a d}\right )+2 \log (a+b x-c-d x) \log \left (\frac{c+d x}{a+b x}\right )}{h (2 b c-2 a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.065, size = 42, normalized size = 1.3 \begin{align*}{\frac{1}{h \left ( ad-bc \right ) }{\it dilog} \left ( -{\frac{ad-bc}{b \left ( bx+a \right ) }}+{\frac{d}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((d*x+c)/(b*x+a))/(b*x+a)/((a-c)*h+(b-d)*h*x),x)

[Out]

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

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Maxima [B]  time = 1.27618, size = 482, normalized size = 14.61 \begin{align*}{\left (\frac{\log \left (-{\left (b - d\right )} x - a + c\right )}{{\left (b c - a d\right )} h} - \frac{\log \left (b x + a\right )}{{\left (b c - a d\right )} h}\right )} \log \left (\frac{d x + c}{b x + a}\right ) + \frac{2 \, \log \left (-{\left (b - d\right )} x - a + c\right ) \log \left (b x + a\right ) - \log \left (b x + a\right )^{2}}{2 \,{\left (b c h - a d h\right )}} + \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 h - a d h} - \frac{\log \left (b x + a\right ) \log \left (-\frac{a{\left (b - d\right )} +{\left (b^{2} - b d\right )} x}{b c - a d} + 1\right ) +{\rm Li}_2\left (\frac{a{\left (b - d\right )} +{\left (b^{2} - b d\right )} x}{b c - a d}\right )}{b c h - a d h} - \frac{\log \left (-{\left (b - d\right )} x - a + c\right ) \log \left (\frac{a d - c d +{\left (b d - d^{2}\right )} x}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{a d - c d +{\left (b d - d^{2}\right )} x}{b c - a d}\right )}{b c h - a d h} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.461014, size = 68, normalized size = 2.06 \begin{align*} -\frac{{\rm Li}_2\left (-\frac{d x + c}{b x + a} + 1\right )}{{\left (b c - a d\right )} h} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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