3.104 \(\int \frac{\log ^2(\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)})}{a+b x} \, dx\)

Optimal. Leaf size=140 \[ -\frac{2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b}+\frac{2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b}-\frac{\log \left (\frac{a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b} \]

[Out]

-((Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2)/b) - (2*Log[((b*e
- a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b + (2*PolyLog[3, (b*(c +
d*x))/(d*(a + b*x))])/b

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Rubi [A]  time = 0.180268, antiderivative size = 149, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2488, 2506, 6610} \[ -\frac{2 \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b}+\frac{2 \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b}-\frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2/(a + b*x),x]

[Out]

-((Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2)/b) - (2*Log[((b*e
- a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/b + (2*PolyLog[3, 1 + (
b*c - a*d)/(d*(a + b*x))])/b

Rule 2488

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_)),
x_Symbol] :> -Simp[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/h, x] + Dist[(p
*r*s*(b*c - a*d))/h, Int[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a
+ b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]

Rule 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \frac{\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{a+b x} \, dx &=-\frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b}-\frac{(2 (b c-a d)) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b}\\ &=-\frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b}-\frac{2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}-\frac{(2 (b c-a d)) \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b}\\ &=-\frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b}-\frac{2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}+\frac{2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0421648, size = 135, normalized size = 0.96 \[ \frac{-2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )+2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )-\log \left (\frac{a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2/(a + b*x),x]

[Out]

(-(Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2) - 2*Log[((b*e - a
*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] + 2*PolyLog[3, (b*(c + d*x))/(
d*(a + b*x))])/b

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Maple [B]  time = 0.065, size = 879, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a),x)

[Out]

-1/(b*c*f-b*d*e)*ln(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f-b*e)/b)^2*ln(1-(b*c*f-b*d*e)/(a*
d*f-b*d*e)*(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f-b*e)/b))*c*f+1/(b*c*f-b*d*e)*ln(-(a*f-b*
e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f-b*e)/b)^2*ln(1-(b*c*f-b*d*e)/(a*d*f-b*d*e)*(-(a*f-b*e)*(a*d-
b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f-b*e)/b))*d*e-2/(b*c*f-b*d*e)*ln(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*
x+a)+d/(c*f-d*e)*(a*f-b*e)/b)*polylog(2,(b*c*f-b*d*e)/(a*d*f-b*d*e)*(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+
d/(c*f-d*e)*(a*f-b*e)/b))*c*f+2/(b*c*f-b*d*e)*ln(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f-b*e
)/b)*polylog(2,(b*c*f-b*d*e)/(a*d*f-b*d*e)*(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*e)*(a*f-b*e)/b))
*d*e+2/(b*c*f-b*d*e)*polylog(3,(b*c*f-b*d*e)/(a*d*f-b*d*e)*(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(b*x+a)+d/(c*f-d*
e)*(a*f-b*e)/b))*c*f-2/(b*c*f-b*d*e)*polylog(3,(b*c*f-b*d*e)/(a*d*f-b*d*e)*(-(a*f-b*e)*(a*d-b*c)/b/(c*f-d*e)/(
b*x+a)+d/(c*f-d*e)*(a*f-b*e)/b))*d*e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\log \left (d x + c\right )^{3}}{a} - \int -\frac{{\left (\log \left (-b e + a f\right )^{2} - 2 \, \log \left (-b e + a f\right ) \log \left (-d e + c f\right ) + \log \left (-d e + c f\right )^{2}\right )} b d x +{\left (\log \left (-b e + a f\right )^{2} - 2 \, \log \left (-b e + a f\right ) \log \left (-d e + c f\right ) + \log \left (-d e + c f\right )^{2}\right )} b c +{\left (b d x + b c\right )} \log \left (b x + a\right )^{2} - 2 \,{\left (b d x{\left (\log \left (-b e + a f\right ) - \log \left (-d e + c f\right )\right )} + b c{\left (\log \left (-b e + a f\right ) - \log \left (-d e + c f\right )\right )}\right )} \log \left (b x + a\right ) + 2 \,{\left (b d x{\left (\log \left (-b e + a f\right ) - \log \left (-d e + c f\right )\right )} + b c{\left (\log \left (-b e + a f\right ) - \log \left (-d e + c f\right )\right )} -{\left (2 \, b d x + b c + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b^{2} d x^{2} + a b c +{\left (b^{2} c + a b d\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a),x, algorithm="maxima")

[Out]

log(d*x + c)^3/a - integrate(-((log(-b*e + a*f)^2 - 2*log(-b*e + a*f)*log(-d*e + c*f) + log(-d*e + c*f)^2)*b*d
*x + (log(-b*e + a*f)^2 - 2*log(-b*e + a*f)*log(-d*e + c*f) + log(-d*e + c*f)^2)*b*c + (b*d*x + b*c)*log(b*x +
a)^2 - 2*(b*d*x*(log(-b*e + a*f) - log(-d*e + c*f)) + b*c*(log(-b*e + a*f) - log(-d*e + c*f)))*log(b*x + a) +
2*(b*d*x*(log(-b*e + a*f) - log(-d*e + c*f)) + b*c*(log(-b*e + a*f) - log(-d*e + c*f)) - (2*b*d*x + b*c + a*d
)*log(b*x + a))*log(d*x + c))/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\frac{b c e - a c f +{\left (b d e - a d f\right )} x}{a d e - a c f +{\left (b d e - b c f\right )} x}\right )^{2}}{b x + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a),x, algorithm="fricas")

[Out]

integral(log((b*c*e - a*c*f + (b*d*e - a*d*f)*x)/(a*d*e - a*c*f + (b*d*e - b*c*f)*x))^2/(b*x + a), x)

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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((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))**2/(b*x+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{{\left (b e - a f\right )}{\left (d x + c\right )}}{{\left (d e - c f\right )}{\left (b x + a\right )}}\right )^{2}}{b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2/(b*x+a),x, algorithm="giac")

[Out]

integrate(log((b*e - a*f)*(d*x + c)/((d*e - c*f)*(b*x + a)))^2/(b*x + a), x)