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

Optimal. Leaf size=433 \[ -\frac{\text{PolyLog}\left (3,\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}-\frac{\text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}+\frac{\log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \text{PolyLog}\left (2,\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}+\frac{\text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b c-a d}-\frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}-\frac{\log \left (\frac{b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}+\frac{\log \left (1-\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)} \]

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

________________________________________________________________________________________

Rubi [A]  time = 0.594495, antiderivative size = 445, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 6, integrand size = 65, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.092, Rules used = {2507, 2489, 2488, 2506, 6610, 2503} \[ -\frac{\text{PolyLog}\left (3,1-\frac{(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right )}{b c-a d}+\frac{\text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}-\frac{\log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \text{PolyLog}\left (2,1-\frac{(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right )}{b c-a d}+\frac{\text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}-\frac{\log \left (\frac{b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}+\frac{\log \left (\frac{(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2507

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_
.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_Symbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[(k*Log[i*(j
*(g + h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] - Dist[(k*h*t*u)/
(p*r*(s + 1)*(b*c - a*d)), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]]
/; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s,
-1]

Rule 2489

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

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]

Rule 2503

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Simp[(Log[e*(f*(
a + b*x)^p*(c + d*x)^q)^r]^s*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/(b*g - a*h), x] + Dist[(
p*r*s*(b*c - a*d))/(b*g - a*h), Int[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)*Log[-(((b*c - a*d)*(g + h*x)
)/((d*g - c*h)*(a + b*x)))])/((a + b*x)*(c + d*x)), x], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]] /; 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] && LinearQ[Simplify[1/
(u*(a + b*x))], x]

Rubi steps

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

Mathematica [B]  time = 0.798781, size = 908, normalized size = 2.1 \[ \frac{\log \left (\frac{b (e+f x)}{b e-a f}\right ) \log ^2(c+d x)-\log \left (\frac{d (e+f x)}{d e-c f}\right ) \log ^2(c+d x)-2 \log (a+b x) \log \left (\frac{b (e+f x)}{b e-a f}\right ) \log (c+d x)-2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right ) \log (c+d x)+2 \log (a+b x) \log \left (\frac{d (e+f x)}{d e-c f}\right ) \log (c+d x)+2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right ) \log (c+d x)-\log \left (\frac{a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )-\log ^2\left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right )-\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right )+2 \log (a+b x) \log \left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right )+2 \log \left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right )+\log ^2\left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right )-2 \log (a+b x) \log \left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right )-2 \log \left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right )+\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{(a d-b c) (e+f x)}{(d e-c f) (a+b x)}\right )-2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )+2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{PolyLog}\left (2,\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )-2 \text{PolyLog}\left (3,\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 b c-2 a d} \]

Antiderivative was successfully verified.

[In]

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

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Maple [F]  time = 2.336, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({\frac{ \left ( -af+be \right ) \left ( dx+c \right ) }{ \left ( -cf+de \right ) \left ( bx+a \right ) }} \right ) \ln \left ({\frac{b \left ( fx+e \right ) }{-af+be}} \right ) }\, dx \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))*ln(b*(f*x+e)/(-a*f+b*e))/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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))*log(b*(f*x+e)/(-a*f+b*e))/(b*x+a)/(d*x+c),x, algorithm="m
axima")

[Out]

Exception raised: RuntimeError

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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 ) \log \left (\frac{b f x + b e}{b e - a f}\right )}{b d x^{2} + a c +{\left (b c + a d\right )} x}, 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))*log(b*(f*x+e)/(-a*f+b*e))/(b*x+a)/(d*x+c),x, algorithm="f
ricas")

[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))*log((b*f*x + b*e)/(b*e -
a*f))/(b*d*x^2 + a*c + (b*c + a*d)*x), 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))*ln(b*(f*x+e)/(-a*f+b*e))/(b*x+a)/(d*x+c),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 (f x + e\right )} b}{b e - a f}\right ) \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 )}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{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))*log(b*(f*x+e)/(-a*f+b*e))/(b*x+a)/(d*x+c),x, algorithm="g
iac")

[Out]

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