\(\int \frac {\log (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}) \log (\frac {b (e+f x)}{b e-a f})}{(a+b x) (c+d x)} \, dx\) [108]

Optimal result

Integrand size = 65, antiderivative size = 433 \[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b c-a d)}-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{2 (b c-a d)}+\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (1-\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b c-a d)}-\frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b c-a d}+\frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b c-a d}+\frac {\operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b c-a d}-\frac {\operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b c-a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.092, Rules used = {2589, 2554, 2404, 2354, 2421, 6724} \[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=-\frac {\operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b c-a d}-\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}+\frac {\operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}-\frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}+\frac {\log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}+\frac {\operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b c-a d} \]

[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]

-1/2*(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*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)

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2554

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.), x_Symbol] :> Dist[b*c - a*d, Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^
(m + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGt
Q[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rule 2589

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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1855\) vs. \(2(433)=866\).

Time = 0.44 (sec) , antiderivative size = 1855, normalized size of antiderivative = 4.28 \[ \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 {2 \log \left (\frac {c}{d}+x\right ) \log \left (\frac {e}{f}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {e}{f}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )-2 (\log (a+b x)-\log (c+d x)) \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )+\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )\right ) \left (\log \left (\frac {e}{f}+x\right )-\log \left (\frac {b (e+f x)}{b e-a f}\right )\right )+\left (\log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log \left (\frac {f (a+b x)}{-b e+a f}\right )\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right ) \left (-2 \log \left (\frac {c}{d}+x\right )+\log \left (\frac {b (e+f x)}{b e-a f}\right )\right )+\log ^2\left (\frac {a}{b}+x\right ) \left (-\log \left (\frac {e}{f}+x\right )+\log \left (\frac {b (e+f x)}{b e-a f}\right )\right )+\left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (\frac {f (c+d x)}{-d e+c f}\right )\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right ) \left (-2 \log \left (\frac {a}{b}+x\right )+\log \left (\frac {d (e+f x)}{d e-c f}\right )\right )+\log ^2\left (\frac {c}{d}+x\right ) \left (-\log \left (\frac {e}{f}+x\right )+\log \left (\frac {d (e+f x)}{d e-c f}\right )\right )+2 \left (-\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {f (c+d x)}{-d e+c f}\right )\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right ) \log \left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )+\left (\log \left (\frac {-b e+a f}{f (a+b x)}\right )+\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )\right ) \log ^2\left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )+2 \left (-\log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {f (a+b x)}{-b e+a f}\right )\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right ) \log \left (\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )+\left (\log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {-d e+c f}{f (c+d x)}\right )-\log \left (\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )+2 \left (\log \left (\frac {e}{f}+x\right )-\log \left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+2 \log \left (\frac {a}{b}+x\right ) \operatorname {PolyLog}\left (2,\frac {f (a+b x)}{-b e+a f}\right )+2 \left (\log \left (\frac {e}{f}+x\right )-\log \left (\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )+\left (\log \left (\frac {e}{f}+x\right )-\log \left (\frac {b (e+f x)}{b e-a f}\right )\right ) \left (\log ^2\left (\frac {a}{b}+x\right )+\log ^2\left (\frac {c}{d}+x\right )-2 \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-2 \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )+2 \log \left (\frac {c}{d}+x\right ) \operatorname {PolyLog}\left (2,\frac {f (c+d x)}{-d e+c f}\right )+2 \left (\log \left (\frac {c}{d}+x\right )+\log \left (\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (e+f x)}{b e-a f}\right )+2 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )+\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )\right ) \left (\log \left (\frac {e}{f}+x\right ) \left (\log \left (\frac {f (a+b x)}{-b e+a f}\right )-\log \left (\frac {f (c+d x)}{-d e+c f}\right )\right )+\operatorname {PolyLog}\left (2,\frac {b (e+f x)}{b e-a f}\right )-\operatorname {PolyLog}\left (2,\frac {d (e+f x)}{d e-c f}\right )\right )+2 \left (\log \left (\frac {a}{b}+x\right )+\log \left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (e+f x)}{d e-c f}\right )+2 \log \left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right ) \left (\operatorname {PolyLog}\left (2,\frac {b (e+f x)}{f (a+b x)}\right )-\operatorname {PolyLog}\left (2,-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )\right )+2 \log \left (\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right ) \left (\operatorname {PolyLog}\left (2,\frac {d (e+f x)}{f (c+d x)}\right )-\operatorname {PolyLog}\left (2,\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )\right )-2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{-b c+a d}\right )-2 \operatorname {PolyLog}\left (3,\frac {f (a+b x)}{-b e+a f}\right )-2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )-2 \operatorname {PolyLog}\left (3,\frac {f (c+d x)}{-d e+c f}\right )-2 \operatorname {PolyLog}\left (3,\frac {b (e+f x)}{b e-a f}\right )-2 \operatorname {PolyLog}\left (3,\frac {d (e+f x)}{d e-c f}\right )-2 \operatorname {PolyLog}\left (3,\frac {b (e+f x)}{f (a+b x)}\right )+2 \operatorname {PolyLog}\left (3,-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )-2 \operatorname {PolyLog}\left (3,\frac {d (e+f x)}{f (c+d x)}\right )+2 \operatorname {PolyLog}\left (3,\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{2 (b c-a d)} \]

[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]

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

Maple [F]

\[\int \frac {\ln \left (\frac {\left (-a f +b e \right ) \left (d x +c \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right ) \ln \left (\frac {b \left (f x +e \right )}{-a f +b e}\right )}{\left (b x +a \right ) \left (d x +c \right )}d x\]

[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)

Fricas [F]

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

[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)

Sympy [F(-1)]

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

[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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[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 >> ECL says: Memory limit reached. Please jump to an outer pointer, quit progra
m and enlarge thememory limits before executing the program again.

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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