\(\int \frac {\log ^2(\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)})}{(a+b x) (e+f x)} \, dx\) [106]

Optimal result

Integrand size = 49, antiderivative size = 204 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (1-\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f}-\frac {2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f}+\frac {2 \operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {2566, 2354, 2421, 6724} \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\frac {2 \operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f}-\frac {2 \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f}-\frac {\log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b e-a f} \]

[In]

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

[Out]

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

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 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 2566

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(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] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, q] && IGtQ
[p, 0] && EqQ[d*h - c*i, 0]

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}& = \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 ) \\ & = -\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 )}{b e-a f}+\frac {2 \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 e-a f} \\ & = -\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 )}{b e-a f}-\frac {2 \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 e-a f}+\frac {2 \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 e-a f} \\ & = -\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 )}{b e-a f}-\frac {2 \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 e-a f}+\frac {2 \text {Li}_3\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b e-a f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1636\) vs. \(2(204)=408\).

Time = 0.63 (sec) , antiderivative size = 1636, normalized size of antiderivative = 8.02 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\frac {-2 \log ^3\left (\frac {a}{b}+x\right )+3 \log ^2\left (\frac {a}{b}+x\right ) \log (a+b x)-6 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log (a+b x)+3 \log ^2\left (\frac {c}{d}+x\right ) \log (a+b x)+6 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )-3 \log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+3 \log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )-3 \log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+6 \log \left (\frac {a}{b}+x\right ) \log (a+b x) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )-6 \log \left (\frac {c}{d}+x\right ) \log (a+b x) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+6 \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+3 \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 )+3 \log (a+b x) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )-3 \log ^2\left (\frac {a}{b}+x\right ) \log (e+f x)+6 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log (e+f x)-3 \log ^2\left (\frac {c}{d}+x\right ) \log (e+f x)-6 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log (e+f x)+6 \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 ) \log (e+f x)-3 \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log (e+f x)+3 \log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )-6 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {f (c+d x)}{-d e+c f}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )+3 \log ^2\left (\frac {f (c+d x)}{-d e+c f}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )+6 \log \left (\frac {a}{b}+x\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 )-6 \log \left (\frac {f (c+d x)}{-d e+c f}\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 )+3 \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 )-6 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )+3 \log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )+6 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {f (c+d x)}{-d e+c f}\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )-3 \log ^2\left (\frac {f (c+d x)}{-d e+c f}\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )-6 \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 ) \log \left (\frac {d (e+f x)}{d e-c f}\right )+6 \log \left (\frac {f (c+d x)}{-d e+c f}\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 )-3 \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)}{(d e-c f) (a+b x)}\right )+6 \log \left (\frac {a}{b}+x\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+6 \left (\log \left (\frac {a}{b}+x\right )+\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )+6 \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 )-6 \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 )-6 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{-b c+a d}\right )-6 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )-6 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{3 b e-3 a f} \]

[In]

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

[Out]

(-2*Log[a/b + x]^3 + 3*Log[a/b + x]^2*Log[a + b*x] - 6*Log[a/b + x]*Log[c/d + x]*Log[a + b*x] + 3*Log[c/d + x]
^2*Log[a + b*x] + 6*Log[a/b + x]*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] - 3*Log[c/d + x]^2*Log[(d*(a +
b*x))/(-(b*c) + a*d)] + 3*Log[a/b + x]^2*Log[(b*(c + d*x))/(b*c - a*d)] - 3*Log[a/b + x]^2*Log[((b*e - a*f)*(
c + d*x))/((d*e - c*f)*(a + b*x))] + 6*Log[a/b + x]*Log[a + b*x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a +
b*x))] - 6*Log[c/d + x]*Log[a + b*x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))] + 6*Log[c/d + x]*Lo
g[(d*(a + b*x))/(-(b*c) + a*d)]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))] + 3*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 + 3*Log[a + b*x]*Log[((b*e - a*f)*(c + d*x
))/((d*e - c*f)*(a + b*x))]^2 - 3*Log[a/b + x]^2*Log[e + f*x] + 6*Log[a/b + x]*Log[c/d + x]*Log[e + f*x] - 3*L
og[c/d + x]^2*Log[e + f*x] - 6*Log[a/b + x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[e + f*x]
+ 6*Log[c/d + x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[e + f*x] - 3*Log[((b*e - a*f)*(c + d
*x))/((d*e - c*f)*(a + b*x))]^2*Log[e + f*x] + 3*Log[a/b + x]^2*Log[(b*(e + f*x))/(b*e - a*f)] - 6*Log[a/b + x
]*Log[(f*(c + d*x))/(-(d*e) + c*f)]*Log[(b*(e + f*x))/(b*e - a*f)] + 3*Log[(f*(c + d*x))/(-(d*e) + c*f)]^2*Log
[(b*(e + f*x))/(b*e - a*f)] + 6*Log[a/b + 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)] - 6*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)] + 3*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)] - 6*Log[a/b + x]*Log[c/d + x]*Log[(d*(e + f*x))/(d*e - c*f)] + 3*Log[c/d + x]^2*Log[(d*(e + f*x))/(
d*e - c*f)] + 6*Log[a/b + x]*Log[(f*(c + d*x))/(-(d*e) + c*f)]*Log[(d*(e + f*x))/(d*e - c*f)] - 3*Log[(f*(c +
d*x))/(-(d*e) + c*f)]^2*Log[(d*(e + f*x))/(d*e - c*f)] - 6*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)] + 6*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)] - 3*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))] + 6*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*
c) + a*d)] + 6*(Log[a/b + x] + Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])*PolyLog[2, (b*(c + d*x))/
(b*c - a*d)] + 6*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))]
- 6*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))] - 6*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 6*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] - 6*PolyLog[3, (
b*(c + d*x))/(d*(a + b*x))] + 6*PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(3*b*e - 3*a*f)

Maple [A] (verified)

Time = 5.45 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.66

method	result	size
derivativedivides	\(\frac {\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}-\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}+1\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{a f -b e}\)	\(338\)
default	\(\frac {\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}-\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}+1\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{a f -b e}\)	\(338\)
risch	\(\frac {\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}-\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}+1\right )}{a f -b e}+\frac {2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{a f -b e}-\frac {2 \,\operatorname {Li}_{3}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{a f -b e}\)	\(357\)

[In]

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

[Out]

1/(a*f-b*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((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+1)+2*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,-(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*polylog(3,-(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))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.29 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=-\frac {\log \left (\frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right )^{2} \log \left (-\frac {{\left (b c - a d\right )} f x + {\left (b c - a d\right )} e}{a d e - a c f + {\left (b d e - b c f\right )} x}\right ) + 2 \, {\rm Li}_2\left (\frac {{\left (b c - a d\right )} f x + {\left (b c - a d\right )} e}{a d e - a c f + {\left (b d e - b c f\right )} x} + 1\right ) \log \left (\frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right ) - 2 \, {\rm polylog}\left (3, \frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right )}{b e - a f} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\int \frac {\log {\left (- \frac {a c f}{- a c f + a d e - b c f x + b d e x} - \frac {a d f x}{- a c f + a d e - b c f x + b d e x} + \frac {b c e}{- a c f + a d e - b c f x + b d e x} + \frac {b d e x}{- a c f + a d e - b c f x + b d e x} \right )}^{2}}{\left (a + b x\right ) \left (e + f x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[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]

\[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\int { \frac {\log \left (\frac {{\left (b e - a f\right )} {\left (d x + c\right )}}{{\left (d e - c f\right )} {\left (b x + a\right )}}\right )^{2}}{{\left (b x + a\right )} {\left (f x + e\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (e+f x)} \, dx=\int \frac {{\ln \left (\frac {\left (a\,f-b\,e\right )\,\left (c+d\,x\right )}{\left (c\,f-d\,e\right )\,\left (a+b\,x\right )}\right )}^2}{\left (e+f\,x\right )\,\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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