3.2.91 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^2 (c i+d i x)} \, dx\) [191]

Optimal. Leaf size=199 \[ -\frac {2 b B^2 n^2 (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}-\frac {2 b B n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {b (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^2 g^2 i (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^2 g^2 i n} \]

[Out]

-2*b*B^2*n^2*(d*x+c)/(-a*d+b*c)^2/g^2/i/(b*x+a)-2*b*B*n*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g
^2/i/(b*x+a)-b*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^2/g^2/i/(b*x+a)-1/3*d*(A+B*ln(e*((b*x+a)/(
d*x+c))^n))^3/B/(-a*d+b*c)^2/g^2/i/n

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2561, 2395, 2342, 2341, 2339, 30} \begin {gather*} -\frac {d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g^2 i n (b c-a d)^2}-\frac {b (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^2 i (a+b x) (b c-a d)^2}-\frac {2 b B n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i (a+b x) (b c-a d)^2}-\frac {2 b B^2 n^2 (c+d x)}{g^2 i (a+b x) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^2*(c*i + d*i*x)),x]

[Out]

(-2*b*B^2*n^2*(c + d*x))/((b*c - a*d)^2*g^2*i*(a + b*x)) - (2*b*B*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]))/((b*c - a*d)^2*g^2*i*(a + b*x)) - (b*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^
2*g^2*i*(a + b*x)) - (d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^2*g^2*i*n)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[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, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(191 c+191 d x) (a g+b g x)^2} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {(b d) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {d^2 \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{191 (b c-a d)^2 g^2}+\frac {b \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{191 (b c-a d) g^2}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {(2 B d n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{191 (b c-a d)^2 g^2}-\frac {(2 B d n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {(2 B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{191 (b c-a d) g^2}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {(2 B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{191 g^2}+\frac {(2 B d n) \int \frac {(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{191 (b c-a d)^2 g^2}-\frac {(2 B d n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{191 (b c-a d)^2 g^2}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {(2 B n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{191 g^2}+\frac {(2 B d n) \int \frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{191 (b c-a d) g^2}-\frac {(2 B d n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{191 (b c-a d) g^2}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {(2 b B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{191 (b c-a d)^2 g^2}+\frac {(2 b B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{191 (b c-a d) g^2}+\frac {(2 B d n) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{191 (b c-a d) g^2}-\frac {(2 B d n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{191 (b c-a d) g^2}\\ &=-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {(2 b B d n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B d^2 n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{c+d x} \, dx}{191 (b c-a d)^2 g^2}+\frac {(2 A B d n) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{191 (b c-a d) g^2}+\frac {\left (2 B^2 d n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{191 (b c-a d) g^2}+\frac {\left (2 B^2 d n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{191 (b c-a d) g^2}\\ &=\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {\left (b B^2 d\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{191 (b c-a d)^2 g^2}-\frac {(2 b B d n) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B d^2 n\right ) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{191 (b c-a d)^2 g^2}+\frac {(2 A B d n) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{191 b (b c-a d) g^2}+\frac {\left (2 B^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{191 g^2}+\frac {\left (2 B^2 d n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{191 (b c-a d)^2 g^2}\\ &=\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {(2 A B d n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{191 (b c-a d)^2 g^2}-\frac {(2 A b B d n) \int \frac {\log (c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}-\frac {\left (2 b B^2 d n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 A B d^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{191 (b c-a d)^2 g^2}-\frac {\left (2 A B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{191 b (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{191 (b c-a d) g^2}+\frac {\left (2 B^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{191 g^2}+\frac {\left (2 b B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}-\frac {\left (2 b B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{191 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2 n^2}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d n^2 \log (a+b x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}+\frac {(2 A B d n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}+\frac {(2 A B d n) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (2 b B^2 d n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}-\frac {\left (2 b B^2 d n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 A B d^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (b B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 b B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (B^2 d^2 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{191 (b c-a d) g^2}-\frac {\left (2 b B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{191 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2 n^2}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d n^2 \log (a+b x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}+\frac {(2 A B d n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{191 (b c-a d)^2 g^2}+\frac {\left (B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{191 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2 n^2}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d n^2 \log (a+b x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}+\frac {\left (B^2 d^2\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{191 b (b c-a d)^2 g^2}+\frac {\left (2 B^2 d^2 n\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{191 b (b c-a d)^2 g^2}+\frac {\left (B^2 d n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{191 b (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2 n^2}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d n^2 \log (a+b x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^3(c+d x)}{573 (b c-a d)^2 g^2}-\frac {2 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2 n^2}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d n^2 \log (a+b x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^3(c+d x)}{573 (b c-a d)^2 g^2}-\frac {2 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}+\frac {\left (b B^2\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}+\frac {\left (b B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{191 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2 n^2}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d n^2 \log (a+b x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^3(c+d x)}{573 (b c-a d)^2 g^2}-\frac {2 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}+\frac {\left (2 B^2 d n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2 n^2}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d n^2 \log (a+b x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^3(c+d x)}{573 (b c-a d)^2 g^2}-\frac {2 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}-2 \frac {\left (2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{191 (b c-a d)^2 g^2}\\ &=-\frac {2 B^2 n^2}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B^2 d n^2 \log (a+b x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(a+b x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{191 (b c-a d)^2 g^2}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d) g^2 (a+b x)}-\frac {2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{191 (b c-a d)^2 g^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n^2 \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{191 (b c-a d)^2 g^2}+\frac {A B d n \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^2(c+d x)}{191 (b c-a d)^2 g^2}-\frac {B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{191 (b c-a d)^2 g^2}+\frac {B^2 d n^2 \log ^3(c+d x)}{573 (b c-a d)^2 g^2}-\frac {2 A B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}-\frac {B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 A B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}+\frac {2 B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{191 (b c-a d)^2 g^2}-\frac {2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{191 (b c-a d)^2 g^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(793\) vs. \(2(199)=398\).
time = 0.45, size = 793, normalized size = 3.98 \begin {gather*} -\frac {B^2 d n^2 \log ^3\left (\frac {a+b x}{c+d x}\right )}{3 (b c-a d)^2 g^2 i}+\frac {2 B n \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B n+B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )}{(-b c+a d) g^2 i (a+b x)}+\frac {\log ^2\left (\frac {a+b x}{c+d x}\right ) \left (-a A B d n-b B^2 c n^2-A b B d n x-b B^2 d n^2 x-a B^2 d n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )-b B^2 d n x \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )}{(-b c+a d)^2 g^2 i (a+b x)}+\frac {-A^2-2 A B n-2 B^2 n^2-2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )-2 B^2 n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )-B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2}{(b c-a d) g^2 i (a+b x)}-\frac {d \log (a+b x) \left (A^2+2 A B n+2 B^2 n^2+2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+2 B^2 n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2\right )}{(b c-a d)^2 g^2 i}+\frac {d \left (A^2+2 A B n+2 B^2 n^2+2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+2 B^2 n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2\right ) \log (c+d x)}{(b c-a d)^2 g^2 i} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^2*(c*i + d*i*x)),x]

[Out]

-1/3*(B^2*d*n^2*Log[(a + b*x)/(c + d*x)]^3)/((b*c - a*d)^2*g^2*i) + (2*B*n*Log[(a + b*x)/(c + d*x)]*(A + B*n +
B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])))/((-(b*c) + a*d)*g^2*i*(a + b*x)) + (Log[(a
+ b*x)/(c + d*x)]^2*(-(a*A*B*d*n) - b*B^2*c*n^2 - A*b*B*d*n*x - b*B^2*d*n^2*x - a*B^2*d*n*(Log[e*((a + b*x)/(c
+ d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) - b*B^2*d*n*x*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c +
d*x)])))/((-(b*c) + a*d)^2*g^2*i*(a + b*x)) + (-A^2 - 2*A*B*n - 2*B^2*n^2 - 2*A*B*(Log[e*((a + b*x)/(c + d*x)
)^n] - n*Log[(a + b*x)/(c + d*x)]) - 2*B^2*n*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) - B
^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2)/((b*c - a*d)*g^2*i*(a + b*x)) - (d*Log[a +
b*x]*(A^2 + 2*A*B*n + 2*B^2*n^2 + 2*A*B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + 2*B^2
*n*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + B^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log
[(a + b*x)/(c + d*x)])^2))/((b*c - a*d)^2*g^2*i) + (d*(A^2 + 2*A*B*n + 2*B^2*n^2 + 2*A*B*(Log[e*((a + b*x)/(c
+ d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + 2*B^2*n*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)
]) + B^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2)*Log[c + d*x])/((b*c - a*d)^2*g^2*i)

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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{2} \left (d i x +c i \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (188) = 376\).
time = 0.45, size = 1028, normalized size = 5.17 \begin {gather*} B^{2} {\left (\frac {1}{{\left (-i \, b^{2} c + i \, a b d\right )} g^{2} x + {\left (-i \, a b c + i \, a^{2} d\right )} g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )^{2} + 2 \, A B {\left (\frac {1}{{\left (-i \, b^{2} c + i \, a b d\right )} g^{2} x + {\left (-i \, a b c + i \, a^{2} d\right )} g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - \frac {1}{3} \, {\left (\frac {{\left ({\left (-i \, b d x - i \, a d\right )} \log \left (b x + a\right )^{3} + {\left (i \, b d x + i \, a d\right )} \log \left (d x + c\right )^{3} - 3 \, {\left (-i \, b d x - i \, a d\right )} \log \left (b x + a\right )^{2} - 3 \, {\left (-i \, b d x - i \, a d + {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} - 6 i \, b c + 6 i \, a d - 6 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right ) - 3 \, {\left (-2 i \, b d x + {\left (-i \, b d x - i \, a d\right )} \log \left (b x + a\right )^{2} - 2 i \, a d + 2 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n^{2}}{a b^{2} c^{2} g^{2} - 2 \, a^{2} b c d g^{2} + a^{3} d^{2} g^{2} + {\left (b^{3} c^{2} g^{2} - 2 \, a b^{2} c d g^{2} + a^{2} b d^{2} g^{2}\right )} x} + \frac {3 \, {\left ({\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )^{2} + {\left (i \, b d x + i \, a d\right )} \log \left (d x + c\right )^{2} - 2 i \, b c + 2 i \, a d - 2 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right ) - 2 \, {\left (-i \, b d x - i \, a d + {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{a b^{2} c^{2} g^{2} - 2 \, a^{2} b c d g^{2} + a^{3} d^{2} g^{2} + {\left (b^{3} c^{2} g^{2} - 2 \, a b^{2} c d g^{2} + a^{2} b d^{2} g^{2}\right )} x}\right )} B^{2} - \frac {{\left ({\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )^{2} + {\left (i \, b d x + i \, a d\right )} \log \left (d x + c\right )^{2} - 2 i \, b c + 2 i \, a d - 2 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right ) - 2 \, {\left (-i \, b d x - i \, a d + {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} A B n}{a b^{2} c^{2} g^{2} - 2 \, a^{2} b c d g^{2} + a^{3} d^{2} g^{2} + {\left (b^{3} c^{2} g^{2} - 2 \, a b^{2} c d g^{2} + a^{2} b d^{2} g^{2}\right )} x} + A^{2} {\left (\frac {1}{{\left (-i \, b^{2} c + i \, a b d\right )} g^{2} x + {\left (-i \, a b c + i \, a^{2} d\right )} g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x, algorithm="maxima")

[Out]

B^2*(1/((-I*b^2*c + I*a*b*d)*g^2*x + (-I*a*b*c + I*a^2*d)*g^2) - d*log(b*x + a)/((I*b^2*c^2 - 2*I*a*b*c*d + I*
a^2*d^2)*g^2) + d*log(d*x + c)/((I*b^2*c^2 - 2*I*a*b*c*d + I*a^2*d^2)*g^2))*log((b*x/(d*x + c) + a/(d*x + c))^
n*e)^2 + 2*A*B*(1/((-I*b^2*c + I*a*b*d)*g^2*x + (-I*a*b*c + I*a^2*d)*g^2) - d*log(b*x + a)/((I*b^2*c^2 - 2*I*a
*b*c*d + I*a^2*d^2)*g^2) + d*log(d*x + c)/((I*b^2*c^2 - 2*I*a*b*c*d + I*a^2*d^2)*g^2))*log((b*x/(d*x + c) + a/
(d*x + c))^n*e) - 1/3*(((-I*b*d*x - I*a*d)*log(b*x + a)^3 + (I*b*d*x + I*a*d)*log(d*x + c)^3 - 3*(-I*b*d*x - I
*a*d)*log(b*x + a)^2 - 3*(-I*b*d*x - I*a*d + (I*b*d*x + I*a*d)*log(b*x + a))*log(d*x + c)^2 - 6*I*b*c + 6*I*a*
d - 6*(I*b*d*x + I*a*d)*log(b*x + a) - 3*(-2*I*b*d*x + (-I*b*d*x - I*a*d)*log(b*x + a)^2 - 2*I*a*d + 2*(I*b*d*
x + I*a*d)*log(b*x + a))*log(d*x + c))*n^2/(a*b^2*c^2*g^2 - 2*a^2*b*c*d*g^2 + a^3*d^2*g^2 + (b^3*c^2*g^2 - 2*a
*b^2*c*d*g^2 + a^2*b*d^2*g^2)*x) + 3*((I*b*d*x + I*a*d)*log(b*x + a)^2 + (I*b*d*x + I*a*d)*log(d*x + c)^2 - 2*
I*b*c + 2*I*a*d - 2*(I*b*d*x + I*a*d)*log(b*x + a) - 2*(-I*b*d*x - I*a*d + (I*b*d*x + I*a*d)*log(b*x + a))*log
(d*x + c))*n*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(a*b^2*c^2*g^2 - 2*a^2*b*c*d*g^2 + a^3*d^2*g^2 + (b^3*c^2*
g^2 - 2*a*b^2*c*d*g^2 + a^2*b*d^2*g^2)*x))*B^2 - ((I*b*d*x + I*a*d)*log(b*x + a)^2 + (I*b*d*x + I*a*d)*log(d*x
+ c)^2 - 2*I*b*c + 2*I*a*d - 2*(I*b*d*x + I*a*d)*log(b*x + a) - 2*(-I*b*d*x - I*a*d + (I*b*d*x + I*a*d)*log(b
*x + a))*log(d*x + c))*A*B*n/(a*b^2*c^2*g^2 - 2*a^2*b*c*d*g^2 + a^3*d^2*g^2 + (b^3*c^2*g^2 - 2*a*b^2*c*d*g^2 +
a^2*b*d^2*g^2)*x) + A^2*(1/((-I*b^2*c + I*a*b*d)*g^2*x + (-I*a*b*c + I*a^2*d)*g^2) - d*log(b*x + a)/((I*b^2*c
^2 - 2*I*a*b*c*d + I*a^2*d^2)*g^2) + d*log(d*x + c)/((I*b^2*c^2 - 2*I*a*b*c*d + I*a^2*d^2)*g^2))

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Fricas [A]
time = 0.40, size = 366, normalized size = 1.84 \begin {gather*} \frac {{\left (i \, B^{2} b d n^{2} x + i \, B^{2} a d n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} - 3 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} b c - 3 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a d - 6 \, {\left (-i \, B^{2} b c + i \, B^{2} a d\right )} n^{2} - 3 \, {\left (-i \, B^{2} b c n^{2} + {\left (-i \, A B - i \, B^{2}\right )} a d n + {\left (-i \, B^{2} b d n^{2} + {\left (-i \, A B - i \, B^{2}\right )} b d n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left ({\left (-i \, A B - i \, B^{2}\right )} b c + {\left (i \, A B + i \, B^{2}\right )} a d\right )} n - 3 \, {\left (-2 i \, B^{2} b c n^{2} + 2 \, {\left (-i \, A B - i \, B^{2}\right )} b c n + {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a d + {\left (-2 i \, B^{2} b d n^{2} + 2 \, {\left (-i \, A B - i \, B^{2}\right )} b d n + {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} b d\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{3 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} x + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x, algorithm="fricas")

[Out]

1/3*((I*B^2*b*d*n^2*x + I*B^2*a*d*n^2)*log((b*x + a)/(d*x + c))^3 - 3*(-I*A^2 - 2*I*A*B - I*B^2)*b*c - 3*(I*A^
2 + 2*I*A*B + I*B^2)*a*d - 6*(-I*B^2*b*c + I*B^2*a*d)*n^2 - 3*(-I*B^2*b*c*n^2 + (-I*A*B - I*B^2)*a*d*n + (-I*B
^2*b*d*n^2 + (-I*A*B - I*B^2)*b*d*n)*x)*log((b*x + a)/(d*x + c))^2 - 6*((-I*A*B - I*B^2)*b*c + (I*A*B + I*B^2)
*a*d)*n - 3*(-2*I*B^2*b*c*n^2 + 2*(-I*A*B - I*B^2)*b*c*n + (-I*A^2 - 2*I*A*B - I*B^2)*a*d + (-2*I*B^2*b*d*n^2
+ 2*(-I*A*B - I*B^2)*b*d*n + (-I*A^2 - 2*I*A*B - I*B^2)*b*d)*x)*log((b*x + a)/(d*x + c)))/((b^3*c^2 - 2*a*b^2*
c*d + a^2*b*d^2)*g^2*x + (a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*g^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**2/(d*i*x+c*i),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x, algorithm="giac")

[Out]

integrate((B*log(((b*x + a)/(d*x + c))^n*e) + A)^2/((b*g*x + a*g)^2*(I*d*x + I*c)), x)

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Mupad [B]
time = 5.85, size = 361, normalized size = 1.81 \begin {gather*} {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {B^2}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}-\frac {B\,d\,\left (A+B\,n\right )}{g^2\,i\,n\,{\left (a\,d-b\,c\right )}^2}\right )+\frac {A^2+2\,A\,B\,n+2\,B^2\,n^2}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}+\frac {2\,B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (A+B\,n\right )}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}-\frac {B^2\,d\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3\,g^2\,i\,n\,{\left (a\,d-b\,c\right )}^2}+\frac {d\,\mathrm {atan}\left (\frac {d\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g^2\,i-b^2\,c^2\,g^2\,i}{g^2\,i\,\left (a\,d-b\,c\right )}\right )\,\left (A^2+2\,A\,B\,n+2\,B^2\,n^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (d\,A^2+2\,d\,A\,B\,n+2\,d\,B^2\,n^2\right )}\right )\,\left (A^2+2\,A\,B\,n+2\,B^2\,n^2\right )\,2{}\mathrm {i}}{g^2\,i\,{\left (a\,d-b\,c\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/((a*g + b*g*x)^2*(c*i + d*i*x)),x)

[Out]

log(e*((a + b*x)/(c + d*x))^n)^2*(B^2/((a*d - b*c)*(a*g^2*i + b*g^2*i*x)) - (B*d*(A + B*n))/(g^2*i*n*(a*d - b*
c)^2)) + (A^2 + 2*B^2*n^2 + 2*A*B*n)/((a*d - b*c)*(a*g^2*i + b*g^2*i*x)) + (2*B*log(e*((a + b*x)/(c + d*x))^n)
*(A + B*n))/((a*d - b*c)*(a*g^2*i + b*g^2*i*x)) + (d*atan((d*(2*b*d*x + (a^2*d^2*g^2*i - b^2*c^2*g^2*i)/(g^2*i
*(a*d - b*c)))*(A^2 + 2*B^2*n^2 + 2*A*B*n)*1i)/((a*d - b*c)*(A^2*d + 2*B^2*d*n^2 + 2*A*B*d*n)))*(A^2 + 2*B^2*n
^2 + 2*A*B*n)*2i)/(g^2*i*(a*d - b*c)^2) - (B^2*d*log(e*((a + b*x)/(c + d*x))^n)^3)/(3*g^2*i*n*(a*d - b*c)^2)

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