3.144 \(\int x^2 (a+b \tanh ^{-1}(\frac{c}{x}))^2 \, dx\)

Optimal. Leaf size=142 \[ \frac{1}{3} b^2 c^3 \text{PolyLog}\left (2,\frac{2}{\frac{c}{x}+1}-1\right )-\frac{1}{3} c^3 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2-\frac{2}{3} b c^3 \log \left (2-\frac{2}{\frac{c}{x}+1}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{3} b c x^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{3} x^3 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{3} b^2 c^2 x-\frac{1}{3} b^2 c^3 \coth ^{-1}\left (\frac{x}{c}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 1.38924, antiderivative size = 695, normalized size of antiderivative = 4.89, number of steps used = 73, number of rules used = 34, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.125, Rules used = {6099, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2319, 44, 2455, 263, 43, 6742, 30, 2557, 12, 2466, 2448, 2462, 260, 2416, 2394, 2393, 2391, 193, 2410, 2395, 36, 29, 2390} \[ -\frac{1}{6} b^2 c^3 \text{PolyLog}\left (2,\frac{c-x}{2 c}\right )+\frac{1}{6} b^2 c^3 \text{PolyLog}\left (2,-\frac{c}{x}\right )-\frac{1}{6} b^2 c^3 \text{PolyLog}\left (2,\frac{c}{x}\right )+\frac{1}{6} b^2 c^3 \text{PolyLog}\left (2,\frac{c+x}{2 c}\right )+\frac{1}{6} b^2 c^3 \text{PolyLog}\left (2,1-\frac{x}{c}\right )-\frac{1}{6} b^2 c^3 \text{PolyLog}\left (2,\frac{x}{c}+1\right )-\frac{1}{3} a b c^2 x-\frac{1}{12} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{3} a b c^3 \log (x)+\frac{1}{3} a b c^3 \log (c+x)+\frac{1}{6} b c^2 x \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{6} a b c x^2+\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{3} a b x^3 \log \left (\frac{c}{x}+1\right )+\frac{1}{3} b^2 c^2 x+\frac{1}{12} b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c^3 \log \left (1-\frac{c}{x}\right )-\frac{1}{12} b^2 c^3 \log (c-x)+\frac{1}{6} b^2 c^3 \log \left (\frac{c}{x}+1\right ) \log (c-x)+\frac{1}{6} b^2 c^3 \log (c-x) \log \left (\frac{x}{c}\right )+\frac{1}{12} b^2 c^3 \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{6} b^2 c^3 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{6} b^2 c^3 \log (c-x) \log \left (\frac{c+x}{2 c}\right )-\frac{1}{4} b^2 c^3 \log \left (\frac{c+x}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (\frac{c}{x}+1\right )-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{12} b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c}{x}+1\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (\frac{c}{x}+1\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[x^2*(a + b*ArcTanh[c/x])^2,x]

[Out]

-(a*b*c^2*x)/3 + (b^2*c^2*x)/3 + (a*b*c*x^2)/6 + (b^2*c^3*Log[1 - c/x])/12 + (b^2*c^2*x*Log[1 - c/x])/6 - (b^2
*c*x^2*Log[1 - c/x])/12 + (b*c^2*(1 - c/x)*x*(2*a - b*Log[1 - c/x]))/6 + (b*c*x^2*(2*a - b*Log[1 - c/x]))/12 -
(c^3*(2*a - b*Log[1 - c/x])^2)/12 + (x^3*(2*a - b*Log[1 - c/x])^2)/12 + (b^2*c^2*x*Log[1 + c/x])/6 + (b^2*c*x
^2*Log[1 + c/x])/12 + (a*b*x^3*Log[1 + c/x])/3 - (b^2*x^3*Log[1 - c/x]*Log[1 + c/x])/6 - (b^2*c^3*Log[c - x])/
12 + (b^2*c^3*Log[1 + c/x]*Log[c - x])/6 + (a*b*c^3*Log[x])/3 + (b^2*c^3*Log[c - x]*Log[x/c])/6 + (a*b*c^3*Log
[c + x])/3 + (b^2*c^3*Log[c + x])/12 - (b^2*c^3*Log[1 - c/x]*Log[c + x])/6 + (b^2*c^3*Log[(c - x)/(2*c)]*Log[c
+ x])/6 - (b^2*c^3*Log[-(x/c)]*Log[c + x])/6 - (b^2*c^3*Log[c - x]*Log[(c + x)/(2*c)])/6 - (b^2*c^3*Log[(c +
x)/x])/4 - (b^2*c^2*x*Log[(c + x)/x])/6 + (b^2*c*x^2*Log[(c + x)/x])/12 + (b^2*c^3*Log[(c + x)/x]^2)/12 + (b^2
*x^3*Log[(c + x)/x]^2)/12 - (b^2*c^3*PolyLog[2, (c - x)/(2*c)])/6 + (b^2*c^3*PolyLog[2, -(c/x)])/6 - (b^2*c^3*
PolyLog[2, c/x])/6 + (b^2*c^3*PolyLog[2, (c + x)/(2*c)])/6 + (b^2*c^3*PolyLog[2, 1 - x/c])/6 - (b^2*c^3*PolyLo
g[2, 1 + x/c])/6

Rule 6099

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^
m*(a + (b*Log[1 + c*x^n])/2 - (b*Log[1 - c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] &&
IntegerQ[m] && IntegerQ[n]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2316

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2314

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2455

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 30

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

Rule 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2466

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_.) + (g_.)*(x_))^(r_.), x_S
ymbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e,
f, g, n, p, q}, x] && IntegerQ[m] && IntegerQ[r]

Rule 2448

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

Rule 2462

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

Rule 260

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

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 2410

Int[(Log[(c_.)*((d_) + (e_.)*(x_))]*(x_)^(m_.))/((f_) + (g_.)*(x_)), x_Symbol] :> Int[ExpandIntegrand[Log[c*(d
+ e*x)], x^m/(f + g*x), x], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m
]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2390

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

Rubi steps

\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} b x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 x^2 \log ^2\left (1+\frac{c}{x}\right )\right ) \, dx\\ &=\frac{1}{4} \int x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \, dx+\frac{1}{2} b \int x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{4} b^2 \int x^2 \log ^2\left (1+\frac{c}{x}\right ) \, dx\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x))^2}{x^4} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} b \int \left (2 a x^2 \log \left (1+\frac{c}{x}\right )-b x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )\right ) \, dx-\frac{1}{4} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+(a b) \int x^2 \log \left (1+\frac{c}{x}\right ) \, dx-\frac{1}{2} b^2 \int x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right ) \, dx-\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{x^3 (1-c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{6} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3 (1+c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{2} b^2 \int \frac{c x^2 \log \left (1-\frac{c}{x}\right )}{3 (-c-x)} \, dx+\frac{1}{2} b^2 \int \frac{c x^2 \log \left (1+\frac{c}{x}\right )}{-3 c+3 x} \, dx+\frac{1}{3} (a b c) \int \frac{x}{1+\frac{c}{x}} \, dx-\frac{1}{6} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^3}-\frac{c \log (1+c x)}{x^2}+\frac{c^2 \log (1+c x)}{x}-\frac{c^3 \log (1+c x)}{1+c x}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{3} (a b c) \int \frac{x^2}{c+x} \, dx+\frac{1}{6} \left (b^2 c\right ) \int \frac{x^2 \log \left (1-\frac{c}{x}\right )}{-c-x} \, dx-\frac{1}{6} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3} \, dx,x,\frac{1}{x}\right )+\frac{1}{2} \left (b^2 c\right ) \int \frac{x^2 \log \left (1+\frac{c}{x}\right )}{-3 c+3 x} \, dx+\frac{1}{6} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{6} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{3} (a b c) \int \left (-c+x+\frac{c^2}{c+x}\right ) \, dx+\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{6} \left (b^2 c\right ) \int \left (c \log \left (1-\frac{c}{x}\right )-x \log \left (1-\frac{c}{x}\right )-\frac{c^2 \log \left (1-\frac{c}{x}\right )}{c+x}\right ) \, dx+\frac{1}{2} \left (b^2 c\right ) \int \left (\frac{1}{3} c \log \left (1+\frac{c}{x}\right )-\frac{c^2 \log \left (1+\frac{c}{x}\right )}{3 (c-x)}+\frac{1}{3} x \log \left (1+\frac{c}{x}\right )\right ) \, dx+\frac{1}{6} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{c}{x}\right )\\ &=-\frac{1}{3} a b c^2 x+\frac{1}{6} a b c x^2+\frac{1}{6} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{3} a b c^3 \log (c+x)-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{c^2}{(-1+x)^2}-\frac{c^2}{-1+x}+\frac{c^2}{x}\right ) \, dx,x,1-\frac{c}{x}\right )-\frac{1}{6} \left (b^2 c\right ) \int x \log \left (1-\frac{c}{x}\right ) \, dx+\frac{1}{6} \left (b^2 c\right ) \int x \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{6} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c}{x}+\frac{c^2}{1+c x}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (b^2 c^2\right ) \int \log \left (1-\frac{c}{x}\right ) \, dx+\frac{1}{6} \left (b^2 c^2\right ) \int \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{6} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{6} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{c+x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{c-x} \, dx+\frac{1}{6} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{6} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3} a b c^2 x+\frac{1}{6} b^2 c^2 x+\frac{1}{6} a b c x^2+\frac{1}{12} b^2 c^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{12} b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{12} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{6} b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{6} b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{3} a b c^3 \log (x)+\frac{1}{3} a b c^3 \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{4} b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{12} \left (b^2 c^2\right ) \int \frac{1}{1-\frac{c}{x}} \, dx+\frac{1}{12} \left (b^2 c^2\right ) \int \frac{1}{1+\frac{c}{x}} \, dx-\frac{1}{6} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{1}{\left (1-\frac{c}{x}\right ) x} \, dx+\frac{1}{6} \left (b^2 c^3\right ) \int \frac{1}{\left (1+\frac{c}{x}\right ) x} \, dx+\frac{1}{6} \left (b^2 c^4\right ) \int \frac{\log (c-x)}{\left (1+\frac{c}{x}\right ) x^2} \, dx+\frac{1}{6} \left (b^2 c^4\right ) \int \frac{\log (c+x)}{\left (1-\frac{c}{x}\right ) x^2} \, dx\\ &=-\frac{1}{3} a b c^2 x+\frac{1}{6} b^2 c^2 x+\frac{1}{6} a b c x^2+\frac{1}{12} b^2 c^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{12} b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{12} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{6} b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{6} b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{3} a b c^3 \log (x)+\frac{1}{3} a b c^3 \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{4} b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{6} b^2 c^3 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{12} \left (b^2 c^2\right ) \int \frac{x}{-c+x} \, dx+\frac{1}{12} \left (b^2 c^2\right ) \int \frac{x}{c+x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{1}{-c+x} \, dx+\frac{1}{6} \left (b^2 c^3\right ) \int \frac{1}{c+x} \, dx+\frac{1}{6} \left (b^2 c^4\right ) \int \left (\frac{\log (c-x)}{c x}-\frac{\log (c-x)}{c (c+x)}\right ) \, dx+\frac{1}{6} \left (b^2 c^4\right ) \int \left (-\frac{\log (c+x)}{c (c-x)}-\frac{\log (c+x)}{c x}\right ) \, dx\\ &=-\frac{1}{3} a b c^2 x+\frac{1}{6} b^2 c^2 x+\frac{1}{6} a b c x^2+\frac{1}{12} b^2 c^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{12} b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{12} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{6} b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 c^3 \log (c-x)+\frac{1}{6} b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{3} a b c^3 \log (x)+\frac{1}{3} a b c^3 \log (c+x)+\frac{1}{6} b^2 c^3 \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{4} b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{6} b^2 c^3 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{12} \left (b^2 c^2\right ) \int \left (1-\frac{c}{c-x}\right ) \, dx+\frac{1}{12} \left (b^2 c^2\right ) \int \left (1-\frac{c}{c+x}\right ) \, dx+\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log (c-x)}{x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log (c-x)}{c+x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log (c+x)}{c-x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log (c+x)}{x} \, dx\\ &=-\frac{1}{3} a b c^2 x+\frac{1}{3} b^2 c^2 x+\frac{1}{6} a b c x^2+\frac{1}{12} b^2 c^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{12} b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{12} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{6} b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{12} b^2 c^3 \log (c-x)+\frac{1}{6} b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} b^2 c^3 \log (c-x) \log \left (\frac{x}{c}\right )+\frac{1}{3} a b c^3 \log (c+x)+\frac{1}{12} b^2 c^3 \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{6} b^2 c^3 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{6} b^2 c^3 \log (c-x) \log \left (\frac{c+x}{2 c}\right )-\frac{1}{4} b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{6} b^2 c^3 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (-\frac{-c-x}{2 c}\right )}{c-x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (\frac{c-x}{2 c}\right )}{c+x} \, dx+\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (-\frac{x}{c}\right )}{c+x} \, dx+\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (\frac{x}{c}\right )}{c-x} \, dx\\ &=-\frac{1}{3} a b c^2 x+\frac{1}{3} b^2 c^2 x+\frac{1}{6} a b c x^2+\frac{1}{12} b^2 c^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{12} b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{12} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{6} b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{12} b^2 c^3 \log (c-x)+\frac{1}{6} b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} b^2 c^3 \log (c-x) \log \left (\frac{x}{c}\right )+\frac{1}{3} a b c^3 \log (c+x)+\frac{1}{12} b^2 c^3 \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{6} b^2 c^3 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{6} b^2 c^3 \log (c-x) \log \left (\frac{c+x}{2 c}\right )-\frac{1}{4} b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{6} b^2 c^3 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (1-\frac{x}{c}\right )-\frac{1}{6} b^2 c^3 \text{Li}_2\left (1+\frac{x}{c}\right )+\frac{1}{6} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-x\right )-\frac{1}{6} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+x\right )\\ &=-\frac{1}{3} a b c^2 x+\frac{1}{3} b^2 c^2 x+\frac{1}{6} a b c x^2+\frac{1}{12} b^2 c^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{12} b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{6} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{12} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{6} b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{12} b^2 c^3 \log (c-x)+\frac{1}{6} b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} b^2 c^3 \log (c-x) \log \left (\frac{x}{c}\right )+\frac{1}{3} a b c^3 \log (c+x)+\frac{1}{12} b^2 c^3 \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{6} b^2 c^3 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)-\frac{1}{6} b^2 c^3 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{6} b^2 c^3 \log (c-x) \log \left (\frac{c+x}{2 c}\right )-\frac{1}{4} b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{6} b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{6} b^2 c^3 \text{Li}_2\left (\frac{c-x}{2 c}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{6} b^2 c^3 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (\frac{c+x}{2 c}\right )+\frac{1}{6} b^2 c^3 \text{Li}_2\left (1-\frac{x}{c}\right )-\frac{1}{6} b^2 c^3 \text{Li}_2\left (1+\frac{x}{c}\right )\\ \end{align*}

Mathematica [A]  time = 0.303487, size = 145, normalized size = 1.02 \[ \frac{1}{3} \left (b^2 c^3 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+a^2 x^3+a b c^3 \log \left (1-\frac{c^2}{x^2}\right )+b \tanh ^{-1}\left (\frac{c}{x}\right ) \left (2 a x^3-2 b c^3 \log \left (1-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )-b c^3+b c x^2\right )-2 a b c^3 \log \left (\frac{c}{x}\right )+a b c x^2+b^2 \left (x^3-c^3\right ) \tanh ^{-1}\left (\frac{c}{x}\right )^2+b^2 c^2 x\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcTanh[c/x])^2,x]

[Out]

(b^2*c^2*x + a*b*c*x^2 + a^2*x^3 + b^2*(-c^3 + x^3)*ArcTanh[c/x]^2 + b*ArcTanh[c/x]*(-(b*c^3) + b*c*x^2 + 2*a*
x^3 - 2*b*c^3*Log[1 - E^(-2*ArcTanh[c/x])]) + a*b*c^3*Log[1 - c^2/x^2] - 2*a*b*c^3*Log[c/x] + b^2*c^3*PolyLog[
2, E^(-2*ArcTanh[c/x])])/3

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Maple [B]  time = 0.032, size = 391, normalized size = 2.8 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{{x}^{3}{b}^{2}}{3} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}}+{\frac{{c}^{3}{b}^{2}}{3}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{{b}^{2}c{x}^{2}}{3}{\it Artanh} \left ({\frac{c}{x}} \right ) }-{\frac{2\,{c}^{3}{b}^{2}}{3}\ln \left ({\frac{c}{x}} \right ){\it Artanh} \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{3}{b}^{2}}{3}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }+{\frac{{b}^{2}{c}^{2}x}{3}}+{\frac{{c}^{3}{b}^{2}}{6}\ln \left ({\frac{c}{x}}-1 \right ) }-{\frac{{c}^{3}{b}^{2}}{6}\ln \left ( 1+{\frac{c}{x}} \right ) }-{\frac{{c}^{3}{b}^{2}}{3}{\it dilog} \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }-{\frac{{c}^{3}{b}^{2}}{6}\ln \left ({\frac{c}{x}}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }+{\frac{{c}^{3}{b}^{2}}{12} \left ( \ln \left ({\frac{c}{x}}-1 \right ) \right ) ^{2}}-{\frac{{c}^{3}{b}^{2}}{6}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }+{\frac{{c}^{3}{b}^{2}}{6}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }-{\frac{{c}^{3}{b}^{2}}{12} \left ( \ln \left ( 1+{\frac{c}{x}} \right ) \right ) ^{2}}+{\frac{{c}^{3}{b}^{2}}{3}{\it dilog} \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{3}{b}^{2}}{3}{\it dilog} \left ( 1+{\frac{c}{x}} \right ) }+{\frac{{c}^{3}{b}^{2}}{3}\ln \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }+{\frac{2\,ab{x}^{3}}{3}{\it Artanh} \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{3}ab}{3}\ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{abc{x}^{2}}{3}}-{\frac{2\,{c}^{3}ab}{3}\ln \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{3}ab}{3}\ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctanh(c/x))^2,x)

[Out]

1/3*x^3*a^2+1/3*b^2*x^3*arctanh(c/x)^2+1/3*c^3*b^2*arctanh(c/x)*ln(c/x-1)+1/3*c*b^2*arctanh(c/x)*x^2-2/3*c^3*b
^2*ln(c/x)*arctanh(c/x)+1/3*c^3*b^2*arctanh(c/x)*ln(1+c/x)+1/3*b^2*c^2*x+1/6*c^3*b^2*ln(c/x-1)-1/6*c^3*b^2*ln(
1+c/x)-1/3*c^3*b^2*dilog(1/2+1/2*c/x)-1/6*c^3*b^2*ln(c/x-1)*ln(1/2+1/2*c/x)+1/12*c^3*b^2*ln(c/x-1)^2-1/6*c^3*b
^2*ln(-1/2*c/x+1/2)*ln(1/2+1/2*c/x)+1/6*c^3*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)-1/12*c^3*b^2*ln(1+c/x)^2+1/3*c^3*b^
2*dilog(c/x)+1/3*c^3*b^2*dilog(1+c/x)+1/3*c^3*b^2*ln(c/x)*ln(1+c/x)+2/3*a*b*x^3*arctanh(c/x)+1/3*c^3*a*b*ln(c/
x-1)+1/3*a*b*c*x^2-2/3*c^3*a*b*ln(c/x)+1/3*c^3*a*b*ln(1+c/x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} x^{3} + \frac{1}{3} \,{\left (2 \, x^{3} \operatorname{artanh}\left (\frac{c}{x}\right ) +{\left (c^{2} \log \left (-c^{2} + x^{2}\right ) + x^{2}\right )} c\right )} a b + \frac{1}{12} \,{\left (6 \, c^{4} \int -\frac{\log \left (c + x\right )}{3 \,{\left (c^{2} - x^{2}\right )}}\,{d x} + x^{3} \log \left (c + x\right )^{2} + 6 \, c^{3} \int -\frac{x \log \left (c + x\right )}{3 \,{\left (c^{2} - x^{2}\right )}}\,{d x} -{\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c^{2} -{\left (c^{3} - x^{3}\right )} \log \left (-c + x\right )^{2} +{\left (c^{2} \log \left (-c^{2} + x^{2}\right ) + x^{2}\right )} c + 12 \, c \int -\frac{x^{3} \log \left (c + x\right )}{3 \,{\left (c^{2} - x^{2}\right )}}\,{d x} - 2 \,{\left (c x^{2} +{\left (c^{3} + x^{3}\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x))^2,x, algorithm="maxima")

[Out]

1/3*a^2*x^3 + 1/3*(2*x^3*arctanh(c/x) + (c^2*log(-c^2 + x^2) + x^2)*c)*a*b + 1/12*(6*c^4*integrate(-1/3*log(c
+ x)/(c^2 - x^2), x) + x^3*log(c + x)^2 + 6*c^3*integrate(-1/3*x*log(c + x)/(c^2 - x^2), x) - (c*log(c + x) -
c*log(-c + x) - 2*x)*c^2 - (c^3 - x^3)*log(-c + x)^2 + (c^2*log(-c^2 + x^2) + x^2)*c + 12*c*integrate(-1/3*x^3
*log(c + x)/(c^2 - x^2), x) - 2*(c*x^2 + (c^3 + x^3)*log(c + x))*log(-c + x))*b^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^2*arctanh(c/x)^2 + 2*a*b*x^2*arctanh(c/x) + a^2*x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{atanh}{\left (\frac{c}{x} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atanh(c/x))**2,x)

[Out]

Integral(x**2*(a + b*atanh(c/x))**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c/x) + a)^2*x^2, x)