3.143 \(\int x^3 (a+b \tanh ^{-1}(\frac{c}{x}))^2 \, dx\)

Optimal. Leaf size=123 \[ -\frac{1}{4} c^4 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{2} b c^3 x \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{6} b c x^3 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{12} b^2 c^2 x^2+\frac{1}{3} b^2 c^4 \log \left (1-\frac{c^2}{x^2}\right )+\frac{2}{3} b^2 c^4 \log (x) \]

[Out]

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

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Rubi [C]  time = 1.70313, antiderivative size = 812, normalized size of antiderivative = 6.6, number of steps used = 88, 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}{16} \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 c^4-\frac{1}{16} b^2 \log ^2\left (\frac{c+x}{x}\right ) c^4+\frac{5}{48} b^2 \log \left (1-\frac{c}{x}\right ) c^4+\frac{5}{48} b^2 \log (c-x) c^4+\frac{1}{8} b^2 \log \left (\frac{c}{x}+1\right ) \log (c-x) c^4+\frac{11}{24} b^2 \log (x) c^4+\frac{1}{4} a b \log (x) c^4+\frac{1}{8} b^2 \log (c-x) \log \left (\frac{x}{c}\right ) c^4+\frac{5}{48} b^2 \log (c+x) c^4-\frac{1}{4} a b \log (c+x) c^4+\frac{1}{8} b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x) c^4-\frac{1}{8} b^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x) c^4+\frac{1}{8} b^2 \log \left (-\frac{x}{c}\right ) \log (c+x) c^4-\frac{1}{8} b^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right ) c^4+\frac{11}{48} b^2 \log \left (\frac{c+x}{x}\right ) c^4-\frac{1}{8} b^2 \text{PolyLog}\left (2,\frac{c-x}{2 c}\right ) c^4-\frac{1}{8} b^2 \text{PolyLog}\left (2,-\frac{c}{x}\right ) c^4-\frac{1}{8} b^2 \text{PolyLog}\left (2,\frac{c}{x}\right ) c^4-\frac{1}{8} b^2 \text{PolyLog}\left (2,\frac{c+x}{2 c}\right ) c^4+\frac{1}{8} b^2 \text{PolyLog}\left (2,1-\frac{x}{c}\right ) c^4+\frac{1}{8} b^2 \text{PolyLog}\left (2,\frac{x}{c}+1\right ) c^4+\frac{1}{4} a b x c^3-\frac{1}{8} b^2 x \log \left (1-\frac{c}{x}\right ) c^3+\frac{1}{8} b \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) c^3+\frac{1}{8} b^2 x \log \left (\frac{c}{x}+1\right ) c^3+\frac{1}{8} b^2 x \log \left (\frac{c+x}{x}\right ) c^3+\frac{1}{12} b^2 x^2 c^2-\frac{1}{8} a b x^2 c^2+\frac{1}{16} b^2 x^2 \log \left (1-\frac{c}{x}\right ) c^2+\frac{1}{16} b x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) c^2+\frac{1}{16} b^2 x^2 \log \left (\frac{c}{x}+1\right ) c^2-\frac{1}{16} b^2 x^2 \log \left (\frac{c+x}{x}\right ) c^2+\frac{1}{12} a b x^3 c-\frac{1}{24} b^2 x^3 \log \left (1-\frac{c}{x}\right ) c+\frac{1}{24} b x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) c+\frac{1}{24} b^2 x^3 \log \left (\frac{c}{x}+1\right ) c+\frac{1}{24} b^2 x^3 \log \left (\frac{c+x}{x}\right ) c+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b x^4 \log \left (\frac{c}{x}+1\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (\frac{c}{x}+1\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[x^3*(a + b*ArcTanh[c/x])^2,x]

[Out]

(a*b*c^3*x)/4 - (a*b*c^2*x^2)/8 + (b^2*c^2*x^2)/12 + (a*b*c*x^3)/12 + (5*b^2*c^4*Log[1 - c/x])/48 - (b^2*c^3*x
*Log[1 - c/x])/8 + (b^2*c^2*x^2*Log[1 - c/x])/16 - (b^2*c*x^3*Log[1 - c/x])/24 + (b*c^3*(1 - c/x)*x*(2*a - b*L
og[1 - c/x]))/8 + (b*c^2*x^2*(2*a - b*Log[1 - c/x]))/16 + (b*c*x^3*(2*a - b*Log[1 - c/x]))/24 - (c^4*(2*a - b*
Log[1 - c/x])^2)/16 + (x^4*(2*a - b*Log[1 - c/x])^2)/16 + (b^2*c^3*x*Log[1 + c/x])/8 + (b^2*c^2*x^2*Log[1 + c/
x])/16 + (b^2*c*x^3*Log[1 + c/x])/24 + (a*b*x^4*Log[1 + c/x])/4 - (b^2*x^4*Log[1 - c/x]*Log[1 + c/x])/8 + (5*b
^2*c^4*Log[c - x])/48 + (b^2*c^4*Log[1 + c/x]*Log[c - x])/8 + (a*b*c^4*Log[x])/4 + (11*b^2*c^4*Log[x])/24 + (b
^2*c^4*Log[c - x]*Log[x/c])/8 - (a*b*c^4*Log[c + x])/4 + (5*b^2*c^4*Log[c + x])/48 + (b^2*c^4*Log[1 - c/x]*Log
[c + x])/8 - (b^2*c^4*Log[(c - x)/(2*c)]*Log[c + x])/8 + (b^2*c^4*Log[-(x/c)]*Log[c + x])/8 - (b^2*c^4*Log[c -
x]*Log[(c + x)/(2*c)])/8 + (11*b^2*c^4*Log[(c + x)/x])/48 + (b^2*c^3*x*Log[(c + x)/x])/8 - (b^2*c^2*x^2*Log[(
c + x)/x])/16 + (b^2*c*x^3*Log[(c + x)/x])/24 - (b^2*c^4*Log[(c + x)/x]^2)/16 + (b^2*x^4*Log[(c + x)/x]^2)/16
- (b^2*c^4*PolyLog[2, (c - x)/(2*c)])/8 - (b^2*c^4*PolyLog[2, -(c/x)])/8 - (b^2*c^4*PolyLog[2, c/x])/8 - (b^2*
c^4*PolyLog[2, (c + x)/(2*c)])/8 + (b^2*c^4*PolyLog[2, 1 - x/c])/8 + (b^2*c^4*PolyLog[2, 1 + x/c])/8

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^3 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} b x^3 \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^3 \log ^2\left (1+\frac{c}{x}\right )\right ) \, dx\\ &=\frac{1}{4} \int x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \, dx+\frac{1}{2} b \int x^3 \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^3 \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^5} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} b \int \left (2 a x^3 \log \left (1+\frac{c}{x}\right )-b x^3 \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^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )+(a b) \int x^3 \log \left (1+\frac{c}{x}\right ) \, dx-\frac{1}{2} b^2 \int x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right ) \, dx-\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{x^4 (1-c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^4 (1+c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^4} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{2} b^2 \int \frac{c x^3 \log \left (1-\frac{c}{x}\right )}{4 (-c-x)} \, dx+\frac{1}{2} b^2 \int \frac{c x^3 \log \left (1+\frac{c}{x}\right )}{-4 c+4 x} \, dx+\frac{1}{4} (a b c) \int \frac{x^2}{1+\frac{c}{x}} \, dx-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^4}-\frac{c \log (1+c x)}{x^3}+\frac{c^2 \log (1+c x)}{x^2}-\frac{c^3 \log (1+c x)}{x}+\frac{c^4 \log (1+c x)}{1+c x}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^4} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{8} (b c) \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}{4} (a b c) \int \frac{x^3}{c+x} \, dx+\frac{1}{8} \left (b^2 c\right ) \int \frac{x^3 \log \left (1-\frac{c}{x}\right )}{-c-x} \, dx-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^4} \, dx,x,\frac{1}{x}\right )+\frac{1}{2} \left (b^2 c\right ) \int \frac{x^3 \log \left (1+\frac{c}{x}\right )}{-4 c+4 x} \, dx+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^2 c^5\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} b c x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{8} b^2 c^3 x \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (\frac{c+x}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{8} (b c) \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}{4} (a b c) \int \left (c^2-c x+x^2-\frac{c^3}{c+x}\right ) \, dx+\frac{1}{24} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{8} \left (b^2 c\right ) \int \left (-c^2 \log \left (1-\frac{c}{x}\right )+c x \log \left (1-\frac{c}{x}\right )-x^2 \log \left (1-\frac{c}{x}\right )+\frac{c^3 \log \left (1-\frac{c}{x}\right )}{c+x}\right ) \, dx+\frac{1}{2} \left (b^2 c\right ) \int \left (\frac{1}{4} c^2 \log \left (1+\frac{c}{x}\right )-\frac{c^3 \log \left (1+\frac{c}{x}\right )}{4 (c-x)}+\frac{1}{4} c x \log \left (1+\frac{c}{x}\right )+\frac{1}{4} x^2 \log \left (1+\frac{c}{x}\right )\right ) \, dx+\frac{1}{8} \left (b c^2\right ) \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}{24} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 (1+c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{16} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{c}{x}\right )\\ &=\frac{1}{4} a b c^3 x-\frac{1}{8} a b c^2 x^2+\frac{1}{12} a b c x^3+\frac{1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{4} a b c^4 \log (c+x)+\frac{1}{8} b^2 c^3 x \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{24} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{c^3}{(-1+x)^3}+\frac{c^3}{(-1+x)^2}-\frac{c^3}{-1+x}+\frac{c^3}{x}\right ) \, dx,x,1-\frac{c}{x}\right )-\frac{1}{8} \left (b^2 c\right ) \int x^2 \log \left (1-\frac{c}{x}\right ) \, dx+\frac{1}{8} \left (b^2 c\right ) \int x^2 \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{8} \left (b c^2\right ) \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}{24} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^3}-\frac{c}{x^2}+\frac{c^2}{x}-\frac{c^3}{1+c x}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{16} \left (b^2 c^2\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}{8} \left (b^2 c^2\right ) \int x \log \left (1-\frac{c}{x}\right ) \, dx+\frac{1}{8} \left (b^2 c^2\right ) \int x \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{8} \left (b c^3\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}{16} \left (b^2 c^3\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}{8} \left (b^2 c^3\right ) \int \log \left (1-\frac{c}{x}\right ) \, dx+\frac{1}{8} \left (b^2 c^3\right ) \int \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{c+x} \, dx-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{c-x} \, dx-\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^2 c^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} a b c^3 x-\frac{1}{16} b^2 c^3 x-\frac{1}{8} a b c^2 x^2+\frac{1}{24} b^2 c^2 x^2+\frac{1}{12} a b c x^3+\frac{1}{24} b^2 c^4 \log \left (1-\frac{c}{x}\right )-\frac{1}{8} b^2 c^3 x \log \left (1-\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{c}{x}\right )-\frac{1}{24} b^2 c x^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b c^3 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b^2 c^3 x \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{8} b^2 c^4 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{13}{48} b^2 c^4 \log (x)-\frac{1}{4} a b c^4 \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{11}{48} b^2 c^4 \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 c^3 x \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{24} \left (b^2 c^2\right ) \int \frac{x}{1-\frac{c}{x}} \, dx+\frac{1}{24} \left (b^2 c^2\right ) \int \frac{x}{1+\frac{c}{x}} \, dx+\frac{1}{16} \left (b^2 c^2\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}{8} \left (b c^3\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}{16} \left (b^2 c^3\right ) \int \frac{1}{1-\frac{c}{x}} \, dx+\frac{1}{16} \left (b^2 c^3\right ) \int \frac{1}{1+\frac{c}{x}} \, dx+\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{8} \left (b c^4\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{1}{\left (1-\frac{c}{x}\right ) x} \, dx+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{1}{\left (1+\frac{c}{x}\right ) x} \, dx+\frac{1}{8} \left (b^2 c^5\right ) \int \frac{\log (c-x)}{\left (1+\frac{c}{x}\right ) x^2} \, dx-\frac{1}{8} \left (b^2 c^5\right ) \int \frac{\log (c+x)}{\left (1-\frac{c}{x}\right ) x^2} \, dx\\ &=\frac{1}{4} a b c^3 x-\frac{1}{8} a b c^2 x^2+\frac{1}{24} b^2 c^2 x^2+\frac{1}{12} a b c x^3+\frac{5}{48} b^2 c^4 \log \left (1-\frac{c}{x}\right )-\frac{1}{8} b^2 c^3 x \log \left (1-\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{c}{x}\right )-\frac{1}{24} b^2 c x^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b c^3 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b^2 c^3 x \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{8} b^2 c^4 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{4} a b c^4 \log (x)+\frac{11}{24} b^2 c^4 \log (x)-\frac{1}{4} a b c^4 \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{11}{48} b^2 c^4 \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 c^3 x \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{24} \left (b^2 c^2\right ) \int \frac{x^2}{-c+x} \, dx+\frac{1}{24} \left (b^2 c^2\right ) \int \frac{x^2}{c+x} \, dx-\frac{1}{16} \left (b^2 c^3\right ) \int \frac{x}{-c+x} \, dx+\frac{1}{16} \left (b^2 c^3\right ) \int \frac{x}{c+x} \, dx-\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{1}{-c+x} \, dx+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{1}{c+x} \, dx+\frac{1}{8} \left (b^2 c^5\right ) \int \left (\frac{\log (c-x)}{c x}-\frac{\log (c-x)}{c (c+x)}\right ) \, dx-\frac{1}{8} \left (b^2 c^5\right ) \int \left (-\frac{\log (c+x)}{c (c-x)}-\frac{\log (c+x)}{c x}\right ) \, dx\\ &=\frac{1}{4} a b c^3 x-\frac{1}{8} a b c^2 x^2+\frac{1}{24} b^2 c^2 x^2+\frac{1}{12} a b c x^3+\frac{5}{48} b^2 c^4 \log \left (1-\frac{c}{x}\right )-\frac{1}{8} b^2 c^3 x \log \left (1-\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{c}{x}\right )-\frac{1}{24} b^2 c x^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b c^3 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b^2 c^3 x \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{8} b^2 c^4 \log (c-x)+\frac{1}{8} b^2 c^4 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{4} a b c^4 \log (x)+\frac{11}{24} b^2 c^4 \log (x)-\frac{1}{4} a b c^4 \log (c+x)+\frac{1}{8} b^2 c^4 \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{11}{48} b^2 c^4 \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 c^3 x \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{24} \left (b^2 c^2\right ) \int \left (c-\frac{c^2}{c-x}+x\right ) \, dx+\frac{1}{24} \left (b^2 c^2\right ) \int \left (-c+x+\frac{c^2}{c+x}\right ) \, dx-\frac{1}{16} \left (b^2 c^3\right ) \int \left (1-\frac{c}{c-x}\right ) \, dx+\frac{1}{16} \left (b^2 c^3\right ) \int \left (1-\frac{c}{c+x}\right ) \, dx+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log (c-x)}{x} \, dx-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log (c-x)}{c+x} \, dx+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log (c+x)}{c-x} \, dx+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log (c+x)}{x} \, dx\\ &=\frac{1}{4} a b c^3 x-\frac{1}{8} a b c^2 x^2+\frac{1}{12} b^2 c^2 x^2+\frac{1}{12} a b c x^3+\frac{5}{48} b^2 c^4 \log \left (1-\frac{c}{x}\right )-\frac{1}{8} b^2 c^3 x \log \left (1-\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{c}{x}\right )-\frac{1}{24} b^2 c x^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b c^3 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b^2 c^3 x \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{5}{48} b^2 c^4 \log (c-x)+\frac{1}{8} b^2 c^4 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{4} a b c^4 \log (x)+\frac{11}{24} b^2 c^4 \log (x)+\frac{1}{8} b^2 c^4 \log (c-x) \log \left (\frac{x}{c}\right )-\frac{1}{4} a b c^4 \log (c+x)+\frac{5}{48} b^2 c^4 \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{8} b^2 c^4 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{8} b^2 c^4 \log (c-x) \log \left (\frac{c+x}{2 c}\right )+\frac{11}{48} b^2 c^4 \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 c^3 x \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log \left (-\frac{-c-x}{2 c}\right )}{c-x} \, dx+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log \left (\frac{c-x}{2 c}\right )}{c+x} \, dx-\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log \left (-\frac{x}{c}\right )}{c+x} \, dx+\frac{1}{8} \left (b^2 c^4\right ) \int \frac{\log \left (\frac{x}{c}\right )}{c-x} \, dx\\ &=\frac{1}{4} a b c^3 x-\frac{1}{8} a b c^2 x^2+\frac{1}{12} b^2 c^2 x^2+\frac{1}{12} a b c x^3+\frac{5}{48} b^2 c^4 \log \left (1-\frac{c}{x}\right )-\frac{1}{8} b^2 c^3 x \log \left (1-\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{c}{x}\right )-\frac{1}{24} b^2 c x^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b c^3 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b^2 c^3 x \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{5}{48} b^2 c^4 \log (c-x)+\frac{1}{8} b^2 c^4 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{4} a b c^4 \log (x)+\frac{11}{24} b^2 c^4 \log (x)+\frac{1}{8} b^2 c^4 \log (c-x) \log \left (\frac{x}{c}\right )-\frac{1}{4} a b c^4 \log (c+x)+\frac{5}{48} b^2 c^4 \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{8} b^2 c^4 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{8} b^2 c^4 \log (c-x) \log \left (\frac{c+x}{2 c}\right )+\frac{11}{48} b^2 c^4 \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 c^3 x \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (1-\frac{x}{c}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (1+\frac{x}{c}\right )+\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-x\right )+\frac{1}{8} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+x\right )\\ &=\frac{1}{4} a b c^3 x-\frac{1}{8} a b c^2 x^2+\frac{1}{12} b^2 c^2 x^2+\frac{1}{12} a b c x^3+\frac{5}{48} b^2 c^4 \log \left (1-\frac{c}{x}\right )-\frac{1}{8} b^2 c^3 x \log \left (1-\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1-\frac{c}{x}\right )-\frac{1}{24} b^2 c x^3 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b c^3 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{16} b c^2 x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{24} b c x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} c^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b^2 c^3 x \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^2 c^2 x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{5}{48} b^2 c^4 \log (c-x)+\frac{1}{8} b^2 c^4 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{4} a b c^4 \log (x)+\frac{11}{24} b^2 c^4 \log (x)+\frac{1}{8} b^2 c^4 \log (c-x) \log \left (\frac{x}{c}\right )-\frac{1}{4} a b c^4 \log (c+x)+\frac{5}{48} b^2 c^4 \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{8} b^2 c^4 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)+\frac{1}{8} b^2 c^4 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{8} b^2 c^4 \log (c-x) \log \left (\frac{c+x}{2 c}\right )+\frac{11}{48} b^2 c^4 \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 c^3 x \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^2 x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{24} b^2 c x^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{16} b^2 c^4 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{c-x}{2 c}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{8} b^2 c^4 \text{Li}_2\left (\frac{c+x}{2 c}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (1-\frac{x}{c}\right )+\frac{1}{8} b^2 c^4 \text{Li}_2\left (1+\frac{x}{c}\right )\\ \end{align*}

Mathematica [A]  time = 0.0610989, size = 131, normalized size = 1.07 \[ \frac{1}{12} \left (3 a^2 x^4+2 b x \tanh ^{-1}\left (\frac{c}{x}\right ) \left (3 a x^3+b c \left (3 c^2+x^2\right )\right )+6 a b c^3 x+b c^4 (3 a+4 b) \log (x-c)-3 a b c^4 \log (c+x)+2 a b c x^3+b^2 c^2 x^2+3 b^2 \left (x^4-c^4\right ) \tanh ^{-1}\left (\frac{c}{x}\right )^2+4 b^2 c^4 \log (c+x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*ArcTanh[c/x])^2,x]

[Out]

(6*a*b*c^3*x + b^2*c^2*x^2 + 2*a*b*c*x^3 + 3*a^2*x^4 + 2*b*x*(3*a*x^3 + b*c*(3*c^2 + x^2))*ArcTanh[c/x] + 3*b^
2*(-c^4 + x^4)*ArcTanh[c/x]^2 + b*(3*a + 4*b)*c^4*Log[-c + x] - 3*a*b*c^4*Log[c + x] + 4*b^2*c^4*Log[c + x])/1
2

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Maple [B]  time = 0.007, size = 328, normalized size = 2.7 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{{x}^{4}{b}^{2}}{4} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}}+{\frac{{c}^{4}{b}^{2}}{4}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{{b}^{2}c{x}^{3}}{6}{\it Artanh} \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{3}{b}^{2}x}{2}{\it Artanh} \left ({\frac{c}{x}} \right ) }-{\frac{{c}^{4}{b}^{2}}{4}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }+{\frac{{c}^{4}{b}^{2}}{16} \left ( \ln \left ({\frac{c}{x}}-1 \right ) \right ) ^{2}}-{\frac{{c}^{4}{b}^{2}}{8}\ln \left ({\frac{c}{x}}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }+{\frac{{c}^{4}{b}^{2}}{8}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }-{\frac{{c}^{4}{b}^{2}}{8}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }+{\frac{{c}^{4}{b}^{2}}{16} \left ( \ln \left ( 1+{\frac{c}{x}} \right ) \right ) ^{2}}+{\frac{{c}^{4}{b}^{2}}{3}\ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{{b}^{2}{c}^{2}{x}^{2}}{12}}-{\frac{2\,{c}^{4}{b}^{2}}{3}\ln \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{4}{b}^{2}}{3}\ln \left ( 1+{\frac{c}{x}} \right ) }+{\frac{ab{x}^{4}}{2}{\it Artanh} \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{4}ab}{4}\ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{abc{x}^{3}}{6}}+{\frac{{c}^{3}abx}{2}}-{\frac{{c}^{4}ab}{4}\ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctanh(c/x))^2,x)

[Out]

1/4*a^2*x^4+1/4*b^2*x^4*arctanh(c/x)^2+1/4*c^4*b^2*arctanh(c/x)*ln(c/x-1)+1/6*c*b^2*x^3*arctanh(c/x)+1/2*c^3*b
^2*x*arctanh(c/x)-1/4*c^4*b^2*arctanh(c/x)*ln(1+c/x)+1/16*c^4*b^2*ln(c/x-1)^2-1/8*c^4*b^2*ln(c/x-1)*ln(1/2+1/2
*c/x)+1/8*c^4*b^2*ln(-1/2*c/x+1/2)*ln(1/2+1/2*c/x)-1/8*c^4*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)+1/16*c^4*b^2*ln(1+c/
x)^2+1/3*c^4*b^2*ln(c/x-1)+1/12*b^2*c^2*x^2-2/3*c^4*b^2*ln(c/x)+1/3*c^4*b^2*ln(1+c/x)+1/2*a*b*x^4*arctanh(c/x)
+1/4*c^4*a*b*ln(c/x-1)+1/6*a*b*c*x^3+1/2*c^3*a*b*x-1/4*c^4*a*b*ln(1+c/x)

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Maxima [A]  time = 0.981755, size = 255, normalized size = 2.07 \begin{align*} \frac{1}{4} \, b^{2} x^{4} \operatorname{artanh}\left (\frac{c}{x}\right )^{2} + \frac{1}{4} \, a^{2} x^{4} + \frac{1}{12} \,{\left (6 \, x^{4} \operatorname{artanh}\left (\frac{c}{x}\right ) -{\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c\right )} a b + \frac{1}{48} \,{\left ({\left (3 \, c^{2} \log \left (c + x\right )^{2} + 3 \, c^{2} \log \left (-c + x\right )^{2} + 16 \, c^{2} \log \left (c + x\right ) + 4 \, x^{2} - 2 \,{\left (3 \, c^{2} \log \left (c + x\right ) - 8 \, c^{2}\right )} \log \left (-c + x\right )\right )} c^{2} - 4 \,{\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c \operatorname{artanh}\left (\frac{c}{x}\right )\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c/x))^2,x, algorithm="maxima")

[Out]

1/4*b^2*x^4*arctanh(c/x)^2 + 1/4*a^2*x^4 + 1/12*(6*x^4*arctanh(c/x) - (3*c^3*log(c + x) - 3*c^3*log(-c + x) -
6*c^2*x - 2*x^3)*c)*a*b + 1/48*((3*c^2*log(c + x)^2 + 3*c^2*log(-c + x)^2 + 16*c^2*log(c + x) + 4*x^2 - 2*(3*c
^2*log(c + x) - 8*c^2)*log(-c + x))*c^2 - 4*(3*c^3*log(c + x) - 3*c^3*log(-c + x) - 6*c^2*x - 2*x^3)*c*arctanh
(c/x))*b^2

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Fricas [A]  time = 1.67716, size = 350, normalized size = 2.85 \begin{align*} \frac{1}{2} \, a b c^{3} x + \frac{1}{12} \, b^{2} c^{2} x^{2} + \frac{1}{6} \, a b c x^{3} + \frac{1}{4} \, a^{2} x^{4} - \frac{1}{12} \,{\left (3 \, a b - 4 \, b^{2}\right )} c^{4} \log \left (c + x\right ) + \frac{1}{12} \,{\left (3 \, a b + 4 \, b^{2}\right )} c^{4} \log \left (-c + x\right ) - \frac{1}{16} \,{\left (b^{2} c^{4} - b^{2} x^{4}\right )} \log \left (-\frac{c + x}{c - x}\right )^{2} + \frac{1}{12} \,{\left (3 \, b^{2} c^{3} x + b^{2} c x^{3} + 3 \, a b x^{4}\right )} \log \left (-\frac{c + x}{c - x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c/x))^2,x, algorithm="fricas")

[Out]

1/2*a*b*c^3*x + 1/12*b^2*c^2*x^2 + 1/6*a*b*c*x^3 + 1/4*a^2*x^4 - 1/12*(3*a*b - 4*b^2)*c^4*log(c + x) + 1/12*(3
*a*b + 4*b^2)*c^4*log(-c + x) - 1/16*(b^2*c^4 - b^2*x^4)*log(-(c + x)/(c - x))^2 + 1/12*(3*b^2*c^3*x + b^2*c*x
^3 + 3*a*b*x^4)*log(-(c + x)/(c - x))

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Sympy [A]  time = 3.44274, size = 158, normalized size = 1.28 \begin{align*} \frac{a^{2} x^{4}}{4} - \frac{a b c^{4} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{2} + \frac{a b c^{3} x}{2} + \frac{a b c x^{3}}{6} + \frac{a b x^{4} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{2} + \frac{2 b^{2} c^{4} \log{\left (- c + x \right )}}{3} - \frac{b^{2} c^{4} \operatorname{atanh}^{2}{\left (\frac{c}{x} \right )}}{4} + \frac{2 b^{2} c^{4} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{3} + \frac{b^{2} c^{3} x \operatorname{atanh}{\left (\frac{c}{x} \right )}}{2} + \frac{b^{2} c^{2} x^{2}}{12} + \frac{b^{2} c x^{3} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{6} + \frac{b^{2} x^{4} \operatorname{atanh}^{2}{\left (\frac{c}{x} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*atanh(c/x))**2,x)

[Out]

a**2*x**4/4 - a*b*c**4*atanh(c/x)/2 + a*b*c**3*x/2 + a*b*c*x**3/6 + a*b*x**4*atanh(c/x)/2 + 2*b**2*c**4*log(-c
+ x)/3 - b**2*c**4*atanh(c/x)**2/4 + 2*b**2*c**4*atanh(c/x)/3 + b**2*c**3*x*atanh(c/x)/2 + b**2*c**2*x**2/12
+ b**2*c*x**3*atanh(c/x)/6 + b**2*x**4*atanh(c/x)**2/4

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Giac [A]  time = 1.22244, size = 209, normalized size = 1.7 \begin{align*} \frac{1}{2} \, a b c^{3} x + \frac{1}{12} \, b^{2} c^{2} x^{2} + \frac{1}{6} \, a b c x^{3} + \frac{1}{4} \, a^{2} x^{4} - \frac{1}{16} \,{\left (b^{2} c^{4} - b^{2} x^{4}\right )} \log \left (-\frac{c + x}{c - x}\right )^{2} - \frac{1}{12} \,{\left (3 \, a b c^{4} - 4 \, b^{2} c^{4}\right )} \log \left (c + x\right ) + \frac{1}{12} \,{\left (3 \, a b c^{4} + 4 \, b^{2} c^{4}\right )} \log \left (c - x\right ) + \frac{1}{12} \,{\left (3 \, b^{2} c^{3} x + b^{2} c x^{3} + 3 \, a b x^{4}\right )} \log \left (-\frac{c + x}{c - x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c/x))^2,x, algorithm="giac")

[Out]

1/2*a*b*c^3*x + 1/12*b^2*c^2*x^2 + 1/6*a*b*c*x^3 + 1/4*a^2*x^4 - 1/16*(b^2*c^4 - b^2*x^4)*log(-(c + x)/(c - x)
)^2 - 1/12*(3*a*b*c^4 - 4*b^2*c^4)*log(c + x) + 1/12*(3*a*b*c^4 + 4*b^2*c^4)*log(c - x) + 1/12*(3*b^2*c^3*x +
b^2*c*x^3 + 3*a*b*x^4)*log(-(c + x)/(c - x))