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

Optimal. Leaf size=1172 \[ \text{result too large to display} \]

[Out]

(4*a*b*c*x)/3 - (2*a*b*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (4*b^2*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (I/3)*b^2*c^(3/2)*
ArcTan[x/Sqrt[c]]^2 - (4*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]^2)/3 - (2*b^2*c^(
3/2)*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/3 - (2*b^2*c*x*Log[1 - c/x^2])/3 + (b^2*c^(3/2)*A
rcTan[x/Sqrt[c]]*Log[1 - c/x^2])/3 - (b*c^(3/2)*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/3 + (x^3*(2*a - b
*Log[1 - c/x^2])^2)/12 + (2*b^2*c*x*Log[1 + c/x^2])/3 + (a*b*x^3*Log[1 + c/x^2])/3 - (b^2*c^(3/2)*ArcTan[x/Sqr
t[c]]*Log[1 + c/x^2])/3 - (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2])/3 - (b^2*x^3*Log[1 - c/x^2]*Log[1 +
c/x^2])/6 + (b^2*x^3*Log[1 + c/x^2]^2)/12 + (2*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*x)])
/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)])/3 - (2*b^2*c^(3/2)*ArcTanh[x
/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/(
(Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqr
t[-c] + Sqrt[c])*(Sqrt[c] + x))])/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*
x)])/3 + (2*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (I/3)*b^2*c^(3/2)*PolyLog[2
, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] + (I/3)*b^2*c^(3/2)*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)] + (I/6)*b^
2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] + (b^2*c^(3/2)*PolyLog[2, -(x/Sqrt[c])])/3 +
(I/3)*b^2*c^(3/2)*PolyLog[2, ((-I)*x)/Sqrt[c]] - (I/3)*b^2*c^(3/2)*PolyLog[2, (I*x)/Sqrt[c]] - (b^2*c^(3/2)*P
olyLog[2, x/Sqrt[c]])/3 + (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (b^2*c^(3/2)*PolyLog[2,
-1 + (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[
c])*(Sqrt[c] + x))])/6 - (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c]
+ x))])/6 + (I/6)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]

________________________________________________________________________________________

Rubi [A]  time = 2.20796, antiderivative size = 1172, normalized size of antiderivative = 1., number of steps used = 79, number of rules used = 33, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.063, Rules used = {6099, 2457, 2471, 2448, 263, 207, 2470, 12, 260, 6688, 5988, 5932, 2447, 2455, 193, 321, 6742, 203, 30, 2557, 5992, 5912, 5920, 2402, 2315, 2476, 204, 4928, 4848, 2391, 4856, 4924, 4868} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c/x^2])^2,x]

[Out]

(4*a*b*c*x)/3 - (2*a*b*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (4*b^2*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (I/3)*b^2*c^(3/2)*
ArcTan[x/Sqrt[c]]^2 - (4*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]^2)/3 - (2*b^2*c^(
3/2)*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/3 - (2*b^2*c*x*Log[1 - c/x^2])/3 + (b^2*c^(3/2)*A
rcTan[x/Sqrt[c]]*Log[1 - c/x^2])/3 - (b*c^(3/2)*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/3 + (x^3*(2*a - b
*Log[1 - c/x^2])^2)/12 + (2*b^2*c*x*Log[1 + c/x^2])/3 + (a*b*x^3*Log[1 + c/x^2])/3 - (b^2*c^(3/2)*ArcTan[x/Sqr
t[c]]*Log[1 + c/x^2])/3 - (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2])/3 - (b^2*x^3*Log[1 - c/x^2]*Log[1 +
c/x^2])/6 + (b^2*x^3*Log[1 + c/x^2]^2)/12 + (2*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*x)])
/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)])/3 - (2*b^2*c^(3/2)*ArcTanh[x
/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/(
(Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqr
t[-c] + Sqrt[c])*(Sqrt[c] + x))])/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*
x)])/3 + (2*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (I/3)*b^2*c^(3/2)*PolyLog[2
, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] + (I/3)*b^2*c^(3/2)*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)] + (I/6)*b^
2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] + (b^2*c^(3/2)*PolyLog[2, -(x/Sqrt[c])])/3 +
(I/3)*b^2*c^(3/2)*PolyLog[2, ((-I)*x)/Sqrt[c]] - (I/3)*b^2*c^(3/2)*PolyLog[2, (I*x)/Sqrt[c]] - (b^2*c^(3/2)*P
olyLog[2, x/Sqrt[c]])/3 + (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (b^2*c^(3/2)*PolyLog[2,
-1 + (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[
c])*(Sqrt[c] + x))])/6 - (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c]
+ x))])/6 + (I/6)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]

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 2457

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

Rule 2471

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

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 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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

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

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 6688

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

Rule 5988

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]

Rule 5932

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

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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 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 321

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

Rule 6742

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 5992

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 5912

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

Rule 5920

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])*Log[2/(1
+ c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d +
e*x))/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x], x] + Simp[((a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)
*(1 + c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 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 2476

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 4928

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4924

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*d*(p + 1)), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

Rule 4868

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

Rubi steps

\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2-\frac{1}{2} b x^2 \left (-2 a+b \log \left (1-\frac{c}{x^2}\right )\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} b^2 x^2 \log ^2\left (1+\frac{c}{x^2}\right )\right ) \, dx\\ &=\frac{1}{4} \int x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2 \, dx-\frac{1}{2} b \int x^2 \left (-2 a+b \log \left (1-\frac{c}{x^2}\right )\right ) \log \left (1+\frac{c}{x^2}\right ) \, dx+\frac{1}{4} b^2 \int x^2 \log ^2\left (1+\frac{c}{x^2}\right ) \, dx\\ &=\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )-\frac{1}{2} b \int \left (-2 a x^2 \log \left (1+\frac{c}{x^2}\right )+b x^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )\right ) \, dx+\frac{1}{3} (b c) \int \frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{1-\frac{c}{x^2}} \, dx+\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (1+\frac{c}{x^2}\right )}{1+\frac{c}{x^2}} \, dx\\ &=\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+(a b) \int x^2 \log \left (1+\frac{c}{x^2}\right ) \, dx-\frac{1}{2} b^2 \int x^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right ) \, dx+\frac{1}{3} (b c) \int \left (2 a-b \log \left (1-\frac{c}{x^2}\right )+\frac{c \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{-c+x^2}\right ) \, dx+\frac{1}{3} \left (b^2 c\right ) \int \left (\log \left (1+\frac{c}{x^2}\right )-\frac{c \log \left (1+\frac{c}{x^2}\right )}{c+x^2}\right ) \, dx\\ &=\frac{2}{3} a b c x+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} b^2 \int \frac{2 c x^2 \log \left (1-\frac{c}{x^2}\right )}{3 \left (-c-x^2\right )} \, dx+\frac{1}{2} b^2 \int \frac{2 c \log \left (1+\frac{c}{x^2}\right )}{3 \left (1-\frac{c}{x^2}\right )} \, dx+\frac{1}{3} (2 a b c) \int \frac{1}{1+\frac{c}{x^2}} \, dx-\frac{1}{3} \left (b^2 c\right ) \int \log \left (1-\frac{c}{x^2}\right ) \, dx+\frac{1}{3} \left (b^2 c\right ) \int \log \left (1+\frac{c}{x^2}\right ) \, dx+\frac{1}{3} \left (b c^2\right ) \int \frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{-c+x^2} \, dx-\frac{1}{3} \left (b^2 c^2\right ) \int \frac{\log \left (1+\frac{c}{x^2}\right )}{c+x^2} \, dx\\ &=\frac{2}{3} a b c x-\frac{1}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{3} (2 a b c) \int \frac{x^2}{c+x^2} \, dx+\frac{1}{3} \left (b^2 c\right ) \int \frac{x^2 \log \left (1-\frac{c}{x^2}\right )}{-c-x^2} \, dx+\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (1+\frac{c}{x^2}\right )}{1-\frac{c}{x^2}} \, dx+\frac{1}{3} \left (2 b^2 c^2\right ) \int \frac{1}{\left (1-\frac{c}{x^2}\right ) x^2} \, dx+\frac{1}{3} \left (2 b^2 c^2\right ) \int \frac{1}{\left (1+\frac{c}{x^2}\right ) x^2} \, dx-\frac{1}{3} \left (2 b^2 c^3\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c} \left (1+\frac{c}{x^2}\right ) x^3} \, dx-\frac{1}{3} \left (2 b^2 c^3\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c} \left (1-\frac{c}{x^2}\right ) x^3} \, dx\\ &=\frac{4}{3} a b c x-\frac{1}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{3} \left (b^2 c\right ) \int \left (-\log \left (1-\frac{c}{x^2}\right )-\frac{c \log \left (1-\frac{c}{x^2}\right )}{-c-x^2}\right ) \, dx+\frac{1}{3} \left (b^2 c\right ) \int \left (\log \left (1+\frac{c}{x^2}\right )+\frac{c \log \left (1+\frac{c}{x^2}\right )}{-c+x^2}\right ) \, dx-\frac{1}{3} \left (2 a b c^2\right ) \int \frac{1}{c+x^2} \, dx+\frac{1}{3} \left (2 b^2 c^2\right ) \int \frac{1}{-c+x^2} \, dx+\frac{1}{3} \left (2 b^2 c^2\right ) \int \frac{1}{c+x^2} \, dx-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\left (1+\frac{c}{x^2}\right ) x^3} \, dx-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\left (1-\frac{c}{x^2}\right ) x^3} \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )-\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )-\frac{1}{3} \left (b^2 c\right ) \int \log \left (1-\frac{c}{x^2}\right ) \, dx+\frac{1}{3} \left (b^2 c\right ) \int \log \left (1+\frac{c}{x^2}\right ) \, dx-\frac{1}{3} \left (b^2 c^2\right ) \int \frac{\log \left (1-\frac{c}{x^2}\right )}{-c-x^2} \, dx+\frac{1}{3} \left (b^2 c^2\right ) \int \frac{\log \left (1+\frac{c}{x^2}\right )}{-c+x^2} \, dx-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (c+x^2\right )} \, dx-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (-c+x^2\right )} \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )-\frac{1}{3} \left (2 i b^2 c^{3/2}\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (i+\frac{x}{\sqrt{c}}\right )} \, dx+\frac{1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (1+\frac{x}{\sqrt{c}}\right )} \, dx+\frac{1}{3} \left (2 b^2 c^2\right ) \int \frac{1}{\left (1-\frac{c}{x^2}\right ) x^2} \, dx+\frac{1}{3} \left (2 b^2 c^2\right ) \int \frac{1}{\left (1+\frac{c}{x^2}\right ) x^2} \, dx-\frac{1}{3} \left (2 b^2 c^3\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c} \left (1-\frac{c}{x^2}\right ) x^3} \, dx-\frac{1}{3} \left (2 b^2 c^3\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c} \left (1+\frac{c}{x^2}\right ) x^3} \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} \left (2 b^2 c\right ) \int \frac{\log \left (2-\frac{2}{1-\frac{i x}{\sqrt{c}}}\right )}{1+\frac{x^2}{c}} \, dx-\frac{1}{3} \left (2 b^2 c\right ) \int \frac{\log \left (2-\frac{2}{1+\frac{x}{\sqrt{c}}}\right )}{1-\frac{x^2}{c}} \, dx+\frac{1}{3} \left (2 b^2 c^2\right ) \int \frac{1}{-c+x^2} \, dx+\frac{1}{3} \left (2 b^2 c^2\right ) \int \frac{1}{c+x^2} \, dx-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\left (1-\frac{c}{x^2}\right ) x^3} \, dx-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\left (1+\frac{c}{x^2}\right ) x^3} \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (-c+x^2\right )} \, dx-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (c+x^2\right )} \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \left (-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c x}-\frac{x \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c \left (c-x^2\right )}\right ) \, dx-\frac{1}{3} \left (2 b^2 c^{5/2}\right ) \int \left (\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c x}-\frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c \left (c+x^2\right )}\right ) \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x} \, dx+\frac{1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac{x \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c-x^2} \, dx-\frac{1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x} \, dx+\frac{1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c+x^2} \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )+\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} \left (i b^2 c^{3/2}\right ) \int \frac{\log \left (1-\frac{i x}{\sqrt{c}}\right )}{x} \, dx-\frac{1}{3} \left (i b^2 c^{3/2}\right ) \int \frac{\log \left (1+\frac{i x}{\sqrt{c}}\right )}{x} \, dx+\frac{1}{3} \left (2 b^2 c^{3/2}\right ) \int \left (\frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{2 \left (\sqrt{c}-x\right )}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{2 \left (\sqrt{c}+x\right )}\right ) \, dx+\frac{1}{3} \left (2 b^2 c^{3/2}\right ) \int \left (-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{2 \left (\sqrt{-c}-x\right )}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{2 \left (\sqrt{-c}+x\right )}\right ) \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )+\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-\frac{i x}{\sqrt{c}}\right )-\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (\frac{i x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} \left (b^2 c^{3/2}\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}-x} \, dx-\frac{1}{3} \left (b^2 c^{3/2}\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}+x} \, dx-\frac{1}{3} \left (b^2 c^{3/2}\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{-c}-x} \, dx+\frac{1}{3} \left (b^2 c^{3/2}\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{-c}+x} \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )+\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-\frac{i x}{\sqrt{c}}\right )-\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (\frac{i x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )-2 \left (\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (\frac{2}{1-\frac{i x}{\sqrt{c}}}\right )}{1+\frac{x^2}{c}} \, dx\right )+\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c} \left (1-\frac{i x}{\sqrt{c}}\right )}\right )}{1+\frac{x^2}{c}} \, dx+\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c} \left (1-\frac{i x}{\sqrt{c}}\right )}\right )}{1+\frac{x^2}{c}} \, dx+2 \left (\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (\frac{2}{1+\frac{x}{\sqrt{c}}}\right )}{1-\frac{x^2}{c}} \, dx\right )-\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (\frac{2 \left (\sqrt{-c}-x\right )}{\left (-1+\frac{\sqrt{-c}}{\sqrt{c}}\right ) \sqrt{c} \left (1+\frac{x}{\sqrt{c}}\right )}\right )}{1-\frac{x^2}{c}} \, dx-\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (\frac{2 \left (\sqrt{-c}+x\right )}{\left (1+\frac{\sqrt{-c}}{\sqrt{c}}\right ) \sqrt{c} \left (1+\frac{x}{\sqrt{c}}\right )}\right )}{1-\frac{x^2}{c}} \, dx\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )+\frac{1}{6} i b^2 c^{3/2} \text{Li}_2\left (1-\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )+\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-\frac{i x}{\sqrt{c}}\right )-\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (\frac{i x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )-\frac{1}{6} b^2 c^{3/2} \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )-\frac{1}{6} b^2 c^{3/2} \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )+\frac{1}{6} i b^2 c^{3/2} \text{Li}_2\left (1-\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )-2 \left (\frac{1}{3} \left (i b^2 c^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{i x}{\sqrt{c}}}\right )\right )+2 \left (\frac{1}{3} \left (b^2 c^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{x}{\sqrt{c}}}\right )\right )\\ &=\frac{4}{3} a b c x-\frac{2}{3} a b c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2-\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c x \log \left (1-\frac{c}{x^2}\right )+\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )-\frac{1}{3} b c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{12} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{2}{3} b^2 c x \log \left (1+\frac{c}{x^2}\right )+\frac{1}{3} a b x^3 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{6} b^2 x^3 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )-\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )+\frac{1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )-\frac{1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )+\frac{2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )-\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (1-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )+\frac{1}{6} i b^2 c^{3/2} \text{Li}_2\left (1-\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )+\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )+\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (-\frac{i x}{\sqrt{c}}\right )-\frac{1}{3} i b^2 c^{3/2} \text{Li}_2\left (\frac{i x}{\sqrt{c}}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )+\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (1-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )-\frac{1}{3} b^2 c^{3/2} \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )-\frac{1}{6} b^2 c^{3/2} \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )-\frac{1}{6} b^2 c^{3/2} \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )+\frac{1}{6} i b^2 c^{3/2} \text{Li}_2\left (1-\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )\\ \end{align*}

Mathematica [F]  time = 2.97755, size = 0, normalized size = 0. \[ \int x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.641, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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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^{2}}\right )^{2} + 2 \, a b x^{2} \operatorname{artanh}\left (\frac{c}{x^{2}}\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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Integral(x**2*(a + b*atanh(c/x**2))**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^{2}}\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))^2,x, algorithm="giac")

[Out]

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