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

Optimal. Leaf size=1549 \[ \text{result too large to display} \]

[Out]

a^2*x + 2*a*b*Sqrt[c]*ArcTan[x/Sqrt[c]] - 2*a*b*Sqrt[c]*ArcTanh[x/Sqrt[c]] - a*b*x*Log[1 - c/x^2] - b^2*Sqrt[c
]*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2] + (b^2*x*Log[1 - c/x^2]^2)/4 + a*b*x*Log[1 + c/x^2] - b^2*Sqrt[c]*ArcTanh[x
/Sqrt[c]]*Log[1 + c/x^2] - (b^2*x*Log[1 - c/x^2]*Log[1 + c/x^2])/2 + (b^2*x*Log[1 + c/x^2]^2)/4 - (b^2*Sqrt[-c
]*Log[1 + c/x^2]*Log[Sqrt[-c] - x])/2 + (b^2*Sqrt[-c]*Log[Sqrt[-c] - x]^2)/4 - (b^2*Sqrt[c]*Log[1 - c/x^2]*Log
[Sqrt[c] - x])/2 + (b^2*Sqrt[c]*Log[Sqrt[c] - x]^2)/4 - 2*b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[
c] - I*x)] + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] - b^2*Sqrt[-c]*Log[Sqr
t[-c] - x]*Log[x/Sqrt[-c]] - b^2*Sqrt[c]*Log[Sqrt[c] - x]*Log[x/Sqrt[c]] + (b^2*Sqrt[-c]*Log[1 + c/x^2]*Log[Sq
rt[-c] + x])/2 - (b^2*Sqrt[-c]*Log[(Sqrt[-c] - x)/(2*Sqrt[-c])]*Log[Sqrt[-c] + x])/2 + b^2*Sqrt[-c]*Log[-(x/Sq
rt[-c])]*Log[Sqrt[-c] + x] - (b^2*Sqrt[-c]*Log[Sqrt[-c] + x]^2)/4 + (b^2*Sqrt[-c]*Log[Sqrt[-c] - x]*Log[(Sqrt[
-c] + x)/(2*Sqrt[-c])])/2 - 2*b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)] + b^2*Sqrt[c]*ArcT
anh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] + b^2*Sqrt[c]*ArcTanh[x/Sq
rt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] + (b^2*Sqrt[c]*Log[1 - c/x^2]*Log[
Sqrt[c] + x])/2 - (b^2*Sqrt[c]*Log[(Sqrt[c] - x)/(2*Sqrt[c])]*Log[Sqrt[c] + x])/2 + b^2*Sqrt[c]*Log[-(x/Sqrt[c
])]*Log[Sqrt[c] + x] - (b^2*Sqrt[c]*Log[Sqrt[c] + x]^2)/4 + (b^2*Sqrt[c]*Log[Sqrt[c] - x]*Log[(Sqrt[c] + x)/(2
*Sqrt[c])])/2 + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)] + I*b^2*Sqrt[c]*Pol
yLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - (I/2)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] -
I*x)] + b^2*Sqrt[c]*PolyLog[2, -(x/Sqrt[c])] - I*b^2*Sqrt[c]*PolyLog[2, ((-I)*x)/Sqrt[c]] + I*b^2*Sqrt[c]*Pol
yLog[2, (I*x)/Sqrt[c]] - b^2*Sqrt[c]*PolyLog[2, x/Sqrt[c]] - (b^2*Sqrt[c]*PolyLog[2, (Sqrt[c] + x)/(2*Sqrt[c])
])/2 + (b^2*Sqrt[-c]*PolyLog[2, (1 - x/Sqrt[-c])/2])/2 - b^2*Sqrt[-c]*PolyLog[2, 1 - x/Sqrt[-c]] + b^2*Sqrt[-c
]*PolyLog[2, 1 + x/Sqrt[-c]] - (b^2*Sqrt[-c]*PolyLog[2, (c - Sqrt[-c]*x)/(2*c)])/2 - b^2*Sqrt[c]*PolyLog[2, 1
- x/Sqrt[c]] + (b^2*Sqrt[c]*PolyLog[2, 1/2 - x/(2*Sqrt[c])])/2 + b^2*Sqrt[c]*PolyLog[2, 1 + x/Sqrt[c]] + b^2*S
qrt[c]*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)] - (b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sq
rt[-c] - Sqrt[c])*(Sqrt[c] + x))])/2 - (b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqr
t[c])*(Sqrt[c] + x))])/2 - (I/2)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]

________________________________________________________________________________________

Rubi [A]  time = 2.24891, antiderivative size = 1549, normalized size of antiderivative = 1., number of steps used = 99, number of rules used = 29, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.417, Rules used = {6093, 2448, 263, 207, 2450, 2476, 2462, 260, 2416, 2394, 2315, 2390, 2301, 2393, 2391, 203, 2556, 12, 2470, 6688, 5992, 5912, 5920, 2402, 2447, 204, 4928, 4848, 4856} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])^2,x]

[Out]

a^2*x + 2*a*b*Sqrt[c]*ArcTan[x/Sqrt[c]] - 2*a*b*Sqrt[c]*ArcTanh[x/Sqrt[c]] - a*b*x*Log[1 - c/x^2] - b^2*Sqrt[c
]*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2] + (b^2*x*Log[1 - c/x^2]^2)/4 + a*b*x*Log[1 + c/x^2] - b^2*Sqrt[c]*ArcTanh[x
/Sqrt[c]]*Log[1 + c/x^2] - (b^2*x*Log[1 - c/x^2]*Log[1 + c/x^2])/2 + (b^2*x*Log[1 + c/x^2]^2)/4 - (b^2*Sqrt[-c
]*Log[1 + c/x^2]*Log[Sqrt[-c] - x])/2 + (b^2*Sqrt[-c]*Log[Sqrt[-c] - x]^2)/4 - (b^2*Sqrt[c]*Log[1 - c/x^2]*Log
[Sqrt[c] - x])/2 + (b^2*Sqrt[c]*Log[Sqrt[c] - x]^2)/4 - 2*b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[
c] - I*x)] + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] - b^2*Sqrt[-c]*Log[Sqr
t[-c] - x]*Log[x/Sqrt[-c]] - b^2*Sqrt[c]*Log[Sqrt[c] - x]*Log[x/Sqrt[c]] + (b^2*Sqrt[-c]*Log[1 + c/x^2]*Log[Sq
rt[-c] + x])/2 - (b^2*Sqrt[-c]*Log[(Sqrt[-c] - x)/(2*Sqrt[-c])]*Log[Sqrt[-c] + x])/2 + b^2*Sqrt[-c]*Log[-(x/Sq
rt[-c])]*Log[Sqrt[-c] + x] - (b^2*Sqrt[-c]*Log[Sqrt[-c] + x]^2)/4 + (b^2*Sqrt[-c]*Log[Sqrt[-c] - x]*Log[(Sqrt[
-c] + x)/(2*Sqrt[-c])])/2 - 2*b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)] + b^2*Sqrt[c]*ArcT
anh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] + b^2*Sqrt[c]*ArcTanh[x/Sq
rt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] + (b^2*Sqrt[c]*Log[1 - c/x^2]*Log[
Sqrt[c] + x])/2 - (b^2*Sqrt[c]*Log[(Sqrt[c] - x)/(2*Sqrt[c])]*Log[Sqrt[c] + x])/2 + b^2*Sqrt[c]*Log[-(x/Sqrt[c
])]*Log[Sqrt[c] + x] - (b^2*Sqrt[c]*Log[Sqrt[c] + x]^2)/4 + (b^2*Sqrt[c]*Log[Sqrt[c] - x]*Log[(Sqrt[c] + x)/(2
*Sqrt[c])])/2 + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)] + I*b^2*Sqrt[c]*Pol
yLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - (I/2)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] -
I*x)] + b^2*Sqrt[c]*PolyLog[2, -(x/Sqrt[c])] - I*b^2*Sqrt[c]*PolyLog[2, ((-I)*x)/Sqrt[c]] + I*b^2*Sqrt[c]*Pol
yLog[2, (I*x)/Sqrt[c]] - b^2*Sqrt[c]*PolyLog[2, x/Sqrt[c]] - (b^2*Sqrt[c]*PolyLog[2, (Sqrt[c] + x)/(2*Sqrt[c])
])/2 + (b^2*Sqrt[-c]*PolyLog[2, (1 - x/Sqrt[-c])/2])/2 - b^2*Sqrt[-c]*PolyLog[2, 1 - x/Sqrt[-c]] + b^2*Sqrt[-c
]*PolyLog[2, 1 + x/Sqrt[-c]] - (b^2*Sqrt[-c]*PolyLog[2, (c - Sqrt[-c]*x)/(2*c)])/2 - b^2*Sqrt[c]*PolyLog[2, 1
- x/Sqrt[c]] + (b^2*Sqrt[c]*PolyLog[2, 1/2 - x/(2*Sqrt[c])])/2 + b^2*Sqrt[c]*PolyLog[2, 1 + x/Sqrt[c]] + b^2*S
qrt[c]*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)] - (b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sq
rt[-c] - Sqrt[c])*(Sqrt[c] + x))])/2 - (b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqr
t[c])*(Sqrt[c] + x))])/2 - (I/2)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]

Rule 6093

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

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 2450

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

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 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 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 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]

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 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 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 2556

Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[(x*Log[w]*D[v, x])/v, x],
x] - Int[SimplifyIntegrand[(x*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[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 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 6688

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

Rubi steps

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

Mathematica [A]  time = 3.40588, size = 565, normalized size = 0.36 \[ -\frac{1}{2} b^2 x \sqrt{\frac{c}{x^2}} \left (-\text{PolyLog}\left (2,\frac{1}{2} \left (1-\sqrt{\frac{c}{x^2}}\right )\right )+\text{PolyLog}\left (2,\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{\frac{c}{x^2}}-1\right )\right )+\text{PolyLog}\left (2,\left (-\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{\frac{c}{x^2}}-1\right )\right )+\text{PolyLog}\left (2,\frac{1}{2} \left (\sqrt{\frac{c}{x^2}}+1\right )\right )-\text{PolyLog}\left (2,\left (\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{\frac{c}{x^2}}+1\right )\right )-\text{PolyLog}\left (2,\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{\frac{c}{x^2}}+1\right )\right )-\frac{1}{2} i \text{PolyLog}\left (2,-e^{4 i \tan ^{-1}\left (\sqrt{\frac{c}{x^2}}\right )}\right )-\frac{1}{2} \log ^2\left (1-\sqrt{\frac{c}{x^2}}\right )+\frac{1}{2} \log ^2\left (\sqrt{\frac{c}{x^2}}+1\right )+\log (2) \log \left (1-\sqrt{\frac{c}{x^2}}\right )+\log \left (1-\sqrt{\frac{c}{x^2}}\right ) \log \left (\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{\frac{c}{x^2}}-i\right )\right )-\log \left (\frac{1}{2} \left ((1+i)-(1-i) \sqrt{\frac{c}{x^2}}\right )\right ) \log \left (\sqrt{\frac{c}{x^2}}+1\right )-\log \left (\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{\frac{c}{x^2}}+i\right )\right ) \log \left (\sqrt{\frac{c}{x^2}}+1\right )-\log (2) \log \left (\sqrt{\frac{c}{x^2}}+1\right )+\log \left (1-\sqrt{\frac{c}{x^2}}\right ) \log \left (\frac{1}{2} \left ((1-i) \sqrt{\frac{c}{x^2}}+(1+i)\right )\right )-2 i \tan ^{-1}\left (\sqrt{\frac{c}{x^2}}\right )^2-\frac{2 \tanh ^{-1}\left (\frac{c}{x^2}\right )^2}{\sqrt{\frac{c}{x^2}}}+2 \tan ^{-1}\left (\sqrt{\frac{c}{x^2}}\right ) \log \left (1+e^{4 i \tan ^{-1}\left (\sqrt{\frac{c}{x^2}}\right )}\right )-2 \log \left (1-\sqrt{\frac{c}{x^2}}\right ) \tanh ^{-1}\left (\frac{c}{x^2}\right )+2 \log \left (\sqrt{\frac{c}{x^2}}+1\right ) \tanh ^{-1}\left (\frac{c}{x^2}\right )+4 \tan ^{-1}\left (\sqrt{\frac{c}{x^2}}\right ) \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+a^2 x+2 a b x \tanh ^{-1}\left (\frac{c}{x^2}\right )-2 a b x \sqrt{\frac{c}{x^2}} \left (\tan ^{-1}\left (\sqrt{\frac{c}{x^2}}\right )+\tanh ^{-1}\left (\sqrt{\frac{c}{x^2}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTanh[c/x^2])^2,x]

[Out]

a^2*x - 2*a*b*Sqrt[c/x^2]*x*(ArcTan[Sqrt[c/x^2]] + ArcTanh[Sqrt[c/x^2]]) + 2*a*b*x*ArcTanh[c/x^2] - (b^2*Sqrt[
c/x^2]*x*((-2*I)*ArcTan[Sqrt[c/x^2]]^2 + 4*ArcTan[Sqrt[c/x^2]]*ArcTanh[c/x^2] - (2*ArcTanh[c/x^2]^2)/Sqrt[c/x^
2] + 2*ArcTan[Sqrt[c/x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c/x^2]])] - 2*ArcTanh[c/x^2]*Log[1 - Sqrt[c/x^2]] + Lo
g[2]*Log[1 - Sqrt[c/x^2]] - Log[1 - Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c/x^2])
] + 2*ArcTanh[c/x^2]*Log[1 + Sqrt[c/x^2]] - Log[2]*Log[1 + Sqrt[c/x^2]] - Log[((1 + I) - (1 - I)*Sqrt[c/x^2])/
2]*Log[1 + Sqrt[c/x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c/x^2])]*Log[1 + Sqrt[c/x^2]] + Log[1 + Sqrt[c/x^2]]^2/2
+ Log[1 - Sqrt[c/x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c/x^2])/2] - (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c/x^2]]
)] - PolyLog[2, (1 - Sqrt[c/x^2])/2] + PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (-1/2 + I/2)*(
-1 + Sqrt[c/x^2])] + PolyLog[2, (1 + Sqrt[c/x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c/x^2])] - PolyLog[2,
(1/2 + I/2)*(1 + Sqrt[c/x^2])]))/2

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Maple [F]  time = 0.549, size = 0, normalized size = 0. \begin{align*} \int \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((a+b*arctanh(c/x^2))^2,x)

[Out]

int((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((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} \operatorname{artanh}\left (\frac{c}{x^{2}}\right )^{2} + 2 \, a b \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) + a^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((a+b*atanh(c/x**2))**2,x)

[Out]

Integral((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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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