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

Optimal. Leaf size=942 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.33509, antiderivative size = 942, normalized size of antiderivative = 1., number of steps used = 47, number of rules used = 21, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.313, Rules used = {6099, 2457, 206, 2470, 12, 5984, 5918, 2402, 2315, 2455, 6742, 203, 30, 2557, 5992, 5920, 2447, 4928, 4856, 4920, 4854} \[ i \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2 b^2+\sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2 b^2-\frac{\log ^2\left (c x^2+1\right ) b^2}{4 x}-2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right ) b^2-2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right ) b^2+\sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right ) b^2+2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{i \sqrt{c} x+1}\right ) b^2+2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{\sqrt{c} x+1}\right ) b^2-\sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (-\frac{2 \sqrt{c} \left (1-\sqrt{-c} x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c} x+1\right )}\right ) b^2-\sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c} x+1\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c} x+1\right )}\right ) b^2+\sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{(1-i) \left (\sqrt{c} x+1\right )}{1-i \sqrt{c} x}\right ) b^2-\sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right ) b^2+\sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (c x^2+1\right ) b^2+\sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (c x^2+1\right ) b^2+\frac{\log \left (1-c x^2\right ) \log \left (c x^2+1\right ) b^2}{2 x}-\sqrt{c} \text{PolyLog}\left (2,1-\frac{2}{1-\sqrt{c} x}\right ) b^2+i \sqrt{c} \text{PolyLog}\left (2,1-\frac{2}{1-i \sqrt{c} x}\right ) b^2-\frac{1}{2} i \sqrt{c} \text{PolyLog}\left (2,1-\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right ) b^2+i \sqrt{c} \text{PolyLog}\left (2,1-\frac{2}{i \sqrt{c} x+1}\right ) b^2-\sqrt{c} \text{PolyLog}\left (2,1-\frac{2}{\sqrt{c} x+1}\right ) b^2+\frac{1}{2} \sqrt{c} \text{PolyLog}\left (2,\frac{2 \sqrt{c} \left (1-\sqrt{-c} x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c} x+1\right )}+1\right ) b^2+\frac{1}{2} \sqrt{c} \text{PolyLog}\left (2,1-\frac{2 \sqrt{c} \left (\sqrt{-c} x+1\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c} x+1\right )}\right ) b^2-\frac{1}{2} i \sqrt{c} \text{PolyLog}\left (2,1-\frac{(1-i) \left (\sqrt{c} x+1\right )}{1-i \sqrt{c} x}\right ) b^2+2 a \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) b+\sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right ) b-\frac{a \log \left (c x^2+1\right ) b}{x}-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^2])^2/x^2,x]

[Out]

2*a*b*Sqrt[c]*ArcTan[Sqrt[c]*x] + I*b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]^2 + b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]^2 - 2*b^2
*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)] - 2*b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)]
+ b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + 2*b^2*Sqrt[c]*ArcTan[Sqrt[c
]*x]*Log[2/(1 + I*Sqrt[c]*x)] + 2*b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)] - b^2*Sqrt[c]*ArcTanh[
Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))] - b^2*Sqrt[c]*ArcTanh[Sqr
t[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))] + b^2*Sqrt[c]*ArcTan[Sqrt[c]*
x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] - b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2] + b*Sqrt[c]
*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]) - (2*a - b*Log[1 - c*x^2])^2/(4*x) - (a*b*Log[1 + c*x^2])/x + b^2
*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2] + b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2] + (b^2*Log[1 - c*x^2
]*Log[1 + c*x^2])/(2*x) - (b^2*Log[1 + c*x^2]^2)/(4*x) - b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)] + I*b^2
*Sqrt[c]*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)] - (I/2)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 -
I*Sqrt[c]*x)] + I*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)] - b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + Sqrt[c]
*x)] + (b^2*Sqrt[c]*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/2 + (
b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/2 - (I/2)*b^2
*Sqrt[c]*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 5984

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

Rule 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[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 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 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 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 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 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 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 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 4920

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

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[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 \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x^2} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x^2}-\frac{b \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{2 x^2}+\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x^2}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{x^2} \, dx-\frac{1}{2} b \int \frac{\left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{x^2} \, dx+\frac{1}{4} b^2 \int \frac{\log ^2\left (1+c x^2\right )}{x^2} \, dx\\ &=-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}-\frac{1}{2} b \int \left (-\frac{2 a \log \left (1+c x^2\right )}{x^2}+\frac{b \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^2}\right ) \, dx+(b c) \int \frac{2 a-b \log \left (1-c x^2\right )}{1-c x^2} \, dx+\left (b^2 c\right ) \int \frac{\log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}+(a b) \int \frac{\log \left (1+c x^2\right )}{x^2} \, dx-\frac{1}{2} b^2 \int \frac{\log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^2} \, dx-\left (2 b^2 c^2\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1+c x^2\right )} \, dx-\left (2 b^2 c^2\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1-c x^2\right )} \, dx\\ &=b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}+\frac{1}{2} b^2 \int -\frac{2 c \log \left (1-c x^2\right )}{1+c x^2} \, dx+\frac{1}{2} b^2 \int \frac{2 c \log \left (1+c x^2\right )}{1-c x^2} \, dx+(2 a b c) \int \frac{1}{1+c x^2} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{1+c x^2} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{1-c x^2} \, dx\\ &=2 a b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+i b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}-\left (b^2 c\right ) \int \frac{\log \left (1-c x^2\right )}{1+c x^2} \, dx+\left (b^2 c\right ) \int \frac{\log \left (1+c x^2\right )}{1-c x^2} \, dx+\left (2 b^2 c\right ) \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{i-\sqrt{c} x} \, dx-\left (2 b^2 c\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{c} x} \, dx\\ &=2 a b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+i b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2-2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )+2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )-b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}+\left (2 b^2 c\right ) \int \frac{\log \left (\frac{2}{1-\sqrt{c} x}\right )}{1-c x^2} \, dx-\left (2 b^2 c\right ) \int \frac{\log \left (\frac{2}{1+i \sqrt{c} x}\right )}{1+c x^2} \, dx-\left (2 b^2 c^2\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1-c x^2\right )} \, dx-\left (2 b^2 c^2\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1+c x^2\right )} \, dx\\ &=2 a b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+i b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2-2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )+2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )-b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}+\left (2 i b^2 \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i \sqrt{c} x}\right )-\left (2 b^2 \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt{c} x}\right )-\left (2 b^2 c^{3/2}\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{1-c x^2} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{1+c x^2} \, dx\\ &=2 a b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+i b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2-2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )+2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )-b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )+i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )-\left (2 b^2 c^{3/2}\right ) \int \left (\frac{\tan ^{-1}\left (\sqrt{c} x\right )}{2 \sqrt{c} \left (1-\sqrt{c} x\right )}-\frac{\tan ^{-1}\left (\sqrt{c} x\right )}{2 \sqrt{c} \left (1+\sqrt{c} x\right )}\right ) \, dx-\left (2 b^2 c^{3/2}\right ) \int \left (-\frac{\sqrt{-c} \tanh ^{-1}\left (\sqrt{c} x\right )}{2 c \left (1-\sqrt{-c} x\right )}+\frac{\sqrt{-c} \tanh ^{-1}\left (\sqrt{c} x\right )}{2 c \left (1+\sqrt{-c} x\right )}\right ) \, dx\\ &=2 a b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+i b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2-2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )+2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )-b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )+i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )-\left (b^2 c\right ) \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{c} x} \, dx+\left (b^2 c\right ) \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{1+\sqrt{c} x} \, dx-\frac{\left (b^2 c^{3/2}\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{-c} x} \, dx}{\sqrt{-c}}+\frac{\left (b^2 c^{3/2}\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1+\sqrt{-c} x} \, dx}{\sqrt{-c}}\\ &=2 a b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+i b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2-2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )-2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )+2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )+2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )-b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (-\frac{2 \sqrt{c} \left (1-\sqrt{-c} x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )-b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2 \sqrt{c} \left (1+\sqrt{-c} x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )-b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )+i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )+2 \left (\left (b^2 c\right ) \int \frac{\log \left (\frac{2}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx\right )-\left (b^2 c\right ) \int \frac{\log \left (\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx-2 \left (\left (b^2 c\right ) \int \frac{\log \left (\frac{2}{1+\sqrt{c} x}\right )}{1-c x^2} \, dx\right )+\left (b^2 c\right ) \int \frac{\log \left (\frac{2 \sqrt{c} \left (1-\sqrt{-c} x\right )}{\left (-\sqrt{-c}+\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )}{1-c x^2} \, dx+\left (b^2 c\right ) \int \frac{\log \left (\frac{2 \sqrt{c} \left (1+\sqrt{-c} x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )}{1-c x^2} \, dx-\left (b^2 c\right ) \int \frac{\log \left (\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx\\ &=2 a b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+i b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2-2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )-2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )+2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )+2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )-b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (-\frac{2 \sqrt{c} \left (1-\sqrt{-c} x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )-b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2 \sqrt{c} \left (1+\sqrt{-c} x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )-b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )-\frac{1}{2} i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )+i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )+\frac{1}{2} b^2 \sqrt{c} \text{Li}_2\left (1+\frac{2 \sqrt{c} \left (1-\sqrt{-c} x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )+\frac{1}{2} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (1+\sqrt{-c} x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )-\frac{1}{2} i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} 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-i \sqrt{c} x}\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+\sqrt{c} x}\right )\right )\\ &=2 a b \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )+i b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right )^2+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right )^2-2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )-2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )+2 b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )+2 b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )-b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (-\frac{2 \sqrt{c} \left (1-\sqrt{-c} x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )-b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2 \sqrt{c} \left (1+\sqrt{-c} x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )-b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1-c x^2\right )+b \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x}-\frac{a b \log \left (1+c x^2\right )}{x}+b^2 \sqrt{c} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+b^2 \sqrt{c} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{2 x}-\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x}-b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )+i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-i \sqrt{c} x}\right )-\frac{1}{2} i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )+i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )-b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1+\sqrt{c} x}\right )+\frac{1}{2} b^2 \sqrt{c} \text{Li}_2\left (1+\frac{2 \sqrt{c} \left (1-\sqrt{-c} x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )+\frac{1}{2} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (1+\sqrt{-c} x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (1+\sqrt{c} x\right )}\right )-\frac{1}{2} i b^2 \sqrt{c} \text{Li}_2\left (1-\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )\\ \end{align*}

Mathematica [A]  time = 3.36907, size = 566, normalized size = 0.6 \[ \frac{b^2 \sqrt{c x^2} \left (-\text{PolyLog}\left (2,\frac{1}{2} \left (1-\sqrt{c x^2}\right )\right )+\text{PolyLog}\left (2,\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}-1\right )\right )+\text{PolyLog}\left (2,\left (-\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}-1\right )\right )+\text{PolyLog}\left (2,\frac{1}{2} \left (\sqrt{c x^2}+1\right )\right )-\text{PolyLog}\left (2,\left (\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}+1\right )\right )-\text{PolyLog}\left (2,\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}+1\right )\right )-\frac{1}{2} i \text{PolyLog}\left (2,-e^{4 i \tan ^{-1}\left (\sqrt{c x^2}\right )}\right )-\frac{1}{2} \log ^2\left (1-\sqrt{c x^2}\right )+\frac{1}{2} \log ^2\left (\sqrt{c x^2}+1\right )+\log (2) \log \left (1-\sqrt{c x^2}\right )+\log \left (1-\sqrt{c x^2}\right ) \log \left (\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}-i\right )\right )-\log \left (\frac{1}{2} \left ((1+i)-(1-i) \sqrt{c x^2}\right )\right ) \log \left (\sqrt{c x^2}+1\right )-\log \left (\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}+i\right )\right ) \log \left (\sqrt{c x^2}+1\right )-\log (2) \log \left (\sqrt{c x^2}+1\right )+\log \left (1-\sqrt{c x^2}\right ) \log \left (\frac{1}{2} \left ((1-i) \sqrt{c x^2}+(1+i)\right )\right )-2 i \tan ^{-1}\left (\sqrt{c x^2}\right )^2-\frac{2 \tanh ^{-1}\left (c x^2\right )^2}{\sqrt{c x^2}}+2 \tan ^{-1}\left (\sqrt{c x^2}\right ) \log \left (1+e^{4 i \tan ^{-1}\left (\sqrt{c x^2}\right )}\right )-2 \log \left (1-\sqrt{c x^2}\right ) \tanh ^{-1}\left (c x^2\right )+2 \log \left (\sqrt{c x^2}+1\right ) \tanh ^{-1}\left (c x^2\right )+4 \tan ^{-1}\left (\sqrt{c x^2}\right ) \tanh ^{-1}\left (c x^2\right )\right )-2 a^2-4 a b \tanh ^{-1}\left (c x^2\right )+4 a b \sqrt{c x^2} \left (\tan ^{-1}\left (\sqrt{c x^2}\right )+\tanh ^{-1}\left (\sqrt{c x^2}\right )\right )}{2 x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*a^2 - 4*a*b*ArcTanh[c*x^2] + 4*a*b*Sqrt[c*x^2]*(ArcTan[Sqrt[c*x^2]] + ArcTanh[Sqrt[c*x^2]]) + b^2*Sqrt[c*x
^2]*((-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]] + Log[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]*Lo
g[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*x)

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Maple [F]  time = 0.175, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integral((b^2*arctanh(c*x^2)^2 + 2*a*b*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 \frac{\left (a + b \operatorname{atanh}{\left (c x^{2} \right )}\right )^{2}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (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((a+b*arctanh(c*x^2))^2/x^2,x, algorithm="giac")

[Out]

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