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

Optimal. Leaf size=1117 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.15037, antiderivative size = 1117, normalized size of antiderivative = 1., number of steps used = 71, number of rules used = 29, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.812, Rules used = {6099, 2457, 2476, 2455, 263, 325, 207, 206, 2470, 12, 260, 6688, 5988, 5932, 2447, 6715, 2448, 321, 6742, 203, 2556, 5992, 5920, 2402, 2315, 4928, 4856, 4924, 4868} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])^2/x^2,x]

[Out]

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

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 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 2455

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

Rule 263

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

Rule 325

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

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 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 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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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

Mathematica [A]  time = 3.03704, size = 568, normalized size = 0.51 \[ \frac{\frac{b^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 )^2-2 \sqrt{\frac{c}{x^2}} \tanh ^{-1}\left (\frac{c}{x^2}\right )^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 )}{\sqrt{\frac{c}{x^2}}}-2 a^2-4 a b \tanh ^{-1}\left (\frac{c}{x^2}\right )-\frac{4 a b \left (\tan ^{-1}\left (\sqrt{\frac{c}{x^2}}\right )-\tanh ^{-1}\left (\sqrt{\frac{c}{x^2}}\right )\right )}{\sqrt{\frac{c}{x^2}}}}{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*(ArcTan[Sqrt[c/x^2]] - ArcTanh[Sqrt[c/x^2]]))/Sqrt[c/x^2] - 4*a*b*ArcTanh[c/x^2] + (b^2*((2*I
)*ArcTan[Sqrt[c/x^2]]^2 - 4*ArcTan[Sqrt[c/x^2]]*ArcTanh[c/x^2] - 2*Sqrt[c/x^2]*ArcTanh[c/x^2]^2 - 2*ArcTan[Sqr
t[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]*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])]))/Sqrt[c/x^2])/(2*x)

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Maple [F]  time = 0.656, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{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((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 (\frac{c}{x^{2}}\right )^{2} + 2 \, a b \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((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 (\frac{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 (\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((a+b*arctanh(c/x^2))^2/x^2,x, algorithm="giac")

[Out]

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