3.84 \(\int \frac{(a+b \tan ^{-1}(c x^2))^2}{x^4} \, dx\)

Optimal. Leaf size=1360 \[ \text{result too large to display} \]

[Out]

(-2*a*b*c)/(3*x) - (4*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/3 + ((-1)^(3/4)*b^2*c^(3/2)*ArcTan[
(-1)^(3/4)*Sqrt[c]*x]^2)/3 + (2*(-1)^(3/4)*a*b*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/3 - (4*(-1)^(1/4)*b^2*c^
(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/3 - ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/3 + (2*(-1)^
(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/3 - (2*(-1)^(1/4)*b^2*c^(3/2
)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4
)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 + (2*(-1)^(1/4)*b^2*c^(3/2)
*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/3 - (2*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(
3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*L
og[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)
^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^
(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 - (
(I/3)*b^2*c*Log[1 - I*c*x^2])/x - ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/3 -
(b*c*(2*a + I*b*Log[1 - I*c*x^2]))/(3*x) - ((-1)^(3/4)*b*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1
- I*c*x^2]))/3 - (2*a + I*b*Log[1 - I*c*x^2])^2/(12*x^3) + ((I/3)*a*b*Log[1 + I*c*x^2])/x^3 + (((2*I)/3)*b^2*
c*Log[1 + I*c*x^2])/x - ((-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/3 + ((-1)^(1/4)
*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/3 - (b^2*Log[1 - I*c*x^2]*Log[1 + I*c*x^2])/(6*x^
3) + (b^2*Log[1 + I*c*x^2]^2)/(12*x^3) + ((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])
/3 + ((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 - ((-1)^(3/4)*b^2*c^(3/2)*PolyLog
[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/6 + ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2,
1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/
3 - ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/6 -
((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/6 -
((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/6

________________________________________________________________________________________

Rubi [A]  time = 2.22732, antiderivative size = 1360, normalized size of antiderivative = 1., number of steps used = 64, number of rules used = 25, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.562, Rules used = {5035, 2457, 2476, 2455, 203, 205, 2470, 12, 4920, 4854, 2402, 2315, 325, 6742, 206, 30, 2557, 4928, 4856, 2447, 208, 5992, 5920, 5984, 5918} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^2])^2/x^4,x]

[Out]

(-2*a*b*c)/(3*x) - (4*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/3 + ((-1)^(3/4)*b^2*c^(3/2)*ArcTan[
(-1)^(3/4)*Sqrt[c]*x]^2)/3 + (2*(-1)^(3/4)*a*b*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/3 - (4*(-1)^(1/4)*b^2*c^
(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/3 - ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/3 + (2*(-1)^
(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/3 - (2*(-1)^(1/4)*b^2*c^(3/2
)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4
)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 + (2*(-1)^(1/4)*b^2*c^(3/2)
*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/3 - (2*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(
3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*L
og[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)
^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^
(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 - (
(I/3)*b^2*c*Log[1 - I*c*x^2])/x - ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/3 -
(b*c*(2*a + I*b*Log[1 - I*c*x^2]))/(3*x) - ((-1)^(3/4)*b*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1
- I*c*x^2]))/3 - (2*a + I*b*Log[1 - I*c*x^2])^2/(12*x^3) + ((I/3)*a*b*Log[1 + I*c*x^2])/x^3 + (((2*I)/3)*b^2*
c*Log[1 + I*c*x^2])/x - ((-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/3 + ((-1)^(1/4)
*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/3 - (b^2*Log[1 - I*c*x^2]*Log[1 + I*c*x^2])/(6*x^
3) + (b^2*Log[1 + I*c*x^2]^2)/(12*x^3) + ((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])
/3 + ((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 - ((-1)^(3/4)*b^2*c^(3/2)*PolyLog
[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/6 + ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2,
1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/
3 - ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/6 -
((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/6 -
((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/6

Rule 5035

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^
m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*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 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 205

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

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 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]

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 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 6742

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

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 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 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 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 208

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

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 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]

Rubi steps

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

Mathematica [F]  time = 2.67412, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcTan[c*x^2])^2/x^4,x]

[Out]

Integrate[(a + b*ArcTan[c*x^2])^2/x^4, x]

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Maple [F]  time = 0.2, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( c{x}^{2} \right ) \right ) ^{2}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x^2))^2/x^4,x)

[Out]

int((a+b*arctan(c*x^2))^2/x^4,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*arctan(c*x^2))^2/x^4,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} \arctan \left (c x^{2}\right )^{2} + 2 \, a b \arctan \left (c x^{2}\right ) + a^{2}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^2))^2/x^4,x, algorithm="fricas")

[Out]

integral((b^2*arctan(c*x^2)^2 + 2*a*b*arctan(c*x^2) + a^2)/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atan}{\left (c x^{2} \right )}\right )^{2}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x**2))**2/x**4,x)

[Out]

Integral((a + b*atan(c*x**2))**2/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^2))^2/x^4,x, algorithm="giac")

[Out]

integrate((b*arctan(c*x^2) + a)^2/x^4, x)