3.425 \(\int \frac{(c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^3}{x^2} \, dx\)

Optimal. Leaf size=901 \[ -\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}+\frac{1}{2} a^2 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{x}-\frac{6 a c^2 \sqrt{a^2 x^2+1} \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}+\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{3}{2} a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{6 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{9 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{9 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{6 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{6 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}} \]

[Out]

(-3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (a^2*c*x*Sqrt[c + a^2
*c*x^2]*ArcTan[a*x]^3)/2 - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^
2*c*x^2] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*
x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^2
*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((9*I)/2)*a*c^2*Sqrt[1 +
a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((9*I)/2)*a*c^2*Sqrt[1 + a^
2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*
Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 +
I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sq
rt[c + a^2*c*x^2] - (9*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*
x^2] + (9*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*c^2*
Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[
4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x]
)])/Sqrt[c + a^2*c*x^2]

________________________________________________________________________________________

Rubi [A]  time = 1.23727, antiderivative size = 901, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 14, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4950, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4890, 4888, 4181, 6609, 4880, 4886} \[ -\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}+\frac{1}{2} a^2 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{x}-\frac{6 a c^2 \sqrt{a^2 x^2+1} \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}+\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{3}{2} a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{6 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{9 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{9 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{6 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{6 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^2,x]

[Out]

(-3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (a^2*c*x*Sqrt[c + a^2
*c*x^2]*ArcTan[a*x]^3)/2 - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^
2*c*x^2] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*
x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^2
*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((9*I)/2)*a*c^2*Sqrt[1 +
a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((9*I)/2)*a*c^2*Sqrt[1 + a^
2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*
Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 +
I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sq
rt[c + a^2*c*x^2] - (9*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*
x^2] + (9*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*c^2*
Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[
4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x]
)])/Sqrt[c + a^2*c*x^2]

Rule 4950

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[
d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*
x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] &&
IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

Rule 4944

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 4958

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

Rule 4956

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

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rule 4890

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

Rule 4888

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

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 4880

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

Rule 4886

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

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x^2} \, dx &=c \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^2} \, dx+\left (a^2 c\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{3}{2} a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+c^2 \int \frac{\tan ^{-1}(a x)^3}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{2} \left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\left (3 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{3}{2} a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{1}{2} a^2 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\left (3 a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{\left (a^2 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (a^2 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 a^2 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{3}{2} a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{1}{2} a^2 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{3}{2} a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{1}{2} a^2 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}}-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{3}{2} a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{1}{2} a^2 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}}-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{9 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{9 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{3}{2} a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{1}{2} a^2 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}}-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{9 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{9 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{9 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{9 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{3}{2} a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{1}{2} a^2 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}}-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{9 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{9 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{9 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{9 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 i a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 a c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{3}{2} a c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{1}{2} a^2 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}}-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{9 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{9 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{6 i a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{9 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{9 a c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 a c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{9 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{9 i a c^2 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 6.50192, size = 1387, normalized size = 1.54 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^2,x]

[Out]

(a*c*Sqrt[c + a^2*c*x^2]*((-7*I)*Pi^4*Sqrt[1 + a^2*x^2] - (8*I)*Pi^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - (384*I)*S
qrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - 96*ArcTan[a*x]^2 - 96*a^2*x^2*ArcTan[a*x]^2 + (24*I)*
Pi^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (64*ArcTan[a*x]^3)/(a*x) - 32*a*x*ArcTan[a*x]^3 + 32*a^3*x^3*ArcTan[a*x
]^3 - (32*I)*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 - (64*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*
x]^3 + (16*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^4 + 48*Pi^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*
x])] - 96*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] - 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[1 + I
/E^(I*ArcTan[a*x])] + 64*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] + 192*Sqrt[1 + a^2*x^2]*
ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[1 + I*E^(I*ArcTan[a*x])] - 48*Pi^2*Sqr
t[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + 96*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + I*E^(I
*ArcTan[a*x])] - 64*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - 192*Sqrt[1 + a^2*x^2]*ArcTa
n[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[2*Sqrt[1 + a^2*x^2]*Sin[(Pi + 2*ArcTan[a*x]
)/4]^2] + (192*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] + (48*I)*Pi*Sqrt[1 + a^2*
x^2]*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] + (384*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -
E^(I*ArcTan[a*x])] + (192*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (48*I)*Pi^2*Sqrt[1 + a^2*x
^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[
a*x])] + (288*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*Sqrt[1 + a^2*x^2
]*PolyLog[2, I*E^(I*ArcTan[a*x])] - (96*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])] - (
384*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 384*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog
[3, (-I)/E^(I*ArcTan[a*x])] - 192*Pi*Sqrt[1 + a^2*x^2]*PolyLog[3, I/E^(I*ArcTan[a*x])] - 384*Sqrt[1 + a^2*x^2]
*PolyLog[3, -E^(I*ArcTan[a*x])] + 192*Pi*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 576*Sqrt[1 + a
^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 192*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*A
rcTan[a*x])] + 384*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] - (384*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)
/E^(I*ArcTan[a*x])] - (576*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] + (192*I)*Sqrt[1 + a^2*x^2]
*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(64*(1 + a^2*x^2))

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Maple [A]  time = 2.009, size = 602, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^2,x)

[Out]

1/2*c*(c*(a*x-I)*(a*x+I))^(1/2)*arctan(a*x)^2*(arctan(a*x)*a^2*x^2-3*a*x-2*arctan(a*x))/x-3/2*I*a*c*(c*(a*x-I)
*(a*x+I))^(1/2)*(-4*I*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*I*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/
2))+2*I*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*
I*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2
))+I*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*a
rctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)
)-I*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+4*I*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+4*arctan(a*x)
*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*polylog(2,-I*(
1+I*a*x)/(a^2*x^2+1)^(1/2))+6*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1
/2))-6*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)

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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^2*c*x^2+c)^(3/2)*arctan(a*x)^3/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{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{3}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^2,x, algorithm="fricas")

[Out]

integral((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}^{3}{\left (a x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)**3/x**2,x)

[Out]

Integral((c*(a**2*x**2 + 1))**(3/2)*atan(a*x)**3/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{3}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^2,x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3/x^2, x)