3.432 \(\int \frac{(c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^3}{x} \, dx\)

Optimal. Leaf size=845 \[ \frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2 c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt{a^2 c x^2+c}}+\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{149 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{149 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{3}{2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right ) c^{5/2}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{29}{40} a x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac{29}{20} \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{1}{20} a x \sqrt{a^2 c x^2+c} c^2+\frac{1}{3} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3 c-\frac{3}{20} a x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2 c+\frac{1}{10} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) c+\frac{1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^3 \]

[Out]

-(a*c^2*x*Sqrt[c + a^2*c*x^2])/20 + (29*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/20 + (c*(c + a^2*c*x^2)^(3/2)*Arc
Tan[a*x])/10 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/40 - (3*a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^
2)/20 + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + c^2
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan
[a*x]^3)/5 - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (3*c^(5/
2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/2 + ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I
*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*Arc
Tan[a*x])])/Sqrt[c + a^2*c*x^2] + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x
])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] - (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (149*c
^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) - (149*c^3*Sqrt[1 + a^2*x^2]
*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) + (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E
^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] + ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

________________________________________________________________________________________

Rubi [A]  time = 1.78395, antiderivative size = 845, normalized size of antiderivative = 1., number of steps used = 54, number of rules used = 16, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4950, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181, 4880, 217, 206, 195} \[ \frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2 c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt{a^2 c x^2+c}}+\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{149 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{149 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{3}{2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right ) c^{5/2}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{29}{40} a x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac{29}{20} \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{1}{20} a x \sqrt{a^2 c x^2+c} c^2+\frac{1}{3} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3 c-\frac{3}{20} a x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2 c+\frac{1}{10} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) c+\frac{1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^3 \]

Antiderivative was successfully verified.

[In]

Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x,x]

[Out]

-(a*c^2*x*Sqrt[c + a^2*c*x^2])/20 + (29*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/20 + (c*(c + a^2*c*x^2)^(3/2)*Arc
Tan[a*x])/10 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/40 - (3*a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^
2)/20 + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + c^2
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan
[a*x]^3)/5 - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (3*c^(5/
2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/2 + ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I
*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*Arc
Tan[a*x])])/Sqrt[c + a^2*c*x^2] + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x
])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] - (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (149*c
^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) - (149*c^3*Sqrt[1 + a^2*x^2]
*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) + (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E
^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] + ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, 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 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 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 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 4930

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

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 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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{x} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x} \, dx+\left (a^2 c\right ) \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{1}{5} (3 a c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx+c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x} \, dx+\left (a^2 c^2\right ) \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{1}{10} \left (a c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx-\frac{1}{20} \left (9 a c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx-\left (a c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+c^3 \int \frac{\tan ^{-1}(a x)^3}{x \sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{1}{20} a c^2 x \sqrt{c+a^2 c x^2}+\frac{29}{20} c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{29}{40} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{1}{20} \left (a c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{40} \left (9 a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{20} \left (9 a c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} \left (a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\left (a c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx-\left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20} a c^2 x \sqrt{c+a^2 c x^2}+\frac{29}{20} c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{29}{40} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{1}{20} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )-\frac{1}{20} \left (9 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )-\left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (9 a c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{40 \sqrt{c+a^2 c x^2}}-\frac{\left (a c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20} a c^2 x \sqrt{c+a^2 c x^2}+\frac{29}{20} c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{29}{40} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3}{2} c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )-\frac{\left (9 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{40 \sqrt{c+a^2 c x^2}}-\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20} a c^2 x \sqrt{c+a^2 c x^2}+\frac{29}{20} c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{29}{40} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 \sqrt{c+a^2 c x^2}}+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3}{2} c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \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 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (9 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 \sqrt{c+a^2 c x^2}}+\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20} a c^2 x \sqrt{c+a^2 c x^2}+\frac{29}{20} c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{29}{40} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 \sqrt{c+a^2 c x^2}}+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3}{2} c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}+\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}-\frac{3 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (9 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 \sqrt{c+a^2 c x^2}}-\frac{\left (9 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 \sqrt{c+a^2 c x^2}}+\frac{\left (i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \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 (i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \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 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \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 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \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 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20} a c^2 x \sqrt{c+a^2 c x^2}+\frac{29}{20} c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{29}{40} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 \sqrt{c+a^2 c x^2}}+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3}{2} c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}+\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}-\frac{3 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 i c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (9 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}+\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20} a c^2 x \sqrt{c+a^2 c x^2}+\frac{29}{20} c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{29}{40} a c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 \sqrt{c+a^2 c x^2}}+c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac{1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac{2 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3}{2} c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}+\frac{149 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}-\frac{3 i c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{149 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}-\frac{149 c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt{c+a^2 c x^2}}+\frac{6 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 i c^3 \sqrt{1+a^2 x^2} \text{Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 i c^3 \sqrt{1+a^2 x^2} \text{Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 6.59863, size = 723, normalized size = 0.86 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (2880 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-i \tan ^{-1}(a x)}\right )+2880 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-7152 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+7152 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+5760 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-i \tan ^{-1}(a x)}\right )-5760 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )+7152 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-7152 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-5760 i \text{PolyLog}\left (4,e^{-i \tan ^{-1}(a x)}\right )-5760 i \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )+32 \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x)^3+640 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^3+960 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3-150 \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x)+960 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)-1440 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-6 \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x)^2 \sin \left (2 \tan ^{-1}(a x)\right )-480 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^2 \sin \left (2 \tan ^{-1}(a x)\right )+33 \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x)^2 \sin \left (4 \tan ^{-1}(a x)\right )-12 \left (a^2 x^2+1\right )^{5/2} \sin \left (2 \tan ^{-1}(a x)\right )-6 \left (a^2 x^2+1\right )^{5/2} \sin \left (4 \tan ^{-1}(a x)\right )-160 \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x)^3 \cos \left (2 \tan ^{-1}(a x)\right )-216 \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )+960 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )-66 \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x) \cos \left (4 \tan ^{-1}(a x)\right )+240 i \tan ^{-1}(a x)^4+1392 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2+960 \tan ^{-1}(a x)^3 \log \left (1-e^{-i \tan ^{-1}(a x)}\right )-960 \tan ^{-1}(a x)^3 \log \left (1+e^{i \tan ^{-1}(a x)}\right )-2880 \tan ^{-1}(a x)^2 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )+2880 \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-120 i \pi ^4\right )}{960 \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x,x]

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*((-120*I)*Pi^4 + 960*(1 + a^2*x^2)^(3/2)*ArcTan[a*x] - 150*(1 + a^2*x^2)^(5/2)*ArcTan
[a*x] + (1392*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 960*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 640*(1 + a^2*
x^2)^(3/2)*ArcTan[a*x]^3 + 32*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^3 + (240*I)*ArcTan[a*x]^4 - 1440*ArcTanh[(a*x)/S
qrt[1 + a^2*x^2]] + 960*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 216*(1 + a^2*x^2)^(5/2)*ArcTan[a*
x]*Cos[2*ArcTan[a*x]] - 160*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] - 66*(1 + a^2*x^2)^(5/2)*ArcT
an[a*x]*Cos[4*ArcTan[a*x]] + 960*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 2880*ArcTan[a*x]^2*Log[1 - I*E^
(I*ArcTan[a*x])] + 2880*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 960*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x
])] + (2880*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (2880*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan
[a*x])] - (7152*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (7152*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*Arc
Tan[a*x])] + 5760*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 5760*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x]
)] + 7152*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 7152*PolyLog[3, I*E^(I*ArcTan[a*x])] - (5760*I)*PolyLog[4, E^((
-I)*ArcTan[a*x])] - (5760*I)*PolyLog[4, -E^(I*ArcTan[a*x])] - 12*(1 + a^2*x^2)^(5/2)*Sin[2*ArcTan[a*x]] - 480*
(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] - 6*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]]
- 6*(1 + a^2*x^2)^(5/2)*Sin[4*ArcTan[a*x]] + 33*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/(960*Sq
rt[1 + a^2*x^2])

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Maple [A]  time = 1.787, size = 562, 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)^(5/2)*arctan(a*x)^3/x,x)

[Out]

1/120*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(24*arctan(a*x)^3*x^4*a^4-18*arctan(a*x)^2*x^3*a^3+88*arctan(a*x)^3*x^2*a^
2+12*arctan(a*x)*a^2*x^2-105*arctan(a*x)^2*x*a+184*arctan(a*x)^3-6*a*x+186*arctan(a*x))+1/40*(c*(a*x-I)*(a*x+I
))^(1/2)/(a^2*x^2+1)^(1/2)*(40*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-40*arctan(a*x)^3*ln(1+(1+I*a*x)
/(a^2*x^2+1)^(1/2))-120*I*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+120*I*arctan(a*x)^2*polylog(2,-
(1+I*a*x)/(a^2*x^2+1)^(1/2))+149*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-149*arctan(a*x)^2*ln(1-I*(1
+I*a*x)/(a^2*x^2+1)^(1/2))-298*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+298*I*arctan(a*x)*polyl
og(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+240*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-240*arctan(a*x)*pol
ylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+240*I*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-240*I*polylog(4,-(1+I*a*x)/(
a^2*x^2+1)^(1/2))+120*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))+298*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-298*
polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))*c^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)^(5/2)*arctan(a*x)^3/x,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^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^3/x, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)**3/x,x)

[Out]

Timed out

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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{5}{2}} \arctan \left (a x\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x,x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3/x, x)