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

Optimal. Leaf size=553 \[ \frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{202 a x}{27 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{a^2 c x^2+c}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a x}{27 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

(2*a*x)/(27*c*(c + a^2*c*x^2)^(3/2)) + (202*a*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) - (2*ArcTan[a*x])/(9*c*(c + a^2*
c*x^2)^(3/2)) - (22*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a*x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2))
- (11*a*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x
]^3/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a
^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])
- ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1
+ a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan
[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*Arc
Tan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c +
a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.921598, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {4966, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4898, 191, 4900, 192} \[ \frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{202 a x}{27 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{a^2 c x^2+c}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a x}{27 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*x)/(27*c*(c + a^2*c*x^2)^(3/2)) + (202*a*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) - (2*ArcTan[a*x])/(9*c*(c + a^2*
c*x^2)^(3/2)) - (22*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a*x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2))
- (11*a*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x
]^3/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a
^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])
- ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1
+ a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan
[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*Arc
Tan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c +
a^2*c*x^2])

Rule 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int[
x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] &
& NeQ[p, -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 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 4898

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 4900

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*p*(d + e*x^2)^(q
+ 1)*(a + b*ArcTan[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\left (a^2 \int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-a \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^3}{x \sqrt{c+a^2 c x^2}} \, dx}{c^2}-\frac{a^2 \int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac{2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{c+a^2 c x^2}}+\frac{1}{9} (2 a) \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx-\frac{(2 a) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac{(3 a) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)^3}{x \sqrt{1+a^2 x^2}} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{c+a^2 c x^2}}+\frac{(4 a) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{27 c}+\frac{(4 a) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}+\frac{(6 a) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{202 a x}{27 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (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 )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (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 )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{202 a x}{27 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 i \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 )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 i \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 )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{202 a x}{27 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 \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 )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 \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 )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{202 a x}{27 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 i \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 )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 i \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 )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{202 a x}{27 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{6 i \sqrt{1+a^2 x^2} \text{Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{6 i \sqrt{1+a^2 x^2} \text{Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.77725, size = 347, normalized size = 0.63 \[ \frac{\left (a^2 x^2+1\right )^{3/2} \left (648 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-i \tan ^{-1}(a x)}\right )+648 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )+1296 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-i \tan ^{-1}(a x)}\right )-1296 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )-1296 i \text{PolyLog}\left (4,e^{-i \tan ^{-1}(a x)}\right )-1296 i \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )+\frac{1620 a x}{\sqrt{a^2 x^2+1}}+\frac{270 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{810 a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{1620 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+54 i \tan ^{-1}(a x)^4+216 \tan ^{-1}(a x)^3 \log \left (1-e^{-i \tan ^{-1}(a x)}\right )-216 \tan ^{-1}(a x)^3 \log \left (1+e^{i \tan ^{-1}(a x)}\right )-18 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+4 \sin \left (3 \tan ^{-1}(a x)\right )+18 \tan ^{-1}(a x)^3 \cos \left (3 \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-27 i \pi ^4\right )}{216 c \left (c \left (a^2 x^2+1\right )\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + a^2*x^2)^(3/2)*((-27*I)*Pi^4 + (1620*a*x)/Sqrt[1 + a^2*x^2] - (1620*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (81
0*a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (270*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (54*I)*ArcTan[a*x]^4 - 12*Arc
Tan[a*x]*Cos[3*ArcTan[a*x]] + 18*ArcTan[a*x]^3*Cos[3*ArcTan[a*x]] + 216*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a
*x])] - 216*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (648*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])]
+ (648*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] + 1296*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 1
296*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] - (1296*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (1296*I)*PolyLog[
4, -E^(I*ArcTan[a*x])] + 4*Sin[3*ArcTan[a*x]] - 18*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]]))/(216*c*(c*(1 + a^2*x^2))
^(3/2))

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Maple [A]  time = 0.353, size = 560, normalized size = 1. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/216*(9*I*arctan(a*x)^2+9*arctan(a*x)^3-2*I-6*arctan(a*x))*(I*x^3*a^3+3*a^2*x^2-3*I*a*x-1)*(c*(a*x-I)*(a*x+I
))^(1/2)/(a^2*x^2+1)^2/c^3+5/8*(arctan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(1+I*a*x)*(c*(a*x-I)*(a*x+I
))^(1/2)/c^3/(a^2*x^2+1)-5/8*(c*(a*x-I)*(a*x+I))^(1/2)*(-1+I*a*x)*(arctan(a*x)^3-6*arctan(a*x)-3*I*arctan(a*x)
^2+6*I)/c^3/(a^2*x^2+1)+1/216*(c*(a*x-I)*(a*x+I))^(1/2)*(I*x^3*a^3-3*a^2*x^2-3*I*a*x+1)*(-9*I*arctan(a*x)^2+9*
arctan(a*x)^3+2*I-6*arctan(a*x))/c^3/(a^4*x^4+2*a^2*x^2+1)+I*(I*arctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2)
)-I*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*
arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6
*I*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*polylog(4,(1
+I*a*x)/(a^2*x^2+1)^(1/2)))*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^(1/2)/c^3

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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(arctan(a*x)^3/x/(a^2*c*x^2+c)^(5/2),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{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^3/(a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atan}^{3}{\left (a x \right )}}{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atan(a*x)**3/(x*(c*(a**2*x**2 + 1))**(5/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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