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

Optimal. Leaf size=597 \[ \frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 c x^2}-\frac{3 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c x}+\frac{a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{6 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.684633, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4962, 4944, 4958, 4954, 4956, 4183, 2531, 6609, 2282, 6589} \[ \frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 c x^2}-\frac{3 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c x}+\frac{a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 c x^2+c}}-\frac{6 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4962

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

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 4954

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

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x^3 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 c x^2}+\frac{1}{2} (3 a) \int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{2} a^2 \int \frac{\tan ^{-1}(a x)^3}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{3 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 c x^2}+\left (3 a^2\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{\left (a^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{x \sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 c x^2}-\frac{\left (a^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{3 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 c x^2}+\frac{a^2 \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{6 a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 a^2 \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 )}{2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 a^2 \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 )}{2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 c x^2}+\frac{a^2 \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{6 a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 i a^2 \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 (3 i a^2 \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{3 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 c x^2}+\frac{a^2 \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{6 a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{3 a^2 \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{3 a^2 \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 (3 a^2 \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 (3 a^2 \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{3 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 c x^2}+\frac{a^2 \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{6 a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{3 a^2 \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{3 a^2 \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 (3 i a^2 \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 (3 i a^2 \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{3 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c x}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 c x^2}+\frac{a^2 \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{6 a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-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^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{3 i a^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{3 a^2 \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{3 a^2 \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{3 i a^2 \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{3 i a^2 \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 = 3.78778, size = 345, normalized size = 0.58 \[ \frac{a^2 \sqrt{a^2 x^2+1} \left (-24 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-i \tan ^{-1}(a x)}\right )-48 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-i \tan ^{-1}(a x)}\right )+48 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )-24 i \left (\tan ^{-1}(a x)^2-2\right ) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+48 i \text{PolyLog}\left (4,e^{-i \tan ^{-1}(a x)}\right )+48 i \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )-2 i \tan ^{-1}(a x)^4-12 \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2-8 \tan ^{-1}(a x)^3 \log \left (1-e^{-i \tan ^{-1}(a x)}\right )+8 \tan ^{-1}(a x)^3 \log \left (1+e^{i \tan ^{-1}(a x)}\right )+48 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-48 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )-12 \tan ^{-1}(a x)^2 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-2 \tan ^{-1}(a x)^3 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+2 \tan ^{-1}(a x)^3 \sec ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+i \pi ^4\right )}{16 \sqrt{c \left (a^2 x^2+1\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*Sqrt[1 + a^2*x^2]*(I*Pi^4 - (2*I)*ArcTan[a*x]^4 - 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*C
sc[ArcTan[a*x]/2]^2 - 8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] + 48*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])
] - 48*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + 8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x
]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] - (24*I)*(-2 + ArcTan[a*x]^2)*PolyLog[2, -E^(I*ArcTan[a*x])] - (48*I)*Pol
yLog[2, E^(I*ArcTan[a*x])] - 48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] + 48*ArcTan[a*x]*PolyLog[3, -E^(I
*ArcTan[a*x])] + (48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] + (48*I)*PolyLog[4, -E^(I*ArcTan[a*x])] + 2*ArcTan[a*
x]^3*Sec[ArcTan[a*x]/2]^2 - 12*ArcTan[a*x]^2*Tan[ArcTan[a*x]/2]))/(16*Sqrt[c*(1 + a^2*x^2)])

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Maple [A]  time = 0.508, size = 410, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 3\,ax+\arctan \left ( ax \right ) \right ) \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,c{x}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{{\frac{i}{2}}{a}^{2}}{c} \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,i\arctan \left ( ax \right ) \ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,i\arctan \left ( ax \right ){\it polylog} \left ( 3,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,i\arctan \left ( ax \right ) \ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,i\arctan \left ( ax \right ){\it polylog} \left ( 3,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,{\it polylog} \left ( 2,{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it polylog} \left ( 4,{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,{\it polylog} \left ( 2,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,{\it polylog} \left ( 4,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)^3/x^3/(a^2*c*x^2+c)^(1/2),x)

[Out]

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

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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^3/(a^2*c*x^2+c)^(1/2),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^{2} c x^{5} + c x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^3/(a^2*c*x^5 + c*x^3), 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^{3} \sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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