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

Optimal. Leaf size=443 \[ \frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{6 a x}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}} \]

[Out]

(6*a*x)/(c*Sqrt[c + a^2*c*x^2]) - (6*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (3*a*x*ArcTan[a*x]^2)/(c*Sqrt[c +
a^2*c*x^2]) + ArcTan[a*x]^3/(c*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*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*Sqr
t[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*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*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*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[
4, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c*S
qrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.557865, antiderivative size = 443, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {4966, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4898, 191} \[ \frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}}+\frac{6 a x}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a*x)/(c*Sqrt[c + a^2*c*x^2]) - (6*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (3*a*x*ArcTan[a*x]^2)/(c*Sqrt[c +
a^2*c*x^2]) + ArcTan[a*x]^3/(c*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*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*Sqr
t[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*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*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*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[
4, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c*S
qrt[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]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^3}{x \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=\frac{\tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-(3 a) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)^3}{x \sqrt{1+a^2 x^2}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=-\frac{6 \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}+(6 a) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{6 a x}{c \sqrt{c+a^2 c x^2}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{c \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 \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 \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 \sqrt{c+a^2 c x^2}}\\ &=\frac{6 a x}{c \sqrt{c+a^2 c x^2}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{c \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 \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 \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 \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 \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 \sqrt{c+a^2 c x^2}}\\ &=\frac{6 a x}{c \sqrt{c+a^2 c x^2}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{c \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 \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 \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 \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 \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 \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 \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 \sqrt{c+a^2 c x^2}}\\ &=\frac{6 a x}{c \sqrt{c+a^2 c x^2}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{c \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 \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 \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 \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 \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 \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 \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 \sqrt{c+a^2 c x^2}}\\ &=\frac{6 a x}{c \sqrt{c+a^2 c x^2}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^3}{c \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 \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 \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 \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 \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 \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 \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 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.423663, size = 295, normalized size = 0.67 \[ \frac{\sqrt{a^2 x^2+1} \left (24 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-i \tan ^{-1}(a x)}\right )+24 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )+48 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-i \tan ^{-1}(a x)}\right )-48 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (4,e^{-i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right )+\frac{48 a x}{\sqrt{a^2 x^2+1}}+\frac{8 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{24 a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{48 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+2 i \tan ^{-1}(a x)^4+8 \tan ^{-1}(a x)^3 \log \left (1-e^{-i \tan ^{-1}(a x)}\right )-8 \tan ^{-1}(a x)^3 \log \left (1+e^{i \tan ^{-1}(a x)}\right )-i \pi ^4\right )}{8 c \sqrt{c \left (a^2 x^2+1\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 + a^2*x^2]*((-I)*Pi^4 + (48*a*x)/Sqrt[1 + a^2*x^2] - (48*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (24*a*x*ArcT
an[a*x]^2)/Sqrt[1 + a^2*x^2] + (8*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (2*I)*ArcTan[a*x]^4 + 8*ArcTan[a*x]^3*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)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] + 48*ArcTan[a*x]*PolyLog[3, E^((-I)*Ar
cTan[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])]))/(8*c*Sqrt[c*(1 + a^2*x^2)])

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Maple [A]  time = 0.309, size = 388, normalized size = 0.9 \begin{align*}{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) +3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}-6\,i \right ) \left ( 1+iax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -1+iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) -3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+6\,i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{i}{{c}^{2}} \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 ){\it polylog} \left ( 3,-{(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 ( 4,-{\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/(a^2*c*x^2+c)^(3/2),x)

[Out]

1/2*(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)/(a^2*x^2+1)/c^2-1/
2*(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)/(a^2*x^2+1)/c^2+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*arcta
n(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*polylo
g(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^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(arctan(a*x)^3/x/(a^2*c*x^2+c)^(3/2),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^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} 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)^(3/2),x, algorithm="fricas")

[Out]

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

[Out]

Integral(atan(a*x)**3/(x*(c*(a**2*x**2 + 1))**(3/2)), 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{3}{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)^(3/2),x, algorithm="giac")

[Out]

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