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

Optimal. Leaf size=493 \[ \frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^3 x}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a}{27 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

(2*a)/(27*c*(c + a^2*c*x^2)^(3/2)) + (94*a)/(9*c^2*Sqrt[c + a^2*c*x^2]) + (2*a^2*x*ArcTan[a*x])/(9*c*(c + a^2*
c*x^2)^(3/2)) + (94*a^2*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) - (a*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3
/2)) - (5*a*ArcTan[a*x]^2)/(c^2*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*
a^2*x*ArcTan[a*x]^3)/(3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c^3*x) - (6*a*Sqrt[1 +
a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*a*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Po
lyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])]
)/(c^2*Sqrt[c + a^2*c*x^2]) + (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.951968, antiderivative size = 493, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4966, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4898, 4894, 4900, 4896} \[ \frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^3 x}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a}{27 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a)/(27*c*(c + a^2*c*x^2)^(3/2)) + (94*a)/(9*c^2*Sqrt[c + a^2*c*x^2]) + (2*a^2*x*ArcTan[a*x])/(9*c*(c + a^2*
c*x^2)^(3/2)) + (94*a^2*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) - (a*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3
/2)) - (5*a*ArcTan[a*x]^2)/(c^2*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*
a^2*x*ArcTan[a*x]^3)/(3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c^3*x) - (6*a*Sqrt[1 +
a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*a*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Po
lyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])]
)/(c^2*Sqrt[c + a^2*c*x^2]) + (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, 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 4944

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

Rule 4958

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

Rule 4956

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

Rule 4183

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

Rule 2531

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

Rule 2282

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

Rule 6589

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

Rule 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 4894

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

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 4896

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

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\left (a^2 \int \frac{\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^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac{a \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{1}{3} \left (2 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^2 \sqrt{c+a^2 c x^2}} \, dx}{c^2}-\frac{\left (2 a^2\right ) \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^3 x}+\frac{(3 a) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx}{c^2}+\frac{\left (4 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 c}+\frac{\left (4 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac{\left (6 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^3 x}+\frac{\left (3 a \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^3 x}+\frac{\left (3 a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^3 x}-\frac{6 a \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^3 x}-\frac{6 a \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{6 i a \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{6 i a \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 i a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 i a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^3 x}-\frac{6 a \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{6 i a \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{6 i a \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^3 x}-\frac{6 a \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{6 i a \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{6 i a \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{6 a \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{6 a \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 2.32763, size = 399, normalized size = 0.81 \[ -\frac{a \left (-648 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )+648 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+648 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )-648 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )+54 \sqrt{a^2 x^2+1} \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^3-324 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left (1-e^{i \tan ^{-1}(a x)}\right )+324 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left (1+e^{i \tan ^{-1}(a x)}\right )+9 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \sin \left (3 \tan ^{-1}(a x)\right )-6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \sin \left (3 \tan ^{-1}(a x)\right )+9 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \cos \left (3 \tan ^{-1}(a x)\right )-2 \sqrt{a^2 x^2+1} \cos \left (3 \tan ^{-1}(a x)\right )+189 a x \tan ^{-1}(a x)^3+567 \tan ^{-1}(a x)^2-1134 a x \tan ^{-1}(a x)+27 a x \tan ^{-1}(a x)^3 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-1134\right )}{108 c^2 \sqrt{a^2 c x^2+c}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a*(-1134 - 1134*a*x*ArcTan[a*x] + 567*ArcTan[a*x]^2 + 189*a*x*ArcTan[a*x]^3 - 2*Sqrt[1 + a^2*x^2]*Cos[3*ArcT
an[a*x]] + 9*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Cos[3*ArcTan[a*x]] + 27*a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2 -
324*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 324*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + E
^(I*ArcTan[a*x])] - (648*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (648*I)*Sqrt[1 + a^
2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 648*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])] - 648*
Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] - 6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Sin[3*ArcTan[a*x]] + 9*Sqrt[
1 + a^2*x^2]*ArcTan[a*x]^3*Sin[3*ArcTan[a*x]] + 54*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Tan[ArcTan[a*x]/2]))/(108*c
^2*Sqrt[c + a^2*c*x^2])

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Maple [A]  time = 0.362, size = 528, normalized size = 1.1 \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^2/(a^2*c*x^2+c)^(5/2),x)

[Out]

1/216*a*(9*I*arctan(a*x)^2+9*arctan(a*x)^3-2*I-6*arctan(a*x))*(a^3*x^3-3*I*a^2*x^2-3*a*x+I)*(c*(a*x-I)*(a*x+I)
)^(1/2)/(a^2*x^2+1)^2/c^3-7/8*a*(arctan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(a*x-I)*(c*(a*x-I)*(a*x+I)
)^(1/2)/c^3/(a^2*x^2+1)-7/8*(c*(a*x-I)*(a*x+I))^(1/2)*(a*x+I)*(arctan(a*x)^3-6*arctan(a*x)-3*I*arctan(a*x)^2+6
*I)*a/c^3/(a^2*x^2+1)+1/216*(c*(a*x-I)*(a*x+I))^(1/2)*(a^3*x^3+3*I*a^2*x^2-3*a*x-I)*(-9*I*arctan(a*x)^2+9*arct
an(a*x)^3+2*I-6*arctan(a*x))*a/c^3/(a^4*x^4+2*a^2*x^2+1)-arctan(a*x)^3*(c*(a*x-I)*(a*x+I))^(1/2)/x/c^3-3*a*(ar
ctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*arctan(a*x)*
polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*polylog(3,-(1
+I*a*x)/(a^2*x^2+1)^(1/2))-2*polylog(3,(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^2/(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^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/x^2/(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^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2), 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^{2} \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**2/(a**2*c*x**2+c)**(5/2),x)

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError