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

Optimal. Leaf size=389 \[ \frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{22}{9 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{a^2 c x^2+c}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

-2/(27*c*(c + a^2*c*x^2)^(3/2)) - 22/(9*c^2*Sqrt[c + a^2*c*x^2]) - (2*a*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3
/2)) - (22*a*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a
*x]^2/(c^2*Sqrt[c + a^2*c*x^2]) - (2*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]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])
- ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 +
a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcT
an[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.7834, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4966, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4894, 4896} \[ \frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{22}{9 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{a^2 c x^2+c}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(27*c*(c + a^2*c*x^2)^(3/2)) - 22/(9*c^2*Sqrt[c + a^2*c*x^2]) - (2*a*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3
/2)) - (22*a*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a
*x]^2/(c^2*Sqrt[c + a^2*c*x^2]) - (2*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]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])
- ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 +
a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcT
an[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 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 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 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)^2}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\left (a^2 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{\tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{1}{3} (2 a) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx}{c^2}-\frac{a^2 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac{2}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{(4 a) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 c}-\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{\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)^2}{x \sqrt{1+a^2 x^2}} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \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}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \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 (2 \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 (2 \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}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \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{2 i \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{2 i \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 (2 i \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 (2 i \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}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \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{2 i \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{2 i \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 (2 \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 (2 \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}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22}{9 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \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{2 i \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{2 i \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{2 \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{2 \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 = 0.50098, size = 246, normalized size = 0.63 \[ \frac{\left (a^2 x^2+1\right )^{3/2} \left (216 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-216 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-216 \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )+216 \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )-\frac{270}{\sqrt{a^2 x^2+1}}+\frac{135 \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{270 a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+108 \tan ^{-1}(a x)^2 \log \left (1-e^{i \tan ^{-1}(a x)}\right )-108 \tan ^{-1}(a x)^2 \log \left (1+e^{i \tan ^{-1}(a x)}\right )-6 \tan ^{-1}(a x) \sin \left (3 \tan ^{-1}(a x)\right )+9 \tan ^{-1}(a x)^2 \cos \left (3 \tan ^{-1}(a x)\right )-2 \cos \left (3 \tan ^{-1}(a x)\right )\right )}{108 c \left (c \left (a^2 x^2+1\right )\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + a^2*x^2)^(3/2)*(-270/Sqrt[1 + a^2*x^2] - (270*a*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + (135*ArcTan[a*x]^2)/S
qrt[1 + a^2*x^2] - 2*Cos[3*ArcTan[a*x]] + 9*ArcTan[a*x]^2*Cos[3*ArcTan[a*x]] + 108*ArcTan[a*x]^2*Log[1 - E^(I*
ArcTan[a*x])] - 108*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + (216*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x
])] - (216*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 216*PolyLog[3, -E^(I*ArcTan[a*x])] + 216*PolyLog[3,
E^(I*ArcTan[a*x])] - 6*ArcTan[a*x]*Sin[3*ArcTan[a*x]]))/(108*c*(c*(1 + a^2*x^2))^(3/2))

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Maple [A]  time = 0.345, size = 461, normalized size = 1.2 \begin{align*} -{\frac{ \left ( 6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) \left ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 5\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-10+10\,i\arctan \left ( ax \right ) \right ) \left ( 1+iax \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -5+5\,iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( i{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-3\,iax+1 \right ) \left ( -6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) }{216\,{c}^{3} \left ({a}^{4}{x}^{4}+2\,{a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{1}{{c}^{3}} \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) - \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -2\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +2\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +2\,{\it polylog} \left ( 3,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -2\,{\it polylog} \left ( 3,{\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)^2/x/(a^2*c*x^2+c)^(5/2),x)

[Out]

-1/216*(6*I*arctan(a*x)+9*arctan(a*x)^2-2)*(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)^2-2+2*I*arctan(a*x))*(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)^2-2-2*I*arctan(a*x))/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)*(-6*I*arctan(a*x)+9*arctan(a*x)^2-2)/c^3/(a^4*x^4+2*a^2*x^2+1)-(arctan(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)^2/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 )^{2}}{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)^2/x/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

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

[Out]

Integral(atan(a*x)**2/(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 )^{2}}{{\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)^2/x/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

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