\(\int \frac {\arctan (a x)^3}{x^2 (c+a^2 c x^2)^{5/2}} \, dx\) [457]

Optimal result

Integrand size = 24, antiderivative size = 493 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a}{9 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 a^2 x \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}-\frac {6 a \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

2/27*a/c/(a^2*c*x^2+c)^(3/2)+2/9*a^2*x*arctan(a*x)/c/(a^2*c*x^2+c)^(3/2)-1/3*a*arctan(a*x)^2/c/(a^2*c*x^2+c)^(
3/2)-1/3*a^2*x*arctan(a*x)^3/c/(a^2*c*x^2+c)^(3/2)+94/9*a/c^2/(a^2*c*x^2+c)^(1/2)+94/9*a^2*x*arctan(a*x)/c^2/(
a^2*c*x^2+c)^(1/2)-5*a*arctan(a*x)^2/c^2/(a^2*c*x^2+c)^(1/2)-5/3*a^2*x*arctan(a*x)^3/c^2/(a^2*c*x^2+c)^(1/2)-6
*a*arctan(a*x)^2*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+6*I*a*arctan(a
*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-6*I*a*arctan(a*x)*polylo
g(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-6*a*polylog(3,-(1+I*a*x)/(a^2*x^2+1
)^(1/2))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+6*a*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2
)/c^2/(a^2*c*x^2+c)^(1/2)-arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/c^3/x

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5086, 5064, 5078, 5076, 4268, 2611, 2320, 6724, 5018, 5014, 5020, 5016} \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {6 a \sqrt {a^2 x^2+1} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c^3 x}+\frac {6 i a \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {6 i a \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {6 a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {a^2 c x^2+c}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {a^2 c x^2+c}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {a \arctan (a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 a^2 x \arctan (a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {94 a}{9 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 a}{27 c \left (a^2 c x^2+c\right )^{3/2}} \]

[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 2320

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 2611

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 4268

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 5014

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 5016

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]

Rule 5018

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 5020

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 5064

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 5076

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 5078

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 5086

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 6724

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*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac {\int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c} \\ & = -\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {a^2 x \arctan (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 {\arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac {\int \frac {\arctan (a x)^3}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{c^2}-\frac {\left (2 a^2\right ) \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {a^2 \int \frac {\arctan (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 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}+\frac {(3 a) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx}{c^2}+\frac {\left (4 a^2\right ) \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 c}+\frac {\left (4 a^2\right ) \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac {\left (6 a^2\right ) \int \frac {\arctan (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 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}+\frac {\left (3 a \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (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 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}+\frac {\left (3 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (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 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}-\frac {6 a \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (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 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}-\frac {6 a \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 i a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (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 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}-\frac {6 a \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 a \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (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 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}-\frac {6 a \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.81 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {a \left (-1134-1134 a x \arctan (a x)+567 \arctan (a x)^2+189 a x \arctan (a x)^3-2 \sqrt {1+a^2 x^2} \cos (3 \arctan (a x))+9 \sqrt {1+a^2 x^2} \arctan (a x)^2 \cos (3 \arctan (a x))+27 a x \arctan (a x)^3 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-324 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )+324 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )-648 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+648 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+648 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )-648 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )-6 \sqrt {1+a^2 x^2} \arctan (a x) \sin (3 \arctan (a x))+9 \sqrt {1+a^2 x^2} \arctan (a x)^3 \sin (3 \arctan (a x))+54 \sqrt {1+a^2 x^2} \arctan (a x)^3 \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{108 c^2 \sqrt {c+a^2 c x^2}} \]

[In]

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

[Out]

-1/108*(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*ArcTan[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]))/
(c^2*Sqrt[c + a^2*c*x^2])

Maple [A] (verified)

Time = 3.77 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {a \left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (a^{3} x^{3}-3 i a^{2} x^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} c^{3}}-\frac {7 a \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {7 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right ) a}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i a^{2} x^{2}-3 a x -i\right ) \left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right ) a}{216 c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{c^{3} x}-\frac {3 a \left (\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-\arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3}}\)	\(528\)

[In]

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

[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)*(I+a*x)
)^(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)*(I+a*x)
)^(1/2)/c^3/(a^2*x^2+1)-7/8*(c*(a*x-I)*(I+a*x))^(1/2)*(I+a*x)*(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)*(I+a*x))^(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)*(I+a*x))^(1/2)/c^3/x-3*a*(ar
ctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-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)*(I+a*x))^(1/2)/(a^2*x^2+1)^(1/
2)/c^3

Fricas [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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