\(\int \frac {\arctan (a x)^3}{x (c+a^2 c x^2)^{5/2}} \, dx\) [456]

Optimal result

Integrand size = 24, antiderivative size = 553 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

2/27*a*x/c/(a^2*c*x^2+c)^(3/2)-2/9*arctan(a*x)/c/(a^2*c*x^2+c)^(3/2)-1/3*a*x*arctan(a*x)^2/c/(a^2*c*x^2+c)^(3/
2)+1/3*arctan(a*x)^3/c/(a^2*c*x^2+c)^(3/2)+202/27*a*x/c^2/(a^2*c*x^2+c)^(1/2)-22/3*arctan(a*x)/c^2/(a^2*c*x^2+
c)^(1/2)-11/3*a*x*arctan(a*x)^2/c^2/(a^2*c*x^2+c)^(1/2)+arctan(a*x)^3/c^2/(a^2*c*x^2+c)^(1/2)-2*arctan(a*x)^3*
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)+3*I*arctan(a*x)^2*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)-3*I*arctan(a*x)^2*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*arctan(a*x)*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*arctan(a*x)*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*I*polylog(4,-(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*polylog(4,(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)

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5086, 5078, 5076, 4268, 2611, 6744, 2320, 6724, 5050, 5018, 197, 5020, 198} \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {a^2 x^2+1} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x)^3}{c^2 \sqrt {a^2 c x^2+c}}-\frac {11 a x \arctan (a x)^2}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {22 \arctan (a x)}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {a x \arctan (a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 \arctan (a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 a x}{27 c \left (a^2 c x^2+c\right )^{3/2}} \]

[In]

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

[Out]

(2*a*x)/(27*c*(c + a^2*c*x^2)^(3/2)) + (202*a*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) - (2*ArcTan[a*x])/(9*c*(c + a^2*
c*x^2)^(3/2)) - (22*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a*x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2))
- (11*a*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x
]^3/(c^2*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^2*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^2*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^2*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^2*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^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*Arc
Tan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c +
a^2*c*x^2])

Rule 197

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]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

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 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 5050

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 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]

Rule 6744

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]

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac {\int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c} \\ & = \frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-a \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac {\int \frac {\arctan (a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx}{c^2}-\frac {a^2 \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c} \\ & = -\frac {2 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {a x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{c^2 \sqrt {c+a^2 c x^2}}+\frac {1}{9} (2 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx-\frac {(2 a) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {(3 a) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{c^2 \sqrt {c+a^2 c x^2}}+\frac {(4 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{27 c}+\frac {(4 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}+\frac {(6 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \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 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \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 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \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 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {202 a x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^3}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.63 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (1+a^2 x^2\right )^{3/2} \left (-27 i \pi ^4+\frac {1620 a x}{\sqrt {1+a^2 x^2}}-\frac {1620 \arctan (a x)}{\sqrt {1+a^2 x^2}}-\frac {810 a x \arctan (a x)^2}{\sqrt {1+a^2 x^2}}+\frac {270 \arctan (a x)^3}{\sqrt {1+a^2 x^2}}+54 i \arctan (a x)^4-12 \arctan (a x) \cos (3 \arctan (a x))+18 \arctan (a x)^3 \cos (3 \arctan (a x))+216 \arctan (a x)^3 \log \left (1-e^{-i \arctan (a x)}\right )-216 \arctan (a x)^3 \log \left (1+e^{i \arctan (a x)}\right )+648 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+648 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+1296 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-1296 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )-1296 i \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-1296 i \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )+4 \sin (3 \arctan (a x))-18 \arctan (a x)^2 \sin (3 \arctan (a x))\right )}{216 c \left (c \left (1+a^2 x^2\right )\right )^{3/2}} \]

[In]

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

[Out]

((1 + a^2*x^2)^(3/2)*((-27*I)*Pi^4 + (1620*a*x)/Sqrt[1 + a^2*x^2] - (1620*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (81
0*a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (270*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (54*I)*ArcTan[a*x]^4 - 12*Arc
Tan[a*x]*Cos[3*ArcTan[a*x]] + 18*ArcTan[a*x]^3*Cos[3*ArcTan[a*x]] + 216*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a
*x])] - 216*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (648*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])]
+ (648*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] + 1296*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 1
296*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] - (1296*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (1296*I)*PolyLog[
4, -E^(I*ArcTan[a*x])] + 4*Sin[3*ArcTan[a*x]] - 18*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]]))/(216*c*(c*(1 + a^2*x^2))
^(3/2))

Maple [A] (verified)

Time = 4.06 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {\left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} c^{3}}+\frac {5 \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {5 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right )}{216 c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}+\frac {i \left (i \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3}}\)	\(560\)

[In]

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

[Out]

-1/216*(9*I*arctan(a*x)^2+9*arctan(a*x)^3-2*I-6*arctan(a*x))*(I*a^3*x^3+3*a^2*x^2-3*I*a*x-1)*(c*(a*x-I)*(I+a*x
))^(1/2)/(a^2*x^2+1)^2/c^3+5/8*(arctan(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(1+I*a*x)*(c*(a*x-I)*(I+a*x
))^(1/2)/c^3/(a^2*x^2+1)-5/8*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a*x-1)*(arctan(a*x)^3-6*arctan(a*x)-3*I*arctan(a*x)^
2+6*I)/c^3/(a^2*x^2+1)+1/216*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a^3*x^3-3*a^2*x^2-3*I*a*x+1)*(-9*I*arctan(a*x)^2+9*a
rctan(a*x)^3+2*I-6*arctan(a*x))/c^3/(a^4*x^4+2*a^2*x^2+1)+I*(I*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)
-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*a
rctan(a*x)^2*polylog(2,(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)*polylog(3,(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))+6*polylog(4,(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]

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

[In]

integrate(arctan(a*x)^3/x/(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^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(arctan(a*x)^3/x/(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), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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