\(\int \frac {x^5 \arctan (a x)^3}{(c+a^2 c x^2)^{5/2}} \, dx\) [450]

Optimal result

Integrand size = 24, antiderivative size = 534 \[ \int \frac {x^5 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 x^3}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 x}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {94 \arctan (a x)}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^6 c^3}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

2/27*x^3/a^3/c/(a^2*c*x^2+c)^(3/2)-2/9*x^2*arctan(a*x)/a^4/c/(a^2*c*x^2+c)^(3/2)-1/3*x^3*arctan(a*x)^2/a^3/c/(
a^2*c*x^2+c)^(3/2)+1/3*x^2*arctan(a*x)^3/a^4/c/(a^2*c*x^2+c)^(3/2)+94/9*x/a^5/c^2/(a^2*c*x^2+c)^(1/2)-94/9*arc
tan(a*x)/a^6/c^2/(a^2*c*x^2+c)^(1/2)-5*x*arctan(a*x)^2/a^5/c^2/(a^2*c*x^2+c)^(1/2)+5/3*arctan(a*x)^3/a^6/c^2/(
a^2*c*x^2+c)^(1/2)+6*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+
c)^(1/2)-6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+c)^(1/
2)+6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+c)^(1/2)+6*po
lylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+c)^(1/2)-6*polylog(3,I*(1+I*a*x)/
(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^6/c^2/(a^2*c*x^2+c)^(1/2)+arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^6/c^3

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5084, 5050, 5010, 5008, 4266, 2611, 2320, 6724, 5018, 197, 5060, 5058} \[ \int \frac {x^5 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^6 c^3}-\frac {6 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^6 c^2 \sqrt {a^2 c x^2+c}}-\frac {94 \arctan (a x)}{9 a^6 c^2 \sqrt {a^2 c x^2+c}}-\frac {5 x \arctan (a x)^2}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {94 x}{9 a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3}{27 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]

[In]

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

[Out]

(2*x^3)/(27*a^3*c*(c + a^2*c*x^2)^(3/2)) + (94*x)/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^2*ArcTan[a*x])/(9*a^4
*c*(c + a^2*c*x^2)^(3/2)) - (94*ArcTan[a*x])/(9*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^2)/(3*a^3*c*(c
+ a^2*c*x^2)^(3/2)) - (5*x*ArcTan[a*x]^2)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(
I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^3)/(3*a^4*c*(c + a^2*c*x^2)^(3
/2)) + (5*ArcTan[a*x]^3)/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^6*c^3) - ((6
*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*S
qrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*
x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(
I*ArcTan[a*x])])/(a^6*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 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 4266

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

Rule 5008

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c*Sqrt[d]), Subst
[Int[(a + b*x)^p*Sec[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 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/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 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 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 5058

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

Rule 5060

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

Rule 5084

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/e, Int[
x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*Arc
Tan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m
, 1] && 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}& = -\frac {\int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac {\int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c} \\ & = -\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{3 a^2}+\frac {\int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac {2 \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^4 c}-\frac {\int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c} \\ & = \frac {2 x^3}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^6 c^3}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^5 c^2}-\frac {2 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c}-\frac {3 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c}+\frac {4 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a^4 c} \\ & = \frac {2 x^3}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {94 \arctan (a x)}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^6 c^3}+\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a^5 c}+\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c}+\frac {6 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^5 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 x}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {94 \arctan (a x)}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^6 c^3}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 x}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {94 \arctan (a x)}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^6 c^3}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 x}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {94 \arctan (a x)}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^6 c^3}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 x}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {94 \arctan (a x)}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^6 c^3}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 x}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^4 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {94 \arctan (a x)}{9 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 x \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^3}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 \arctan (a x)^3}{3 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^6 c^3}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.93 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.69 \[ \int \frac {x^5 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {\left (1+a^2 x^2\right )^2 \left (1134 \arctan (a x)-405 \arctan (a x)^3+1128 \arctan (a x) \cos (2 \arctan (a x))-180 \arctan (a x)^3 \cos (2 \arctan (a x))-6 \arctan (a x) \cos (4 \arctan (a x))+9 \arctan (a x)^3 \cos (4 \arctan (a x))+\frac {648 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {648 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {1296 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {1296 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {1296 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {1296 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-1132 \sin (2 \arctan (a x))+558 \arctan (a x)^2 \sin (2 \arctan (a x))+2 \sin (4 \arctan (a x))-9 \arctan (a x)^2 \sin (4 \arctan (a x))\right )}{216 a^6 c \left (c \left (1+a^2 x^2\right )\right )^{3/2}} \]

[In]

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

[Out]

-1/216*((1 + a^2*x^2)^2*(1134*ArcTan[a*x] - 405*ArcTan[a*x]^3 + 1128*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 180*ArcT
an[a*x]^3*Cos[2*ArcTan[a*x]] - 6*ArcTan[a*x]*Cos[4*ArcTan[a*x]] + 9*ArcTan[a*x]^3*Cos[4*ArcTan[a*x]] + (648*Ar
cTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (648*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])
/Sqrt[1 + a^2*x^2] + ((1296*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((1296*I)*A
rcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (1296*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt
[1 + a^2*x^2] + (1296*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 1132*Sin[2*ArcTan[a*x]] + 558*ArcTa
n[a*x]^2*Sin[2*ArcTan[a*x]] + 2*Sin[4*ArcTan[a*x]] - 9*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/(a^6*c*(c*(1 + a^2*x
^2))^(3/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^5*arctan(a*x)^3/(a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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