3.451 \(\int \frac{x^4 \tan ^{-1}(a x)^3}{(c+a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=622 \[ \frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{68}{9 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a^5 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

-2/(27*a^5*c*(c + a^2*c*x^2)^(3/2)) + 68/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^3*ArcTan[a*x])/(9*a^2*c*(c + a
^2*c*x^2)^(3/2)) + (22*x*ArcTan[a*x])/(3*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^2)/(3*a^3*c*(c + a^2*
c*x^2)^(3/2)) - (11*ArcTan[a*x]^2)/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^3)/(3*a^2*c*(c + a^2*c*x
^2)^(3/2)) - (x*ArcTan[a*x]^3)/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x
])]*ArcTan[a*x]^3)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I
*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTa
n[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])
/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sq
rt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]
) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.998001, antiderivative size = 622, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 15, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4964, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4898, 4894, 4944, 4940, 4930, 266, 43} \[ \frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{68}{9 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a^5 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]

[Out]

-2/(27*a^5*c*(c + a^2*c*x^2)^(3/2)) + 68/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^3*ArcTan[a*x])/(9*a^2*c*(c + a
^2*c*x^2)^(3/2)) + (22*x*ArcTan[a*x])/(3*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^2)/(3*a^3*c*(c + a^2*
c*x^2)^(3/2)) - (11*ArcTan[a*x]^2)/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^3)/(3*a^2*c*(c + a^2*c*x
^2)^(3/2)) - (x*ArcTan[a*x]^3)/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x
])]*ArcTan[a*x]^3)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I
*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTa
n[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])
/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sq
rt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]
) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2])

Rule 4964

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 4890

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 4888

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 4181

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 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 6609

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]

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 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 4940

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 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 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^4 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac{\int \frac{x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac{\int \frac{x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\int \frac{x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a}+\frac{\int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac{\int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}\\ &=\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{3 \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{9 a}+\frac{6 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}+\frac{2 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^3 c}+\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{a^4 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{6}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{6 x \tan ^{-1}(a x)}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\operatorname{Subst}\left (\int \frac{x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )}{9 a}+\frac{4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^4 c}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{22}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac{1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{9 a}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{68}{9 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{68}{9 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{68}{9 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (6 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (6 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{68}{9 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{6 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}-\frac{6 i \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{6 i \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 2.62937, size = 691, normalized size = 1.11 \[ -\frac{\sqrt{c \left (a^2 x^2+1\right )} \left (-5184 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{-i \tan ^{-1}(a x)}\right )-5184 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+5184 i \pi \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-10368 \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{-i \tan ^{-1}(a x)}\right )+10368 \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-1296 i \pi \left (\pi -4 \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,i e^{-i \tan ^{-1}(a x)}\right )-1296 i \pi ^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+5184 \pi \text{PolyLog}\left (3,i e^{-i \tan ^{-1}(a x)}\right )-5184 \pi \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+10368 i \text{PolyLog}\left (4,-i e^{-i \tan ^{-1}(a x)}\right )+10368 i \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )-\frac{12960}{\sqrt{a^2 x^2+1}}+\frac{2160 a x \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{6480 \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{12960 a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-432 i \tan ^{-1}(a x)^4+864 i \pi \tan ^{-1}(a x)^3-648 i \pi ^2 \tan ^{-1}(a x)^2+216 i \pi ^3 \tan ^{-1}(a x)-1728 \tan ^{-1}(a x)^3 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right )+1728 \tan ^{-1}(a x)^3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+2592 \pi \tan ^{-1}(a x)^2 \log \left (1-i e^{-i \tan ^{-1}(a x)}\right )-2592 \pi \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-1296 \pi ^2 \tan ^{-1}(a x) \log \left (1-i e^{-i \tan ^{-1}(a x)}\right )+1296 \pi ^2 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+216 \pi ^3 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right )-216 \pi ^3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-216 \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (2 \tan ^{-1}(a x)+\pi \right )\right )\right )-144 \tan ^{-1}(a x)^3 \sin \left (3 \tan ^{-1}(a x)\right )+96 \tan ^{-1}(a x) \sin \left (3 \tan ^{-1}(a x)\right )-144 \tan ^{-1}(a x)^2 \cos \left (3 \tan ^{-1}(a x)\right )+32 \cos \left (3 \tan ^{-1}(a x)\right )+189 i \pi ^4\right )}{1728 a^5 c^3 \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]

[Out]

-(Sqrt[c*(1 + a^2*x^2)]*((189*I)*Pi^4 - 12960/Sqrt[1 + a^2*x^2] + (216*I)*Pi^3*ArcTan[a*x] - (12960*a*x*ArcTan
[a*x])/Sqrt[1 + a^2*x^2] - (648*I)*Pi^2*ArcTan[a*x]^2 + (6480*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (864*I)*Pi*Ar
cTan[a*x]^3 + (2160*a*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] - (432*I)*ArcTan[a*x]^4 + 32*Cos[3*ArcTan[a*x]] - 144
*ArcTan[a*x]^2*Cos[3*ArcTan[a*x]] - 1296*Pi^2*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])] + 2592*Pi*ArcTan[a*x]^2
*Log[1 - I/E^(I*ArcTan[a*x])] + 216*Pi^3*Log[1 + I/E^(I*ArcTan[a*x])] - 1728*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcT
an[a*x])] - 216*Pi^3*Log[1 + I*E^(I*ArcTan[a*x])] + 1296*Pi^2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - 2592*
Pi*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 1728*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - 216*Pi^3*Log
[Tan[(Pi + 2*ArcTan[a*x])/4]] - (5184*I)*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] - (1296*I)*Pi*(Pi -
4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] - (1296*I)*Pi^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (5184*I)*P
i*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (5184*I)*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] -
10368*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])] + 5184*Pi*PolyLog[3, I/E^(I*ArcTan[a*x])] - 5184*Pi*Poly
Log[3, (-I)*E^(I*ArcTan[a*x])] + 10368*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (10368*I)*PolyLog[4, (
-I)/E^(I*ArcTan[a*x])] + (10368*I)*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] + 96*ArcTan[a*x]*Sin[3*ArcTan[a*x]] - 14
4*ArcTan[a*x]^3*Sin[3*ArcTan[a*x]]))/(1728*a^5*c^3*Sqrt[1 + a^2*x^2])

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Maple [F]  time = 0.884, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( \arctan \left ( ax \right ) \right ) ^{3} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(x^4*arctan(a*x)^3/(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} x^{4} \arctan \left (a x\right )^{3}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*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^4*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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \operatorname{atan}^{3}{\left (a x \right )}}{\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(x**4*atan(a*x)**3/(a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(x**4*atan(a*x)**3/(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{x^{4} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

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