3.382 \(\int (c+a^2 c x^2)^3 \tan ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=388 \[ \frac{24 c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{35 a}+\frac{48 i c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{35 a}-\frac{c^3 \left (a^2 x^2+1\right )^2}{140 a}-\frac{13 c^3 \left (a^2 x^2+1\right )}{210 a}-\frac{7 c^3 \log \left (a^2 x^2+1\right )}{15 a}+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3-\frac{c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{14 a}-\frac{9 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{12 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{35 a}+\frac{1}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{16 i c^3 \tan ^{-1}(a x)^3}{35 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{48 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{35 a} \]

[Out]

(-13*c^3*(1 + a^2*x^2))/(210*a) - (c^3*(1 + a^2*x^2)^2)/(140*a) + (14*c^3*x*ArcTan[a*x])/15 + (13*c^3*x*(1 + a
^2*x^2)*ArcTan[a*x])/105 + (c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 - (12*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a
) - (9*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(70*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(14*a) + (((16*I)/35)*c
^3*ArcTan[a*x]^3)/a + (16*c^3*x*ArcTan[a*x]^3)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/35 + (6*c^3*x*(1 + a
^2*x^2)^2*ArcTan[a*x]^3)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/7 + (48*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x
)])/(35*a) - (7*c^3*Log[1 + a^2*x^2])/(15*a) + (((48*I)/35)*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a +
(24*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a)

________________________________________________________________________________________

Rubi [A]  time = 0.340402, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {4880, 4846, 4920, 4854, 4884, 4994, 6610, 260, 4878} \[ \frac{24 c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{35 a}+\frac{48 i c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{35 a}-\frac{c^3 \left (a^2 x^2+1\right )^2}{140 a}-\frac{13 c^3 \left (a^2 x^2+1\right )}{210 a}-\frac{7 c^3 \log \left (a^2 x^2+1\right )}{15 a}+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3-\frac{c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{14 a}-\frac{9 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{12 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{35 a}+\frac{1}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{16 i c^3 \tan ^{-1}(a x)^3}{35 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{48 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{35 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*c^3*(1 + a^2*x^2))/(210*a) - (c^3*(1 + a^2*x^2)^2)/(140*a) + (14*c^3*x*ArcTan[a*x])/15 + (13*c^3*x*(1 + a
^2*x^2)*ArcTan[a*x])/105 + (c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 - (12*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a
) - (9*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(70*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(14*a) + (((16*I)/35)*c
^3*ArcTan[a*x]^3)/a + (16*c^3*x*ArcTan[a*x]^3)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/35 + (6*c^3*x*(1 + a
^2*x^2)^2*ArcTan[a*x]^3)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/7 + (48*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x
)])/(35*a) - (7*c^3*Log[1 + a^2*x^2])/(15*a) + (((48*I)/35)*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a +
(24*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a)

Rule 4880

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> -Simp[(b*p*(d + e*x^2)^q
*(a + b*ArcTan[c*x])^(p - 1))/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*
ArcTan[c*x])^p, x], x] + Dist[(b^2*d*p*(p - 1))/(2*q*(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^(
p - 2), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p)/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && E
qQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 4884

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u
])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*
I)/(I - c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4878

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

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3 \, dx &=-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{14 a}+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3+\frac{1}{7} c \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx+\frac{1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac{1}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{14 a}+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3+\frac{1}{35} \left (4 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx+\frac{1}{35} \left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx+\frac{1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac{c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac{13}{105} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{1}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac{12 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{14 a}+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3+\frac{1}{105} \left (8 c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac{1}{35} \left (6 c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac{1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x)^3 \, dx+\frac{1}{35} \left (24 c^3\right ) \int \tan ^{-1}(a x) \, dx\\ &=-\frac{13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac{c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{1}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac{12 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{14 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3-\frac{1}{105} \left (8 a c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{35} \left (6 a c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{35} \left (24 a c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{35} \left (48 a c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac{c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{1}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac{12 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{14 a}+\frac{16 i c^3 \tan ^{-1}(a x)^3}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3-\frac{7 c^3 \log \left (1+a^2 x^2\right )}{15 a}+\frac{1}{35} \left (48 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx\\ &=-\frac{13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac{c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{1}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac{12 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{14 a}+\frac{16 i c^3 \tan ^{-1}(a x)^3}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3+\frac{48 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{35 a}-\frac{7 c^3 \log \left (1+a^2 x^2\right )}{15 a}-\frac{1}{35} \left (96 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac{c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{1}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac{12 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{14 a}+\frac{16 i c^3 \tan ^{-1}(a x)^3}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3+\frac{48 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{35 a}-\frac{7 c^3 \log \left (1+a^2 x^2\right )}{15 a}+\frac{48 i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{35 a}-\frac{1}{35} \left (48 i c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac{c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)+\frac{1}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)-\frac{12 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac{9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{70 a}-\frac{c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{14 a}+\frac{16 i c^3 \tan ^{-1}(a x)^3}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3+\frac{48 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{35 a}-\frac{7 c^3 \log \left (1+a^2 x^2\right )}{15 a}+\frac{48 i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{35 a}+\frac{24 c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{35 a}\\ \end{align*}

Mathematica [A]  time = 1.12658, size = 243, normalized size = 0.63 \[ \frac{c^3 \left (-576 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+288 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-3 a^4 x^4-32 a^2 x^2-196 \log \left (a^2 x^2+1\right )+60 a^7 x^7 \tan ^{-1}(a x)^3-30 a^6 x^6 \tan ^{-1}(a x)^2+252 a^5 x^5 \tan ^{-1}(a x)^3+12 a^5 x^5 \tan ^{-1}(a x)-144 a^4 x^4 \tan ^{-1}(a x)^2+420 a^3 x^3 \tan ^{-1}(a x)^3+76 a^3 x^3 \tan ^{-1}(a x)-342 a^2 x^2 \tan ^{-1}(a x)^2+420 a x \tan ^{-1}(a x)^3+456 a x \tan ^{-1}(a x)-192 i \tan ^{-1}(a x)^3-228 \tan ^{-1}(a x)^2+576 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-29\right )}{420 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-29 - 32*a^2*x^2 - 3*a^4*x^4 + 456*a*x*ArcTan[a*x] + 76*a^3*x^3*ArcTan[a*x] + 12*a^5*x^5*ArcTan[a*x] - 2
28*ArcTan[a*x]^2 - 342*a^2*x^2*ArcTan[a*x]^2 - 144*a^4*x^4*ArcTan[a*x]^2 - 30*a^6*x^6*ArcTan[a*x]^2 - (192*I)*
ArcTan[a*x]^3 + 420*a*x*ArcTan[a*x]^3 + 420*a^3*x^3*ArcTan[a*x]^3 + 252*a^5*x^5*ArcTan[a*x]^3 + 60*a^7*x^7*Arc
Tan[a*x]^3 + 576*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 196*Log[1 + a^2*x^2] - (576*I)*ArcTan[a*x]*Pol
yLog[2, -E^((2*I)*ArcTan[a*x])] + 288*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(420*a)

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Maple [C]  time = 2.551, size = 1134, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12/35*I/a*c^3*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2+12/35*I/a*c^3*
Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2-12/35*I/a*c^3*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*ar
ctan(a*x)^2+c^3*x*arctan(a*x)^3+38/35*c^3*x*arctan(a*x)-29/420/a*c^3-12/35*I/a*c^3*Pi*csgn(I/((1+I*a*x)^2/(a^2
*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*ar
ctan(a*x)^2-1/140*a^3*x^4*c^3-8/105*a*c^3*x^2-19/35/a*c^3*arctan(a*x)^2+14/15/a*c^3*ln((1+I*a*x)^2/(a^2*x^2+1)
+1)+24/35/a*c^3*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+19/105*a^2*c^3*arctan(a*x)*x^3-57/70*a*c^3*arctan(a*x)^2*x
^2+1/7*a^6*c^3*arctan(a*x)^3*x^7+3/5*a^4*c^3*arctan(a*x)^3*x^5+a^2*c^3*arctan(a*x)^3*x^3-24/35/a*c^3*arctan(a*
x)^2*ln(a^2*x^2+1)+48/35/a*c^3*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+48/35/a*c^3*arctan(a*x)^2*ln(2)-1
/14*a^5*c^3*arctan(a*x)^2*x^6-12/35*a^3*c^3*arctan(a*x)^2*x^4+1/35*a^4*c^3*arctan(a*x)*x^5-14/15*I/a*c^3*arcta
n(a*x)-16/35*I/a*c^3*arctan(a*x)^3+12/35*I/a*c^3*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/
(a^2*x^2+1)+1)^2)*arctan(a*x)^2+24/35*I/a*c^3*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x
^2+1))^2*arctan(a*x)^2+12/35*I/a*c^3*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/(
(1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2-12/35*I/a*c^3*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1
+I*a*x)^2/(a^2*x^2+1))*arctan(a*x)^2+12/35*I/a*c^3*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*
x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2-24/35*I/a*c^3*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csg
n(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2-48/35*I/a*c^3*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1
))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

980*a^8*c^3*integrate(1/1120*x^8*arctan(a*x)^3/(a^2*x^2 + 1), x) + 105*a^8*c^3*integrate(1/1120*x^8*arctan(a*x
)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 60*a^8*c^3*integrate(1/1120*x^8*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2
+ 1), x) - 60*a^7*c^3*integrate(1/1120*x^7*arctan(a*x)^2/(a^2*x^2 + 1), x) + 15*a^7*c^3*integrate(1/1120*x^7*
log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 3920*a^6*c^3*integrate(1/1120*x^6*arctan(a*x)^3/(a^2*x^2 + 1), x) + 420
*a^6*c^3*integrate(1/1120*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 252*a^6*c^3*integrate(1/1120*
x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 252*a^5*c^3*integrate(1/1120*x^5*arctan(a*x)^2/(a^2*x^2 +
1), x) + 63*a^5*c^3*integrate(1/1120*x^5*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 5880*a^4*c^3*integrate(1/1120
*x^4*arctan(a*x)^3/(a^2*x^2 + 1), x) + 630*a^4*c^3*integrate(1/1120*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^
2 + 1), x) + 420*a^4*c^3*integrate(1/1120*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 420*a^3*c^3*int
egrate(1/1120*x^3*arctan(a*x)^2/(a^2*x^2 + 1), x) + 105*a^3*c^3*integrate(1/1120*x^3*log(a^2*x^2 + 1)^2/(a^2*x
^2 + 1), x) + 7/32*c^3*arctan(a*x)^4/a + 3920*a^2*c^3*integrate(1/1120*x^2*arctan(a*x)^3/(a^2*x^2 + 1), x) + 4
20*a^2*c^3*integrate(1/1120*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 420*a^2*c^3*integrate(1/112
0*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 420*a*c^3*integrate(1/1120*x*arctan(a*x)^2/(a^2*x^2 + 1
), x) + 105*a*c^3*integrate(1/1120*x*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 1/280*(5*a^6*c^3*x^7 + 21*a^4*c^3*
x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*arctan(a*x)^3 + 105*c^3*integrate(1/1120*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*
x^2 + 1), x) - 3/1120*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*arctan(a*x)*log(a^2*x^2 + 1
)^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int 3 a^{2} x^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(3*a**2*x**2*atan(a*x)**3, x) + Integral(3*a**4*x**4*atan(a*x)**3, x) + Integral(a**6*x**6*atan(
a*x)**3, x) + Integral(atan(a*x)**3, x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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