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

Optimal. Leaf size=389 \[ -\frac{8 c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{105 a^3}-\frac{16 i c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{105 a^3}-\frac{1}{504} a^3 c^3 x^6+\frac{31 c^3 \log \left (a^2 x^2+1\right )}{945 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}-\frac{47 c^3 x \tan ^{-1}(a x)}{1260 a^2}-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{47 c^3 \tan ^{-1}(a x)^2}{2520 a^3}-\frac{16 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{105 a^3}-\frac{11 a c^3 x^4}{1260}-\frac{107 c^3 x^2}{7560 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{239 c^3 x^3 \tan ^{-1}(a x)}{3780}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a} \]

[Out]

(-107*c^3*x^2)/(7560*a) - (11*a*c^3*x^4)/1260 - (a^3*c^3*x^6)/504 - (47*c^3*x*ArcTan[a*x])/(1260*a^2) + (239*c
^3*x^3*ArcTan[a*x])/3780 + (59*a^2*c^3*x^5*ArcTan[a*x])/1260 + (a^4*c^3*x^7*ArcTan[a*x])/84 + (47*c^3*ArcTan[a
*x]^2)/(2520*a^3) - (8*c^3*x^2*ArcTan[a*x]^2)/(105*a) - (89*a*c^3*x^4*ArcTan[a*x]^2)/420 - (10*a^3*c^3*x^6*Arc
Tan[a*x]^2)/63 - (a^5*c^3*x^8*ArcTan[a*x]^2)/24 - (((16*I)/315)*c^3*ArcTan[a*x]^3)/a^3 + (c^3*x^3*ArcTan[a*x]^
3)/3 + (3*a^2*c^3*x^5*ArcTan[a*x]^3)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^3)/7 + (a^6*c^3*x^9*ArcTan[a*x]^3)/9 - (16
*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(105*a^3) + (31*c^3*Log[1 + a^2*x^2])/(945*a^3) - (((16*I)/105)*c^3*Arc
Tan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (8*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(105*a^3)

________________________________________________________________________________________

Rubi [A]  time = 3.03923, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 132, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4948, 4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610, 266, 43} \[ -\frac{8 c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{105 a^3}-\frac{16 i c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{105 a^3}-\frac{1}{504} a^3 c^3 x^6+\frac{31 c^3 \log \left (a^2 x^2+1\right )}{945 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}-\frac{47 c^3 x \tan ^{-1}(a x)}{1260 a^2}-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{47 c^3 \tan ^{-1}(a x)^2}{2520 a^3}-\frac{16 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{105 a^3}-\frac{11 a c^3 x^4}{1260}-\frac{107 c^3 x^2}{7560 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{239 c^3 x^3 \tan ^{-1}(a x)}{3780}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-107*c^3*x^2)/(7560*a) - (11*a*c^3*x^4)/1260 - (a^3*c^3*x^6)/504 - (47*c^3*x*ArcTan[a*x])/(1260*a^2) + (239*c
^3*x^3*ArcTan[a*x])/3780 + (59*a^2*c^3*x^5*ArcTan[a*x])/1260 + (a^4*c^3*x^7*ArcTan[a*x])/84 + (47*c^3*ArcTan[a
*x]^2)/(2520*a^3) - (8*c^3*x^2*ArcTan[a*x]^2)/(105*a) - (89*a*c^3*x^4*ArcTan[a*x]^2)/420 - (10*a^3*c^3*x^6*Arc
Tan[a*x]^2)/63 - (a^5*c^3*x^8*ArcTan[a*x]^2)/24 - (((16*I)/315)*c^3*ArcTan[a*x]^3)/a^3 + (c^3*x^3*ArcTan[a*x]^
3)/3 + (3*a^2*c^3*x^5*ArcTan[a*x]^3)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^3)/7 + (a^6*c^3*x^9*ArcTan[a*x]^3)/9 - (16
*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(105*a^3) + (31*c^3*Log[1 + a^2*x^2])/(945*a^3) - (((16*I)/105)*c^3*Arc
Tan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (8*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(105*a^3)

Rule 4948

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

Rule 4852

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

Rule 4916

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcTan[c*x])^p)
/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 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 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 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 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 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 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 x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3 \, dx &=\int \left (c^3 x^2 \tan ^{-1}(a x)^3+3 a^2 c^3 x^4 \tan ^{-1}(a x)^3+3 a^4 c^3 x^6 \tan ^{-1}(a x)^3+a^6 c^3 x^8 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^3 \int x^2 \tan ^{-1}(a x)^3 \, dx+\left (3 a^2 c^3\right ) \int x^4 \tan ^{-1}(a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^6 \tan ^{-1}(a x)^3 \, dx+\left (a^6 c^3\right ) \int x^8 \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\left (a c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{5} \left (9 a^3 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{7} \left (9 a^5 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^7 c^3\right ) \int \frac{x^9 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{c^3 \int x \tan ^{-1}(a x)^2 \, dx}{a}+\frac{c^3 \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a}-\frac{1}{5} \left (9 a c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\frac{1}{5} \left (9 a c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{7} \left (9 a^3 c^3\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx+\frac{1}{7} \left (9 a^3 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^5 c^3\right ) \int x^7 \tan ^{-1}(a x)^2 \, dx+\frac{1}{3} \left (a^5 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 x^2 \tan ^{-1}(a x)^2}{2 a}-\frac{9}{20} a c^3 x^4 \tan ^{-1}(a x)^2-\frac{3}{14} a^3 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2-\frac{i c^3 \tan ^{-1}(a x)^3}{3 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3+c^3 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{c^3 \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{a^2}+\frac{\left (9 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx}{5 a}-\frac{\left (9 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{7} \left (9 a c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx-\frac{1}{7} \left (9 a c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac{1}{10} \left (9 a^2 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{3} \left (a^3 c^3\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx-\frac{1}{3} \left (a^3 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac{1}{7} \left (3 a^4 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{12} \left (a^6 c^3\right ) \int \frac{x^8 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{2 c^3 x^2 \tan ^{-1}(a x)^2}{5 a}-\frac{9}{70} a c^3 x^4 \tan ^{-1}(a x)^2-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2+\frac{4 i c^3 \tan ^{-1}(a x)^3}{15 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^3}+\frac{1}{10} \left (9 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{10} \left (9 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (9 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{c^3 \int \tan ^{-1}(a x) \, dx}{a^2}-\frac{c^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}+\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{5 a^2}+\frac{\left (2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac{\left (9 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx}{7 a}+\frac{\left (9 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{7 a}-\frac{1}{3} \left (a c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\frac{1}{3} \left (a c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac{1}{7} \left (3 a^2 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx-\frac{1}{7} \left (3 a^2 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{14} \left (9 a^2 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{12} \left (a^4 c^3\right ) \int x^6 \tan ^{-1}(a x) \, dx-\frac{1}{12} \left (a^4 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{9} \left (a^4 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{c^3 x \tan ^{-1}(a x)}{a^2}+\frac{3}{10} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{35} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{2 a^3}-\frac{17 c^3 x^2 \tan ^{-1}(a x)^2}{70 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2-\frac{17 i c^3 \tan ^{-1}(a x)^3}{105 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3+\frac{4 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^3}-\frac{i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^3}-\frac{1}{7} \left (3 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac{1}{7} \left (3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{14} \left (9 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac{1}{14} \left (9 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{7} \left (9 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{\left (i c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac{\left (9 c^3\right ) \int \tan ^{-1}(a x) \, dx}{10 a^2}+\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{10 a^2}-\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{7 a^2}-\frac{\left (9 c^3\right ) \int \tan ^{-1}(a x) \, dx}{5 a^2}+\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (18 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac{c^3 \int x \tan ^{-1}(a x)^2 \, dx}{3 a}-\frac{c^3 \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{3 a}-\frac{c^3 \int \frac{x}{1+a^2 x^2} \, dx}{a}-\frac{1}{10} \left (3 a c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{12} \left (a^2 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\frac{1}{12} \left (a^2 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{9} \left (a^2 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\frac{1}{9} \left (a^2 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{6} \left (a^2 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{35} \left (3 a^3 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx-\frac{1}{84} \left (a^5 c^3\right ) \int \frac{x^7}{1+a^2 x^2} \, dx\\ &=-\frac{17 c^3 x \tan ^{-1}(a x)}{10 a^2}-\frac{2}{35} c^3 x^3 \tan ^{-1}(a x)+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{17 c^3 \tan ^{-1}(a x)^2}{20 a^3}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{17 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{35 a^3}-\frac{c^3 \log \left (1+a^2 x^2\right )}{2 a^3}+\frac{4 i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^3}-\frac{c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^3}+\frac{1}{12} c^3 \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{12} c^3 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{9} c^3 \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{9} c^3 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{6} c^3 \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{6} c^3 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} c^3 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{\left (9 i c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac{c^3 \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{3 a^2}+\frac{\left (3 c^3\right ) \int \tan ^{-1}(a x) \, dx}{7 a^2}-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (9 c^3\right ) \int \tan ^{-1}(a x) \, dx}{14 a^2}-\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{14 a^2}+\frac{\left (9 c^3\right ) \int \tan ^{-1}(a x) \, dx}{7 a^2}-\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (18 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (9 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx}{10 a}+\frac{\left (9 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{7} \left (a c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{20} \left (3 a c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{14} \left (3 a c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{1}{60} \left (a^3 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx+\frac{1}{45} \left (a^3 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx-\frac{1}{70} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{168} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{23 c^3 x \tan ^{-1}(a x)}{35 a^2}+\frac{239 c^3 x^3 \tan ^{-1}(a x)}{3780}+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac{23 c^3 \tan ^{-1}(a x)^2}{70 a^3}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{16 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{105 a^3}+\frac{17 c^3 \log \left (1+a^2 x^2\right )}{20 a^3}-\frac{17 i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{35 a^3}+\frac{2 c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{5 a^3}+\frac{\left (9 i c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}-\frac{c^3 \int \tan ^{-1}(a x) \, dx}{12 a^2}+\frac{c^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{12 a^2}-\frac{c^3 \int \tan ^{-1}(a x) \, dx}{9 a^2}+\frac{c^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{9 a^2}-\frac{c^3 \int \tan ^{-1}(a x) \, dx}{6 a^2}+\frac{c^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{6 a^2}-\frac{c^3 \int \tan ^{-1}(a x) \, dx}{3 a^2}+\frac{c^3 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^2}-\frac{\left (2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}-\frac{\left (3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx}{7 a}-\frac{\left (9 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx}{14 a}-\frac{\left (9 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx}{7 a}-\frac{1}{36} \left (a c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{27} \left (a c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{18} \left (a c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{1}{14} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{28} \left (3 a c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{20} \left (3 a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{1}{120} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{90} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{70} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{168} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^6}-\frac{x}{a^4}+\frac{x^2}{a^2}-\frac{1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{19 c^3 x^2}{168 a}-\frac{31 a c^3 x^4}{1680}-\frac{1}{504} a^3 c^3 x^6-\frac{47 c^3 x \tan ^{-1}(a x)}{1260 a^2}+\frac{239 c^3 x^3 \tan ^{-1}(a x)}{3780}+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{47 c^3 \tan ^{-1}(a x)^2}{2520 a^3}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{16 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{105 a^3}-\frac{181 c^3 \log \left (1+a^2 x^2\right )}{840 a^3}-\frac{16 i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{105 a^3}-\frac{17 c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{70 a^3}-\frac{\left (i c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac{c^3 \int \frac{x}{1+a^2 x^2} \, dx}{12 a}+\frac{c^3 \int \frac{x}{1+a^2 x^2} \, dx}{9 a}+\frac{c^3 \int \frac{x}{1+a^2 x^2} \, dx}{6 a}+\frac{c^3 \int \frac{x}{1+a^2 x^2} \, dx}{3 a}-\frac{1}{72} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{54} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{36} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{14} \left (a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{1}{28} \left (3 a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{1}{120} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{1}{90} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{29 c^3 x^2}{630 a}-\frac{11 a c^3 x^4}{1260}-\frac{1}{504} a^3 c^3 x^6-\frac{47 c^3 x \tan ^{-1}(a x)}{1260 a^2}+\frac{239 c^3 x^3 \tan ^{-1}(a x)}{3780}+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{47 c^3 \tan ^{-1}(a x)^2}{2520 a^3}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{16 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{105 a^3}-\frac{23 c^3 \log \left (1+a^2 x^2\right )}{840 a^3}-\frac{16 i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{105 a^3}-\frac{8 c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{105 a^3}-\frac{1}{72} \left (a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{54} \left (a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{36} \left (a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{107 c^3 x^2}{7560 a}-\frac{11 a c^3 x^4}{1260}-\frac{1}{504} a^3 c^3 x^6-\frac{47 c^3 x \tan ^{-1}(a x)}{1260 a^2}+\frac{239 c^3 x^3 \tan ^{-1}(a x)}{3780}+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{47 c^3 \tan ^{-1}(a x)^2}{2520 a^3}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{16 c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{105 a^3}+\frac{31 c^3 \log \left (1+a^2 x^2\right )}{945 a^3}-\frac{16 i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{105 a^3}-\frac{8 c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{105 a^3}\\ \end{align*}

Mathematica [A]  time = 1.86585, size = 281, normalized size = 0.72 \[ \frac{c^3 \left (1152 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-576 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-15 a^6 x^6-66 a^4 x^4-107 a^2 x^2+248 \log \left (a^2 x^2+1\right )+840 a^9 x^9 \tan ^{-1}(a x)^3-315 a^8 x^8 \tan ^{-1}(a x)^2+3240 a^7 x^7 \tan ^{-1}(a x)^3+90 a^7 x^7 \tan ^{-1}(a x)-1200 a^6 x^6 \tan ^{-1}(a x)^2+4536 a^5 x^5 \tan ^{-1}(a x)^3+354 a^5 x^5 \tan ^{-1}(a x)-1602 a^4 x^4 \tan ^{-1}(a x)^2+2520 a^3 x^3 \tan ^{-1}(a x)^3+478 a^3 x^3 \tan ^{-1}(a x)-576 a^2 x^2 \tan ^{-1}(a x)^2-282 a x \tan ^{-1}(a x)+384 i \tan ^{-1}(a x)^3+141 \tan ^{-1}(a x)^2-1152 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-56\right )}{7560 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-56 - 107*a^2*x^2 - 66*a^4*x^4 - 15*a^6*x^6 - 282*a*x*ArcTan[a*x] + 478*a^3*x^3*ArcTan[a*x] + 354*a^5*x^
5*ArcTan[a*x] + 90*a^7*x^7*ArcTan[a*x] + 141*ArcTan[a*x]^2 - 576*a^2*x^2*ArcTan[a*x]^2 - 1602*a^4*x^4*ArcTan[a
*x]^2 - 1200*a^6*x^6*ArcTan[a*x]^2 - 315*a^8*x^8*ArcTan[a*x]^2 + (384*I)*ArcTan[a*x]^3 + 2520*a^3*x^3*ArcTan[a
*x]^3 + 4536*a^5*x^5*ArcTan[a*x]^3 + 3240*a^7*x^7*ArcTan[a*x]^3 + 840*a^9*x^9*ArcTan[a*x]^3 - 1152*ArcTan[a*x]
^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 248*Log[1 + a^2*x^2] + (1152*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*
x])] - 576*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(7560*a^3)

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Maple [C]  time = 8.589, size = 1181, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

59/1260*a^2*c^3*x^5*arctan(a*x)+1/84*a^4*c^3*x^7*arctan(a*x)-107/7560*c^3*x^2/a-11/1260*a*c^3*x^4-1/504*a^3*c^
3*x^6+239/3780*c^3*x^3*arctan(a*x)+1/3*c^3*x^3*arctan(a*x)^3+47/2520*c^3*arctan(a*x)^2/a^3-62/945/a^3*c^3*ln((
1+I*a*x)^2/(a^2*x^2+1)+1)-8/105/a^3*c^3*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-16/105/a^3*c^3*arctan(a*x)^2*ln(2)
+8/105/a^3*c^3*arctan(a*x)^2*ln(a^2*x^2+1)-16/105/a^3*c^3*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+62/945
*I/a^3*c^3*arctan(a*x)+16/315*I/a^3*c^3*arctan(a*x)^3+4/105*I/a^3*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)*arctan(a*x)^2-4
7/1260*c^3*x*arctan(a*x)/a^2-8/105*c^3*x^2*arctan(a*x)^2/a-89/420*a*c^3*x^4*arctan(a*x)^2-10/63*a^3*c^3*x^6*ar
ctan(a*x)^2-1/24*a^5*c^3*x^8*arctan(a*x)^2+3/5*a^2*c^3*x^5*arctan(a*x)^3+3/7*a^4*c^3*x^7*arctan(a*x)^3+1/9*a^6
*c^3*x^9*arctan(a*x)^3+16/105*I/a^3*c^3*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-4/105*I/a^3*c^3*Pi*csg
n(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)^2-8/105*I/a^3*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)^2-4/105*I/a^3*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+4/105
*I/a^3*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)^2-4/105*I/a^3*
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+8/105*I/a^3*c^3*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a
*x)^2+4/105*I/a^3*c^3*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2+4/105*I/a^3*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)^2-4/105*I/a^3*c^3*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)
+1)^2)^3*arctan(a*x)^2-1/135/a^3*c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2520} \,{\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right )^{3} - \frac{1}{3360} \,{\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{2940 \,{\left (a^{8} c^{3} x^{10} + 4 \, a^{6} c^{3} x^{8} + 6 \, a^{4} c^{3} x^{6} + 4 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{3} - 4 \,{\left (35 \, a^{7} c^{3} x^{9} + 135 \, a^{5} c^{3} x^{7} + 189 \, a^{3} c^{3} x^{5} + 105 \, a c^{3} x^{3}\right )} \arctan \left (a x\right )^{2} + 4 \,{\left (35 \, a^{8} c^{3} x^{10} + 135 \, a^{6} c^{3} x^{8} + 189 \, a^{4} c^{3} x^{6} + 105 \, a^{2} c^{3} x^{4}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) +{\left (35 \, a^{7} c^{3} x^{9} + 135 \, a^{5} c^{3} x^{7} + 189 \, a^{3} c^{3} x^{5} + 105 \, a c^{3} x^{3} + 315 \,{\left (a^{8} c^{3} x^{10} + 4 \, a^{6} c^{3} x^{8} + 6 \, a^{4} c^{3} x^{6} + 4 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{3360 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(35*a^6*c^3*x^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*arctan(a*x)^3 - 1/3360*(35*a^6*c^3*x
^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/3360*(2940*
(a^8*c^3*x^10 + 4*a^6*c^3*x^8 + 6*a^4*c^3*x^6 + 4*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)^3 - 4*(35*a^7*c^3*x^9 + 1
35*a^5*c^3*x^7 + 189*a^3*c^3*x^5 + 105*a*c^3*x^3)*arctan(a*x)^2 + 4*(35*a^8*c^3*x^10 + 135*a^6*c^3*x^8 + 189*a
^4*c^3*x^6 + 105*a^2*c^3*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (35*a^7*c^3*x^9 + 135*a^5*c^3*x^7 + 189*a^3*c^3*x
^5 + 105*a*c^3*x^3 + 315*(a^8*c^3*x^10 + 4*a^6*c^3*x^8 + 6*a^4*c^3*x^6 + 4*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x))
*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int x^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{4} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{6} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{8} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x**2*atan(a*x)**3, x) + Integral(3*a**2*x**4*atan(a*x)**3, x) + Integral(3*a**4*x**6*atan(a*x)*
*3, x) + Integral(a**6*x**8*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} x^{2} \arctan \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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