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

Optimal. Leaf size=381 \[ \frac{26 i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{525 a^4}-\frac{1}{840} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}-\frac{107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac{389 c^3 x}{12600 a^3}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}-\frac{389 c^3 \tan ^{-1}(a x)}{12600 a^4}+\frac{52 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{525 a^4}-\frac{1}{252} a c^3 x^5-\frac{17 c^3 x^3}{9450 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a} \]

[Out]

(389*c^3*x)/(12600*a^3) - (17*c^3*x^3)/(9450*a) - (a*c^3*x^5)/252 - (a^3*c^3*x^7)/840 - (389*c^3*ArcTan[a*x])/
(12600*a^4) - (107*c^3*x^2*ArcTan[a*x])/(4200*a^2) + (53*c^3*x^4*ArcTan[a*x])/2100 + (71*a^2*c^3*x^6*ArcTan[a*
x])/2520 + (a^4*c^3*x^8*ArcTan[a*x])/120 + (((26*I)/525)*c^3*ArcTan[a*x]^2)/a^4 + (3*c^3*x*ArcTan[a*x]^2)/(40*
a^3) - (c^3*x^3*ArcTan[a*x]^2)/(40*a) - (27*a*c^3*x^5*ArcTan[a*x]^2)/200 - (33*a^3*c^3*x^7*ArcTan[a*x]^2)/280
- (a^5*c^3*x^9*ArcTan[a*x]^2)/30 - (c^3*ArcTan[a*x]^3)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^3)/4 + (a^2*c^3*x^6*Arc
Tan[a*x]^3)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^3)/8 + (a^6*c^3*x^10*ArcTan[a*x]^3)/10 + (52*c^3*ArcTan[a*x]*Log[2/
(1 + I*a*x)])/(525*a^4) + (((26*I)/525)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4

________________________________________________________________________________________

Rubi [A]  time = 3.72532, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 184, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884, 302} \[ \frac{26 i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{525 a^4}-\frac{1}{840} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}-\frac{107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac{389 c^3 x}{12600 a^3}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}-\frac{389 c^3 \tan ^{-1}(a x)}{12600 a^4}+\frac{52 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{525 a^4}-\frac{1}{252} a c^3 x^5-\frac{17 c^3 x^3}{9450 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(389*c^3*x)/(12600*a^3) - (17*c^3*x^3)/(9450*a) - (a*c^3*x^5)/252 - (a^3*c^3*x^7)/840 - (389*c^3*ArcTan[a*x])/
(12600*a^4) - (107*c^3*x^2*ArcTan[a*x])/(4200*a^2) + (53*c^3*x^4*ArcTan[a*x])/2100 + (71*a^2*c^3*x^6*ArcTan[a*
x])/2520 + (a^4*c^3*x^8*ArcTan[a*x])/120 + (((26*I)/525)*c^3*ArcTan[a*x]^2)/a^4 + (3*c^3*x*ArcTan[a*x]^2)/(40*
a^3) - (c^3*x^3*ArcTan[a*x]^2)/(40*a) - (27*a*c^3*x^5*ArcTan[a*x]^2)/200 - (33*a^3*c^3*x^7*ArcTan[a*x]^2)/280
- (a^5*c^3*x^9*ArcTan[a*x]^2)/30 - (c^3*ArcTan[a*x]^3)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^3)/4 + (a^2*c^3*x^6*Arc
Tan[a*x]^3)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^3)/8 + (a^6*c^3*x^10*ArcTan[a*x]^3)/10 + (52*c^3*ArcTan[a*x]*Log[2/
(1 + I*a*x)])/(525*a^4) + (((26*I)/525)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

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 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3 \, dx &=\int \left (c^3 x^3 \tan ^{-1}(a x)^3+3 a^2 c^3 x^5 \tan ^{-1}(a x)^3+3 a^4 c^3 x^7 \tan ^{-1}(a x)^3+a^6 c^3 x^9 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^3 \int x^3 \tan ^{-1}(a x)^3 \, dx+\left (3 a^2 c^3\right ) \int x^5 \tan ^{-1}(a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^7 \tan ^{-1}(a x)^3 \, dx+\left (a^6 c^3\right ) \int x^9 \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac{1}{4} \left (3 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{8} \left (9 a^5 c^3\right ) \int \frac{x^8 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{10} \left (3 a^7 c^3\right ) \int \frac{x^{10} \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac{\left (3 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{4 a}+\frac{\left (3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a}-\frac{1}{2} \left (3 a c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\frac{1}{2} \left (3 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{8} \left (9 a^3 c^3\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx+\frac{1}{8} \left (9 a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{10} \left (3 a^5 c^3\right ) \int x^8 \tan ^{-1}(a x)^2 \, dx+\frac{1}{10} \left (3 a^5 c^3\right ) \int \frac{x^8 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{4 a}-\frac{3}{10} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{9}{56} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac{1}{2} c^3 \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{\left (3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{4 a^3}-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a^3}+\frac{\left (3 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{2 a}-\frac{\left (3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a}+\frac{1}{8} \left (9 a c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx-\frac{1}{8} \left (9 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac{1}{5} \left (3 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^3 c^3\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx-\frac{1}{10} \left (3 a^3 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac{1}{28} \left (9 a^4 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{15} \left (a^6 c^3\right ) \int \frac{x^9 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{3 c^3 x \tan ^{-1}(a x)^2}{4 a^3}+\frac{c^3 x^3 \tan ^{-1}(a x)^2}{4 a}-\frac{3}{40} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac{1}{5} \left (3 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{5} \left (3 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-c^3 \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{\left (3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{2 a^3}+\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a^3}+\frac{c^3 \int x \tan ^{-1}(a x) \, dx}{2 a^2}-\frac{c^3 \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac{\left (3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac{\left (9 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{8 a}+\frac{\left (9 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{8 a}-\frac{1}{10} \left (3 a c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\frac{1}{10} \left (3 a c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac{1}{28} \left (9 a^2 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx-\frac{1}{28} \left (9 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{20} \left (9 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{15} \left (a^4 c^3\right ) \int x^7 \tan ^{-1}(a x) \, dx-\frac{1}{15} \left (a^4 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{35} \left (3 a^4 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{c^3 x^2 \tan ^{-1}(a x)}{4 a^2}+\frac{3}{20} c^3 x^4 \tan ^{-1}(a x)+\frac{3}{56} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{i c^3 \tan ^{-1}(a x)^2}{a^4}-\frac{3 c^3 x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{8 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac{c^3 \tan ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac{1}{28} \left (9 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{28} \left (9 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{20} \left (9 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{20} \left (9 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{4} \left (3 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{c^3 \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac{\left (9 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{8 a^3}-\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{8 a^3}+\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}-\frac{\left (3 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{5 a^2}+\frac{\left (3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{c^3 \int x \tan ^{-1}(a x) \, dx}{a^2}+\frac{c^3 \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}+\frac{\left (3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}-\frac{c^3 \int \frac{x^2}{1+a^2 x^2} \, dx}{4 a}+\frac{\left (3 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{10 a}-\frac{\left (3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{10 a}-\frac{1}{20} \left (3 a c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{15} \left (a^2 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx+\frac{1}{15} \left (a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{35} \left (3 a^2 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx+\frac{1}{35} \left (3 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{25} \left (3 a^2 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{56} \left (3 a^3 c^3\right ) \int \frac{x^6}{1+a^2 x^2} \, dx-\frac{1}{120} \left (a^5 c^3\right ) \int \frac{x^8}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 x}{4 a^3}-\frac{11 c^3 x^2 \tan ^{-1}(a x)}{20 a^2}-\frac{3}{70} c^3 x^4 \tan ^{-1}(a x)+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac{13 i c^3 \tan ^{-1}(a x)^2}{10 a^4}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{8 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac{2 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4}+\frac{1}{15} c^3 \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{15} c^3 \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{35} \left (3 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{35} \left (3 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{25} \left (3 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{25} \left (3 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} c^3 \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{4 a^3}-\frac{\left (3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{10 a^3}+\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{10 a^3}-\frac{c^3 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}-\frac{c^3 \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{a^3}-\frac{\left (3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{a^3}+\frac{\left (9 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{28 a^2}-\frac{\left (9 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{28 a^2}+\frac{\left (9 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{20 a^2}-\frac{\left (9 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{20 a^2}+\frac{\left (3 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{4 a^2}-\frac{\left (3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^2}-\frac{\left (9 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^2}+\frac{\left (3 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx}{10 a}+\frac{c^3 \int \frac{x^2}{1+a^2 x^2} \, dx}{2 a}+\frac{1}{112} \left (9 a c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{80} \left (9 a c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{20} \left (3 a c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac{1}{90} \left (a^3 c^3\right ) \int \frac{x^6}{1+a^2 x^2} \, dx+\frac{1}{70} \left (a^3 c^3\right ) \int \frac{x^6}{1+a^2 x^2} \, dx-\frac{1}{56} \left (3 a^3 c^3\right ) \int \left (\frac{1}{a^6}-\frac{x^2}{a^4}+\frac{x^4}{a^2}-\frac{1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx-\frac{1}{120} \left (a^5 c^3\right ) \int \left (-\frac{1}{a^8}+\frac{x^2}{a^6}-\frac{x^4}{a^4}+\frac{x^6}{a^2}+\frac{1}{a^8 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{55 c^3 x}{84 a^3}-\frac{11 c^3 x^3}{315 a}-\frac{19 a c^3 x^5}{2100}-\frac{1}{840} a^3 c^3 x^7+\frac{c^3 \tan ^{-1}(a x)}{4 a^4}+\frac{59 c^3 x^2 \tan ^{-1}(a x)}{280 a^2}+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{41 i c^3 \tan ^{-1}(a x)^2}{70 a^4}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac{13 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{5 a^4}+\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{2 a^4}+\frac{\left (3 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{2 a^4}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{120 a^3}+\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{56 a^3}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{20 a^3}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{10 a^3}+\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{28 a^3}+\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{20 a^3}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{2 a^3}+\frac{\left (3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{4 a^3}+\frac{c^3 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3}+\frac{\left (9 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{4 a^3}+\frac{\left (3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3}-\frac{c^3 \int x \tan ^{-1}(a x) \, dx}{15 a^2}+\frac{c^3 \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{15 a^2}-\frac{\left (3 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{35 a^2}+\frac{\left (3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{35 a^2}-\frac{\left (3 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{25 a^2}+\frac{\left (3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{25 a^2}-\frac{c^3 \int x \tan ^{-1}(a x) \, dx}{5 a^2}+\frac{c^3 \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}+\frac{\left (3 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (9 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx}{56 a}-\frac{\left (9 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx}{40 a}-\frac{\left (3 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx}{8 a}-\frac{1}{60} \left (a c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{140} \left (3 a c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{100} \left (3 a c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{112} \left (9 a c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac{1}{80} \left (9 a c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac{1}{90} \left (a^3 c^3\right ) \int \left (\frac{1}{a^6}-\frac{x^2}{a^4}+\frac{x^4}{a^2}-\frac{1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx+\frac{1}{70} \left (a^3 c^3\right ) \int \left (\frac{1}{a^6}-\frac{x^2}{a^4}+\frac{x^4}{a^2}-\frac{1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac{689 c^3 x}{2520 a^3}+\frac{79 c^3 x^3}{3780 a}-\frac{1}{252} a c^3 x^5-\frac{1}{840} a^3 c^3 x^7-\frac{55 c^3 \tan ^{-1}(a x)}{84 a^4}-\frac{107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac{41 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{35 a^4}+\frac{i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^4}-\frac{\left (3 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^4}-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^4}-\frac{\left (3 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^4}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{90 a^3}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{70 a^3}-\frac{c^3 \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{15 a^3}+\frac{\left (9 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{112 a^3}-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{35 a^3}+\frac{\left (9 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{80 a^3}-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{25 a^3}+\frac{\left (9 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{56 a^3}-\frac{c^3 \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}+\frac{\left (9 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{40 a^3}-\frac{\left (9 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{28 a^3}+\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{8 a^3}-\frac{\left (9 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{20 a^3}-\frac{\left (3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}-\frac{\left (3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}-\frac{\left (9 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}+\frac{c^3 \int \frac{x^2}{1+a^2 x^2} \, dx}{30 a}+\frac{\left (3 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx}{70 a}+\frac{\left (3 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx}{50 a}+\frac{c^3 \int \frac{x^2}{1+a^2 x^2} \, dx}{10 a}-\frac{1}{60} \left (a c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac{1}{140} \left (3 a c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac{1}{100} \left (3 a c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{389 c^3 x}{12600 a^3}-\frac{17 c^3 x^3}{9450 a}-\frac{1}{252} a c^3 x^5-\frac{1}{840} a^3 c^3 x^7+\frac{689 c^3 \tan ^{-1}(a x)}{2520 a^4}-\frac{107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac{52 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{525 a^4}-\frac{13 i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{10 a^4}+\frac{\left (9 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{28 a^4}+\frac{\left (9 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{20 a^4}+\frac{\left (3 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{4 a^4}+\frac{\left (9 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{4 a^4}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{60 a^3}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{140 a^3}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{100 a^3}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{30 a^3}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{70 a^3}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{50 a^3}+\frac{c^3 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{15 a^3}+\frac{\left (3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{35 a^3}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{10 a^3}+\frac{\left (3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{25 a^3}+\frac{c^3 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac{\left (3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}\\ &=\frac{389 c^3 x}{12600 a^3}-\frac{17 c^3 x^3}{9450 a}-\frac{1}{252} a c^3 x^5-\frac{1}{840} a^3 c^3 x^7-\frac{389 c^3 \tan ^{-1}(a x)}{12600 a^4}-\frac{107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac{52 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{525 a^4}+\frac{41 i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{70 a^4}-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{15 a^4}-\frac{\left (3 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{35 a^4}-\frac{\left (3 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{25 a^4}-\frac{\left (i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^4}-\frac{\left (3 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^4}\\ &=\frac{389 c^3 x}{12600 a^3}-\frac{17 c^3 x^3}{9450 a}-\frac{1}{252} a c^3 x^5-\frac{1}{840} a^3 c^3 x^7-\frac{389 c^3 \tan ^{-1}(a x)}{12600 a^4}-\frac{107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac{52 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{525 a^4}+\frac{26 i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{525 a^4}\\ \end{align*}

Mathematica [A]  time = 1.9998, size = 191, normalized size = 0.5 \[ \frac{c^3 \left (-1872 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-a x \left (45 a^6 x^6+150 a^4 x^4+68 a^2 x^2-1167\right )+945 \left (4 a^2 x^2-1\right ) \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^3-9 \left (140 a^9 x^9+495 a^7 x^7+567 a^5 x^5+105 a^3 x^3-315 a x+208 i\right ) \tan ^{-1}(a x)^2+3 \tan ^{-1}(a x) \left (105 a^8 x^8+355 a^6 x^6+318 a^4 x^4-321 a^2 x^2+1248 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-389\right )\right )}{37800 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-(a*x*(-1167 + 68*a^2*x^2 + 150*a^4*x^4 + 45*a^6*x^6)) - 9*(208*I - 315*a*x + 105*a^3*x^3 + 567*a^5*x^5
+ 495*a^7*x^7 + 140*a^9*x^9)*ArcTan[a*x]^2 + 945*(1 + a^2*x^2)^4*(-1 + 4*a^2*x^2)*ArcTan[a*x]^3 + 3*ArcTan[a*x
]*(-389 - 321*a^2*x^2 + 318*a^4*x^4 + 355*a^6*x^6 + 105*a^8*x^8 + 1248*Log[1 + E^((2*I)*ArcTan[a*x])]) - (1872
*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(37800*a^4)

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Maple [A]  time = 0.099, size = 471, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

71/2520*a^2*c^3*x^6*arctan(a*x)+1/120*a^4*c^3*x^8*arctan(a*x)-13/525*I/a^4*c^3*ln(a*x+I)*ln(1/2*I*(a*x-I))-13/
525*I/a^4*c^3*ln(a*x-I)*ln(a^2*x^2+1)+13/525*I/a^4*c^3*ln(a*x-I)*ln(-1/2*I*(a*x+I))+13/525*I/a^4*c^3*ln(a*x+I)
*ln(a^2*x^2+1)+13/1050*I/a^4*c^3*ln(a*x-I)^2-13/525*I/a^4*c^3*dilog(1/2*I*(a*x-I))+13/525*I/a^4*c^3*dilog(-1/2
*I*(a*x+I))-26/525/a^4*c^3*arctan(a*x)*ln(a^2*x^2+1)-13/1050*I/a^4*c^3*ln(a*x+I)^2+389/12600*c^3*x/a^3-17/9450
*c^3*x^3/a-1/252*a*c^3*x^5-1/840*a^3*c^3*x^7-389/12600*c^3*arctan(a*x)/a^4+53/2100*c^3*x^4*arctan(a*x)-1/40*c^
3*arctan(a*x)^3/a^4+1/4*c^3*x^4*arctan(a*x)^3-107/4200*c^3*x^2*arctan(a*x)/a^2+3/40*c^3*x*arctan(a*x)^2/a^3-1/
40*c^3*x^3*arctan(a*x)^2/a-27/200*a*c^3*x^5*arctan(a*x)^2-33/280*a^3*c^3*x^7*arctan(a*x)^2-1/30*a^5*c^3*x^9*ar
ctan(a*x)^2+1/2*a^2*c^3*x^6*arctan(a*x)^3+3/8*a^4*c^3*x^8*arctan(a*x)^3+1/10*a^6*c^3*x^10*arctan(a*x)^3

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^9 + 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 + c^3*x^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 x^{3} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{5} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{7} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{9} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x**3*atan(a*x)**3, x) + Integral(3*a**2*x**5*atan(a*x)**3, x) + Integral(3*a**4*x**7*atan(a*x)*
*3, x) + Integral(a**6*x**9*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^{3} \arctan \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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