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

Optimal. Leaf size=370 \[ -2 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{4} i c^2 \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{4} i c^2 \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} i c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} c^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3-\frac{1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c^2 x^2 \tan ^{-1}(a x)-\frac{1}{4} a c^2 x-\frac{9}{4} a c^2 x \tan ^{-1}(a x)^2+\frac{3}{4} c^2 \tan ^{-1}(a x)^3-2 i c^2 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 \tan ^{-1}(a x)-4 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

[Out]

-(a*c^2*x)/4 + (c^2*ArcTan[a*x])/4 + (a^2*c^2*x^2*ArcTan[a*x])/4 - (2*I)*c^2*ArcTan[a*x]^2 - (9*a*c^2*x*ArcTan
[a*x]^2)/4 - (a^3*c^2*x^3*ArcTan[a*x]^2)/4 + (3*c^2*ArcTan[a*x]^3)/4 + a^2*c^2*x^2*ArcTan[a*x]^3 + (a^4*c^2*x^
4*ArcTan[a*x]^3)/4 + 2*c^2*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] - (
2*I)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c^2*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)
*c^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c^2*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c
^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c^2*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c^2
*PolyLog[4, -1 + 2/(1 + I*a*x)]

________________________________________________________________________________________

Rubi [A]  time = 0.970399, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 16, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {4948, 4850, 4988, 4884, 4994, 4998, 6610, 4852, 4916, 4846, 4920, 4854, 2402, 2315, 321, 203} \[ -2 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{4} i c^2 \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{4} i c^2 \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} i c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} c^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3-\frac{1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c^2 x^2 \tan ^{-1}(a x)-\frac{1}{4} a c^2 x-\frac{9}{4} a c^2 x \tan ^{-1}(a x)^2+\frac{3}{4} c^2 \tan ^{-1}(a x)^3-2 i c^2 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 \tan ^{-1}(a x)-4 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*c^2*x)/4 + (c^2*ArcTan[a*x])/4 + (a^2*c^2*x^2*ArcTan[a*x])/4 - (2*I)*c^2*ArcTan[a*x]^2 - (9*a*c^2*x*ArcTan
[a*x]^2)/4 - (a^3*c^2*x^3*ArcTan[a*x]^2)/4 + (3*c^2*ArcTan[a*x]^3)/4 + a^2*c^2*x^2*ArcTan[a*x]^3 + (a^4*c^2*x^
4*ArcTan[a*x]^3)/4 + 2*c^2*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] - (
2*I)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c^2*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)
*c^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c^2*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c
^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c^2*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c^2
*PolyLog[4, -1 + 2/(1 + I*a*x)]

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 4850

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

Rule 4988

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[(
Log[1 + u]*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[1 - u]*(a + b*ArcTan[c*x])^p)/(d +
e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I - c*x))^
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 4998

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

Rubi steps

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

Mathematica [A]  time = 0.558051, size = 302, normalized size = 0.82 \[ \frac{1}{64} c^2 \left (96 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+32 i \left (3 \tan ^{-1}(a x)^2+4\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+96 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-96 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (4,e^{-2 i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (4,-e^{2 i \tan ^{-1}(a x)}\right )+16 a^4 x^4 \tan ^{-1}(a x)^3-16 a^3 x^3 \tan ^{-1}(a x)^2+64 a^2 x^2 \tan ^{-1}(a x)^3+16 a^2 x^2 \tan ^{-1}(a x)-16 a x-144 a x \tan ^{-1}(a x)^2+32 i \tan ^{-1}(a x)^4+48 \tan ^{-1}(a x)^3+128 i \tan ^{-1}(a x)^2+16 \tan ^{-1}(a x)+64 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-64 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-256 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-i \pi ^4\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*((-I)*Pi^4 - 16*a*x + 16*ArcTan[a*x] + 16*a^2*x^2*ArcTan[a*x] + (128*I)*ArcTan[a*x]^2 - 144*a*x*ArcTan[a*
x]^2 - 16*a^3*x^3*ArcTan[a*x]^2 + 48*ArcTan[a*x]^3 + 64*a^2*x^2*ArcTan[a*x]^3 + 16*a^4*x^4*ArcTan[a*x]^3 + (32
*I)*ArcTan[a*x]^4 + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - 256*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan
[a*x])] - 64*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a
*x])] + (32*I)*(4 + 3*ArcTan[a*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*PolyLog[3, E^((-2*I)*
ArcTan[a*x])] - 96*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])]
- (48*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/64

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Maple [A]  time = 1.497, size = 566, normalized size = 1.5 \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)^2*arctan(a*x)^3/x,x)

[Out]

1/4*c^2*(-3*I*arctan(a*x)^3+3*arctan(a*x)^3*a*x-I*arctan(a*x)^3*a^2*x^2+arctan(a*x)^3*a^3*x^3-8*arctan(a*x)^2+
I*arctan(a*x)^2*a*x-arctan(a*x)^2*x^2*a^2-I*arctan(a*x)+arctan(a*x)*x*a-1)*(a*x+I)-3*I*c^2*arctan(a*x)^2*polyl
og(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3/2*c^2*arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+6*I*c^2*polylog(4,(1
+I*a*x)/(a^2*x^2+1)^(1/2))+c^2*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*c^2*arctan(a*x)^2*polylog(2
,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*c^2*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+c^2*arctan(a*x)^3*ln(1
+(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*c^2*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*c^2*arctan(a*x)*polylog(3,-(1+
I*a*x)/(a^2*x^2+1)^(1/2))-c^2*arctan(a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+2*I*c^2*polylog(2,-(1+I*a*x)^2/(a^2*
x^2+1))-4*c^2*arctan(a*x)*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+3/2*I*c^2*arctan(a*x)^2*polylog(2,-(1+I*a*x)^2/(a^2*x^
2+1))+4*I*c^2*arctan(a*x)^2-3/4*I*c^2*polylog(4,-(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*} \frac{1}{32} \,{\left (a^{4} c^{2} x^{4} + 4 \, a^{2} c^{2} x^{2}\right )} \arctan \left (a x\right )^{3} - \frac{3}{128} \,{\left (a^{4} c^{2} x^{4} + 4 \, a^{2} c^{2} x^{2}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{112 \,{\left (a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{3} - 12 \,{\left (a^{5} c^{2} x^{5} + 4 \, a^{3} c^{2} x^{3}\right )} \arctan \left (a x\right )^{2} + 12 \,{\left (a^{6} c^{2} x^{6} + 4 \, a^{4} c^{2} x^{4}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) + 3 \,{\left (a^{5} c^{2} x^{5} + 4 \, a^{3} c^{2} x^{3} + 4 \,{\left (a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{128 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(a^4*c^2*x^4 + 4*a^2*c^2*x^2)*arctan(a*x)^3 - 3/128*(a^4*c^2*x^4 + 4*a^2*c^2*x^2)*arctan(a*x)*log(a^2*x^2
+ 1)^2 + integrate(1/128*(112*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*arctan(a*x)^3 - 12*(a^5*c^2
*x^5 + 4*a^3*c^2*x^3)*arctan(a*x)^2 + 12*(a^6*c^2*x^6 + 4*a^4*c^2*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + 3*(a^5*c
^2*x^5 + 4*a^3*c^2*x^3 + 4*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*arctan(a*x))*log(a^2*x^2 + 1)^2
)/(a^2*x^3 + x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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