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

Optimal. Leaf size=287 \[ -\frac{1}{2} c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^3 \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-i c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )+\frac{1}{60} a^4 c^3 x^4+\frac{29}{180} a^2 c^3 x^2+\frac{34}{45} c^3 \log \left (a^2 x^2+1\right )+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2-\frac{7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2-\frac{11}{6} a c^3 x \tan ^{-1}(a x)+\frac{11}{12} c^3 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

[Out]

(29*a^2*c^3*x^2)/180 + (a^4*c^3*x^4)/60 - (11*a*c^3*x*ArcTan[a*x])/6 - (7*a^3*c^3*x^3*ArcTan[a*x])/18 - (a^5*c
^3*x^5*ArcTan[a*x])/15 + (11*c^3*ArcTan[a*x]^2)/12 + (3*a^2*c^3*x^2*ArcTan[a*x]^2)/2 + (3*a^4*c^3*x^4*ArcTan[a
*x]^2)/4 + (a^6*c^3*x^6*ArcTan[a*x]^2)/6 + 2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (34*c^3*Log[1 + a^
2*x^2])/45 - I*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)
] - (c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c^3*PolyLog[3, -1 + 2/(1 + I*a*x)])/2

________________________________________________________________________________________

Rubi [A]  time = 0.743414, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4948, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260, 266, 43} \[ -\frac{1}{2} c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^3 \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-i c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )+\frac{1}{60} a^4 c^3 x^4+\frac{29}{180} a^2 c^3 x^2+\frac{34}{45} c^3 \log \left (a^2 x^2+1\right )+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2-\frac{7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2-\frac{11}{6} a c^3 x \tan ^{-1}(a x)+\frac{11}{12} c^3 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(29*a^2*c^3*x^2)/180 + (a^4*c^3*x^4)/60 - (11*a*c^3*x*ArcTan[a*x])/6 - (7*a^3*c^3*x^3*ArcTan[a*x])/18 - (a^5*c
^3*x^5*ArcTan[a*x])/15 + (11*c^3*ArcTan[a*x]^2)/12 + (3*a^2*c^3*x^2*ArcTan[a*x]^2)/2 + (3*a^4*c^3*x^4*ArcTan[a
*x]^2)/4 + (a^6*c^3*x^6*ArcTan[a*x]^2)/6 + 2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (34*c^3*Log[1 + a^
2*x^2])/45 - I*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)
] - (c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c^3*PolyLog[3, -1 + 2/(1 + I*a*x)])/2

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 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 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 266

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

Rule 43

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

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2}{x} \, dx &=\int \left (\frac{c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+3 a^4 c^3 x^3 \tan ^{-1}(a x)^2+a^6 c^3 x^5 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)^2}{x} \, dx+\left (3 a^2 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx\\ &=\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\left (4 a c^3\right ) \int \frac{\tan ^{-1}(a x) \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^5 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^7 c^3\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\left (2 a c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 a c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a c^3\right ) \int \tan ^{-1}(a x) \, dx+\left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^3 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac{1}{2} \left (3 a^3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^5 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\frac{1}{3} \left (a^5 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-3 a c^3 x \tan ^{-1}(a x)-\frac{1}{2} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{2} c^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )+\left (i a c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (i a c^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{2} \left (3 a c^3\right ) \int \tan ^{-1}(a x) \, dx-\frac{1}{2} \left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (3 a^2 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx+\frac{1}{3} \left (a^3 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{3} \left (a^3 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{2} \left (a^4 c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{1}{15} \left (a^6 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} a c^3 x \tan ^{-1}(a x)-\frac{7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{4} c^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{2} c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^3 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{3} \left (a c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac{1}{3} \left (a c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^2 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{9} \left (a^4 c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{1}{4} \left (a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{30} \left (a^6 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{11}{6} a c^3 x \tan ^{-1}(a x)-\frac{7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{11}{12} c^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{3}{4} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{2} c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^3 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{1}{3} \left (a^2 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{18} \left (a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{4} \left (a^4 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}{30} \left (a^6 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{13}{60} a^2 c^3 x^2+\frac{1}{60} a^4 c^3 x^4-\frac{11}{6} a c^3 x \tan ^{-1}(a x)-\frac{7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{11}{12} c^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{7}{10} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{2} c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^3 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{18} \left (a^4 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{29}{180} a^2 c^3 x^2+\frac{1}{60} a^4 c^3 x^4-\frac{11}{6} a c^3 x \tan ^{-1}(a x)-\frac{7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{11}{12} c^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{34}{45} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{1}{2} c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} c^3 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.528906, size = 252, normalized size = 0.88 \[ \frac{1}{360} c^3 \left (360 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+360 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+180 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-180 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )+6 a^4 x^4+58 a^2 x^2+272 \log \left (a^2 x^2+1\right )+60 a^6 x^6 \tan ^{-1}(a x)^2-24 a^5 x^5 \tan ^{-1}(a x)+270 a^4 x^4 \tan ^{-1}(a x)^2-140 a^3 x^3 \tan ^{-1}(a x)+540 a^2 x^2 \tan ^{-1}(a x)^2-660 a x \tan ^{-1}(a x)+240 i \tan ^{-1}(a x)^3+330 \tan ^{-1}(a x)^2+360 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-360 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-15 i \pi ^3+52\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(52 - (15*I)*Pi^3 + 58*a^2*x^2 + 6*a^4*x^4 - 660*a*x*ArcTan[a*x] - 140*a^3*x^3*ArcTan[a*x] - 24*a^5*x^5*A
rcTan[a*x] + 330*ArcTan[a*x]^2 + 540*a^2*x^2*ArcTan[a*x]^2 + 270*a^4*x^4*ArcTan[a*x]^2 + 60*a^6*x^6*ArcTan[a*x
]^2 + (240*I)*ArcTan[a*x]^3 + 360*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 360*ArcTan[a*x]^2*Log[1 + E^
((2*I)*ArcTan[a*x])] + 272*Log[1 + a^2*x^2] + (360*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (360*I)
*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 180*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 180*PolyLog[3, -E^(
(2*I)*ArcTan[a*x])]))/360

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Maple [C]  time = 3.927, size = 1217, normalized size = 4.2 \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)^2/x,x)

[Out]

13/90*c^3-11/6*a*c^3*x*arctan(a*x)-7/18*a^3*c^3*x^3*arctan(a*x)-1/15*a^5*c^3*x^5*arctan(a*x)+3/2*a^2*c^3*x^2*a
rctan(a*x)^2+3/4*a^4*c^3*x^4*arctan(a*x)^2+1/6*a^6*c^3*x^6*arctan(a*x)^2+29/180*a^2*c^3*x^2+1/60*a^4*c^3*x^4+1
1/12*c^3*arctan(a*x)^2+2*c^3*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/2*c^3*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1)
)-68/45*c^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+2*c^3*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/2*I*c^3*Pi*csgn(((1+
I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2+1/2*I*c^3*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1
)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*arctan(a*x)^2+I*c^3*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+1/2*I*
c^3*Pi*arctan(a*x)^2-2*I*c^3*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*c^3*arctan(a*x)*polylog(2,
-(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/2*I*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))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2+1/2*I*c^3*Pi*csgn(I*((1+I*a
*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*arctan(a*x)^2+c^3*arctan(a*x)^2*ln(a*x)+c^3*arctan(a*x)^2*
ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+c^3*arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+68/45*I*c^3*arctan(a*x)-
1/2*I*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)/((1+I*a*x)^2/(a^2*x^2+1)+1
))^2*arctan(a*x)^2-1/2*I*c^3*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x
)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2+1/2*I*c^3*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1
)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2-1/2
*I*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)/((1+I*a*x)^2/(a^2*x^2+1)+1))^
2*arctan(a*x)^2-c^3*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-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)^2/x,x, algorithm="maxima")

[Out]

36*a^8*c^3*integrate(1/48*x^8*arctan(a*x)^2/(a^2*x^3 + x), x) + 3*a^8*c^3*integrate(1/48*x^8*log(a^2*x^2 + 1)^
2/(a^2*x^3 + x), x) + 2*a^8*c^3*integrate(1/48*x^8*log(a^2*x^2 + 1)/(a^2*x^3 + x), x) - 4*a^7*c^3*integrate(1/
48*x^7*arctan(a*x)/(a^2*x^3 + x), x) + 144*a^6*c^3*integrate(1/48*x^6*arctan(a*x)^2/(a^2*x^3 + x), x) + 12*a^6
*c^3*integrate(1/48*x^6*log(a^2*x^2 + 1)^2/(a^2*x^3 + x), x) + 9*a^6*c^3*integrate(1/48*x^6*log(a^2*x^2 + 1)/(
a^2*x^3 + x), x) - 18*a^5*c^3*integrate(1/48*x^5*arctan(a*x)/(a^2*x^3 + x), x) + 216*a^4*c^3*integrate(1/48*x^
4*arctan(a*x)^2/(a^2*x^3 + x), x) + 18*a^4*c^3*integrate(1/48*x^4*log(a^2*x^2 + 1)^2/(a^2*x^3 + x), x) + 18*a^
4*c^3*integrate(1/48*x^4*log(a^2*x^2 + 1)/(a^2*x^3 + x), x) - 36*a^3*c^3*integrate(1/48*x^3*arctan(a*x)/(a^2*x
^3 + x), x) + 144*a^2*c^3*integrate(1/48*x^2*arctan(a*x)^2/(a^2*x^3 + x), x) + 1/24*c^3*log(a^2*x^2 + 1)^3 + 3
6*c^3*integrate(1/48*arctan(a*x)^2/(a^2*x^3 + x), x) + 3*c^3*integrate(1/48*log(a^2*x^2 + 1)^2/(a^2*x^3 + x),
x) + 1/48*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^4 + 18*a^2*c^3*x^2)*arctan(a*x)^2 - 1/192*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^
4 + 18*a^2*c^3*x^2)*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 (\frac{{\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 )^{2}}{x}, 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)^2/x,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)^2/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname{atan}^{2}{\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)**2/x,x)

[Out]

c**3*(Integral(atan(a*x)**2/x, x) + Integral(3*a**2*x*atan(a*x)**2, x) + Integral(3*a**4*x**3*atan(a*x)**2, x)
+ Integral(a**6*x**5*atan(a*x)**2, 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 )}^{3} \arctan \left (a x\right )^{2}}{x}\,{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)^2/x,x, algorithm="giac")

[Out]

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