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

Optimal. Leaf size=299 \[ -\frac{3}{2} a^2 c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c^3 \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-3 i a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+3 i a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )+\frac{1}{12} a^4 c^3 x^2+\frac{2}{3} a^2 c^3 \log \left (a^2 x^2+1\right )+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2-\frac{1}{6} a^5 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+a^2 c^3 \log (x)-\frac{5}{2} a^3 c^3 x \tan ^{-1}(a x)+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}-\frac{a c^3 \tan ^{-1}(a x)}{x} \]

[Out]

(a^4*c^3*x^2)/12 - (a*c^3*ArcTan[a*x])/x - (5*a^3*c^3*x*ArcTan[a*x])/2 - (a^5*c^3*x^3*ArcTan[a*x])/6 + (3*a^2*
c^3*ArcTan[a*x]^2)/4 - (c^3*ArcTan[a*x]^2)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^2)/2 + (a^6*c^3*x^4*ArcTan[a*x
]^2)/4 + 6*a^2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^3*Log[x] + (2*a^2*c^3*Log[1 + a^2*x^2])/3
- (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I
*a*x)] - (3*a^2*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*a^2*c^3*PolyLog[3, -1 + 2/(1 + I*a*x)])/2

________________________________________________________________________________________

Rubi [A]  time = 0.603183, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 16, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {4948, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610, 4916, 4846, 260, 43} \[ -\frac{3}{2} a^2 c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c^3 \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-3 i a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+3 i a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )+\frac{1}{12} a^4 c^3 x^2+\frac{2}{3} a^2 c^3 \log \left (a^2 x^2+1\right )+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2-\frac{1}{6} a^5 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+a^2 c^3 \log (x)-\frac{5}{2} a^3 c^3 x \tan ^{-1}(a x)+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}-\frac{a c^3 \tan ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*c^3*x^2)/12 - (a*c^3*ArcTan[a*x])/x - (5*a^3*c^3*x*ArcTan[a*x])/2 - (a^5*c^3*x^3*ArcTan[a*x])/6 + (3*a^2*
c^3*ArcTan[a*x]^2)/4 - (c^3*ArcTan[a*x]^2)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^2)/2 + (a^6*c^3*x^4*ArcTan[a*x
]^2)/4 + 6*a^2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^3*Log[x] + (2*a^2*c^3*Log[1 + a^2*x^2])/3
- (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I
*a*x)] - (3*a^2*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*a^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 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 4918

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f^2), 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] && LtQ[m, -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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

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 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 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 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 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^3} \, dx &=\int \left (\frac{c^3 \tan ^{-1}(a x)^2}{x^3}+\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^2+a^6 c^3 x^3 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)^2}{x^3} \, dx+\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x} \, dx+\left (3 a^4 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\left (a c^3\right ) \int \frac{\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (12 a^3 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^5 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{2} \left (a^7 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\left (a c^3\right ) \int \frac{\tan ^{-1}(a x)}{x^2} \, dx-\left (a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \tan ^{-1}(a x) \, dx+\left (3 a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (6 a^3 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 (6 a^3 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-\frac{1}{2} \left (a^5 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac{1}{2} \left (a^5 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{a c^3 \tan ^{-1}(a x)}{x}-3 a^3 c^3 x \tan ^{-1}(a x)-\frac{1}{6} a^5 c^3 x^3 \tan ^{-1}(a x)+a^2 c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 i a^2 c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )+\left (a^2 c^3\right ) \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx+\left (3 i a^3 c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a^3 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 (a^3 c^3\right ) \int \tan ^{-1}(a x) \, dx-\frac{1}{2} \left (a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (3 a^4 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx+\frac{1}{6} \left (a^6 c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=-\frac{a c^3 \tan ^{-1}(a x)}{x}-\frac{5}{2} a^3 c^3 x \tan ^{-1}(a x)-\frac{1}{6} a^5 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c^3 \log \left (1+a^2 x^2\right )-3 i a^2 c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} a^2 c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c^3 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{1}{2} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{2} \left (a^4 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx+\frac{1}{12} \left (a^6 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a c^3 \tan ^{-1}(a x)}{x}-\frac{5}{2} a^3 c^3 x \tan ^{-1}(a x)-\frac{1}{6} a^5 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{5}{4} a^2 c^3 \log \left (1+a^2 x^2\right )-3 i a^2 c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} a^2 c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c^3 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{1}{2} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (a^4 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{12} \left (a^6 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}{12} a^4 c^3 x^2-\frac{a c^3 \tan ^{-1}(a x)}{x}-\frac{5}{2} a^3 c^3 x \tan ^{-1}(a x)-\frac{1}{6} a^5 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+a^2 c^3 \log (x)+\frac{2}{3} a^2 c^3 \log \left (1+a^2 x^2\right )-3 i a^2 c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} a^2 c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c^3 \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.386366, size = 333, normalized size = 1.11 \[ \frac{c^3 \left (72 i a^2 x^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+72 i a^2 x^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+36 a^2 x^2 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-36 a^2 x^2 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )+2 a^4 x^4+2 a^2 x^2-3 i \pi ^3 a^2 x^2+24 a^2 x^2 \log \left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+28 a^2 x^2 \log \left (a^2 x^2+1\right )+6 a^6 x^6 \tan ^{-1}(a x)^2-4 a^5 x^5 \tan ^{-1}(a x)+36 a^4 x^4 \tan ^{-1}(a x)^2-60 a^3 x^3 \tan ^{-1}(a x)+48 i a^2 x^2 \tan ^{-1}(a x)^3+18 a^2 x^2 \tan ^{-1}(a x)^2+72 a^2 x^2 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-72 a^2 x^2 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-24 a x \tan ^{-1}(a x)-12 \tan ^{-1}(a x)^2\right )}{24 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(2*a^2*x^2 - (3*I)*a^2*Pi^3*x^2 + 2*a^4*x^4 - 24*a*x*ArcTan[a*x] - 60*a^3*x^3*ArcTan[a*x] - 4*a^5*x^5*Arc
Tan[a*x] - 12*ArcTan[a*x]^2 + 18*a^2*x^2*ArcTan[a*x]^2 + 36*a^4*x^4*ArcTan[a*x]^2 + 6*a^6*x^6*ArcTan[a*x]^2 +
(48*I)*a^2*x^2*ArcTan[a*x]^3 + 72*a^2*x^2*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 72*a^2*x^2*ArcTan[a*
x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 24*a^2*x^2*Log[(a*x)/Sqrt[1 + a^2*x^2]] + 28*a^2*x^2*Log[1 + a^2*x^2] +
(72*I)*a^2*x^2*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (72*I)*a^2*x^2*ArcTan[a*x]*PolyLog[2, -E^((2*I
)*ArcTan[a*x])] + 36*a^2*x^2*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 36*a^2*x^2*PolyLog[3, -E^((2*I)*ArcTan[a*x])
]))/(24*x^2)

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Maple [C]  time = 4.543, size = 1333, normalized size = 4.5 \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)^3*arctan(a*x)^2/x^3,x)

[Out]

-5/2*a^3*c^3*x*arctan(a*x)-1/6*a^5*c^3*x^3*arctan(a*x)+3/2*a^4*c^3*x^2*arctan(a*x)^2+1/4*a^6*c^3*x^4*arctan(a*
x)^2+3*I*a^2*c^3*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-6*I*a^2*c^3*arctan(a*x)*polylog(2,(1+I*a*x)/(
a^2*x^2+1)^(1/2))-6*I*a^2*c^3*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3/2*I*a^2*c^3*Pi*arctan(a*x)
^2+3/4*a^2*c^3*arctan(a*x)^2-1/2*c^3*arctan(a*x)^2/x^2+1/12*a^4*c^3*x^2-a*c^3*arctan(a*x)/x+1/12*c^3*a^2-7/3*a
^2*c^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+6*a^2*c^3*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*a^2*c^3*polylog(3,(1+
I*a*x)/(a^2*x^2+1)^(1/2))-3/2*a^2*c^3*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+a^2*c^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/
2)-1)+a^2*c^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*a^2*c^3*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)+3*a^2*c^
3*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a^2*c^3*arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*
a^2*c^3*arctan(a*x)^2*ln(a*x)+4/3*I*a^2*c^3*arctan(a*x)-3/2*I*a^2*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+3/2*I*a^2*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+3/2*I*a^2*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*a
rctan(a*x)^2+3/2*I*a^2*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)^2-3/2*I*a^2*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)^2-3/2*I*a^2*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+3/2*I*a^2*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)^2-3/2*I*a^2*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

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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^3,x, algorithm="maxima")

[Out]

1/64*(4*(192*a^8*c^3*integrate(1/16*x^8*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 16*a^8*c^3*integrate(1/16*x^8*log(
a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) + 16*a^8*c^3*integrate(1/16*x^8*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) - 32*a
^7*c^3*integrate(1/16*x^7*arctan(a*x)/(a^2*x^5 + x^3), x) + 768*a^6*c^3*integrate(1/16*x^6*arctan(a*x)^2/(a^2*
x^5 + x^3), x) + 64*a^6*c^3*integrate(1/16*x^6*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) + 96*a^6*c^3*integrate(1
/16*x^6*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) - 192*a^5*c^3*integrate(1/16*x^5*arctan(a*x)/(a^2*x^5 + x^3), x)
+ 1152*a^4*c^3*integrate(1/16*x^4*arctan(a*x)^2/(a^2*x^5 + x^3), x) + a^2*c^3*log(a^2*x^2 + 1)^3 + 768*a^2*c^3
*integrate(1/16*x^2*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 64*a^2*c^3*integrate(1/16*x^2*log(a^2*x^2 + 1)^2/(a^2*
x^5 + x^3), x) - 32*a^2*c^3*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) + 64*a*c^3*integrate(1/16*
x*arctan(a*x)/(a^2*x^5 + x^3), x) + 192*c^3*integrate(1/16*arctan(a*x)^2/(a^2*x^5 + x^3), x) + 16*c^3*integrat
e(1/16*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x))*x^2 + 4*(a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 2*c^3)*arctan(a*x)^2 - (
a^6*c^3*x^6 + 6*a^4*c^3*x^4 - 2*c^3)*log(a^2*x^2 + 1)^2)/x^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^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^2/x^3, 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^{3}}\, dx + \int \frac{3 a^{2} \operatorname{atan}^{2}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{3} \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**3,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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