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

Optimal. Leaf size=399 \[ -\frac{3}{2} i a^2 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c^2 \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-3 i a^2 c^2 \tan ^{-1}(a x)^2-3 a^2 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+3 a^2 c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c^2 \tan ^{-1}(a x)^2}{2 x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.796296, antiderivative size = 399, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 18, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818, Rules used = {4948, 4852, 4918, 4924, 4868, 2447, 4884, 4850, 4988, 4994, 4998, 6610, 4916, 4846, 4920, 4854, 2402, 2315} \[ -\frac{3}{2} i a^2 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c^2 \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-3 i a^2 c^2 \tan ^{-1}(a x)^2-3 a^2 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+3 a^2 c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c^2 \tan ^{-1}(a x)^2}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*I)*a^2*c^2*ArcTan[a*x]^2 - (3*a*c^2*ArcTan[a*x]^2)/(2*x) - (3*a^3*c^2*x*ArcTan[a*x]^2)/2 - (c^2*ArcTan[a*x
]^3)/(2*x^2) + (a^4*c^2*x^2*ArcTan[a*x]^3)/2 + 4*a^2*c^2*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 3*a^2*c^2*
ArcTan[a*x]*Log[2/(1 + I*a*x)] + 3*a^2*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^2*PolyLog[2, -
1 + 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)] - (3*I)*a^2*c^2*ArcTan[a*x]^2*PolyLog[2,
1 - 2/(1 + I*a*x)] + (3*I)*a^2*c^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - 3*a^2*c^2*ArcTan[a*x]*PolyLo
g[3, 1 - 2/(1 + I*a*x)] + 3*a^2*c^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)] + ((3*I)/2)*a^2*c^2*PolyLog[4,
1 - 2/(1 + I*a*x)] - ((3*I)/2)*a^2*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 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 4924

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

Rule 4868

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTan[c*x]
)^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2 - 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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, 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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3}{x^3} \, dx &=\int \left (\frac{c^2 \tan ^{-1}(a x)^3}{x^3}+\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^2 \int \frac{\tan ^{-1}(a x)^3}{x^3} \, dx+\left (2 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^3}{x} \, dx+\left (a^4 c^2\right ) \int x \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} \left (3 a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (12 a^3 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-\frac{1}{2} \left (3 a^5 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} \left (3 a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx-\frac{1}{2} \left (3 a^3 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx+\left (6 a^3 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 (6 a^3 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{3 a c^2 \tan ^{-1}(a x)^2}{2 x}-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )+\left (3 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx+\left (6 i a^3 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 (6 i a^3 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 a^4 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-3 i a^2 c^2 \tan ^{-1}(a x)^2-\frac{3 a c^2 \tan ^{-1}(a x)^2}{2 x}-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\left (3 i a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx-\left (3 a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx+\left (3 a^3 c^2\right ) \int \frac{\text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^2\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-3 i a^2 c^2 \tan ^{-1}(a x)^2-\frac{3 a c^2 \tan ^{-1}(a x)^2}{2 x}-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c^2 \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c^2 \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )+\left (3 a^3 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^2\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-3 i a^2 c^2 \tan ^{-1}(a x)^2-\frac{3 a c^2 \tan ^{-1}(a x)^2}{2 x}-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c^2 \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c^2 \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )-\left (3 i a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )\\ &=-3 i a^2 c^2 \tan ^{-1}(a x)^2-\frac{3 a c^2 \tan ^{-1}(a x)^2}{2 x}-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c^2 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-3 a^2 c^2 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+3 a^2 c^2 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c^2 \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c^2 \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.601002, size = 302, normalized size = 0.76 \[ \frac{1}{32} a^2 c^2 \left (96 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+96 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-96 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )+48 i \left (2 \tan ^{-1}(a x)^2+1\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (4,e^{-2 i \tan ^{-1}(a x)}\right )-48 i \text{PolyLog}\left (4,-e^{2 i \tan ^{-1}(a x)}\right )+16 a^2 x^2 \tan ^{-1}(a x)^3-\frac{16 \tan ^{-1}(a x)^3}{a^2 x^2}+32 i \tan ^{-1}(a x)^4-48 a x \tan ^{-1}(a x)^2-\frac{48 \tan ^{-1}(a x)^2}{a x}+64 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-64 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+96 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )-96 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-i \pi ^4\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*c^2*((-I)*Pi^4 - (48*ArcTan[a*x]^2)/(a*x) - 48*a*x*ArcTan[a*x]^2 - (16*ArcTan[a*x]^3)/(a^2*x^2) + 16*a^2*
x^2*ArcTan[a*x]^3 + (32*I)*ArcTan[a*x]^4 + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*L
og[1 - E^((2*I)*ArcTan[a*x])] - 96*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])] + (48*I)*(1 + 2*ArcTan[a*x]^2)*Pol
yLog[2, -E^((2*I)*ArcTan[a*x])] - (48*I)*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])]))/32

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Maple [A]  time = 2.365, size = 682, normalized size = 1.7 \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^3,x)

[Out]

1/2*a^4*c^2*x^2*arctan(a*x)^3-3/2*a^3*c^2*x*arctan(a*x)^2-3/2*a*c^2*arctan(a*x)^2/x-1/2*c^2*arctan(a*x)^3/x^2+
3*a^2*c^2*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+3/2*I*a^2*c^2*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+2*a^
2*c^2*arctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*a^2*c^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+12*a^
2*c^2*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+12*I*a^2*c^2*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))
+2*a^2*c^2*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*a^2*c^2*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*
x^2+1)^(1/2))-3*a^2*c^2*arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-3*I*a^2*c^2*polylog(2,-(1+I*a*x)/(a^2*
x^2+1)^(1/2))-3*a^2*c^2*arctan(a*x)*ln((1+I*a*x)^2/(a^2*x^2+1)+1)-6*I*a^2*c^2*arctan(a*x)^2*polylog(2,-(1+I*a*
x)/(a^2*x^2+1)^(1/2))-2*a^2*c^2*arctan(a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+3*I*a^2*c^2*arctan(a*x)^2*polylog(
2,-(1+I*a*x)^2/(a^2*x^2+1))+12*a^2*c^2*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3/2*I*a^2*c^2*polylo
g(4,-(1+I*a*x)^2/(a^2*x^2+1))+3*a^2*c^2*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+12*I*a^2*c^2*polylog(4,(
1+I*a*x)/(a^2*x^2+1)^(1/2))

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

[Out]

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

[Out]

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