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

Optimal. Leaf size=332 \[ \frac{3 a \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^3}-\frac{3 i a \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{93 a}{128 c^3 \left (a^2 x^2+1\right )}+\frac{3 a}{128 c^3 \left (a^2 x^2+1\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left (a^2 x^2+1\right )}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac{93 a^2 x \tan ^{-1}(a x)}{64 c^3 \left (a^2 x^2+1\right )}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{i a \tan ^{-1}(a x)^3}{c^3}+\frac{93 a \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3} \]

[Out]

(3*a)/(128*c^3*(1 + a^2*x^2)^2) + (93*a)/(128*c^3*(1 + a^2*x^2)) + (3*a^2*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)
^2) + (93*a^2*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) + (93*a*ArcTan[a*x]^2)/(128*c^3) - (3*a*ArcTan[a*x]^2)/(16
*c^3*(1 + a^2*x^2)^2) - (21*a*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^3)/c^3 - ArcTan[a*x]^3/
(c^3*x) - (a^2*x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) - (7*a^2*x*ArcTan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) - (15*
a*ArcTan[a*x]^4)/(32*c^3) + (3*a*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 - ((3*I)*a*ArcTan[a*x]*PolyLog[2, -
1 + 2/(1 - I*a*x)])/c^3 + (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.75429, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {4966, 4918, 4852, 4924, 4868, 4884, 4992, 6610, 4892, 4930, 261, 4900, 4896} \[ \frac{3 a \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^3}-\frac{3 i a \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{93 a}{128 c^3 \left (a^2 x^2+1\right )}+\frac{3 a}{128 c^3 \left (a^2 x^2+1\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left (a^2 x^2+1\right )}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac{93 a^2 x \tan ^{-1}(a x)}{64 c^3 \left (a^2 x^2+1\right )}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{i a \tan ^{-1}(a x)^3}{c^3}+\frac{93 a \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a)/(128*c^3*(1 + a^2*x^2)^2) + (93*a)/(128*c^3*(1 + a^2*x^2)) + (3*a^2*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)
^2) + (93*a^2*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) + (93*a*ArcTan[a*x]^2)/(128*c^3) - (3*a*ArcTan[a*x]^2)/(16
*c^3*(1 + a^2*x^2)^2) - (21*a*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^3)/c^3 - ArcTan[a*x]^3/
(c^3*x) - (a^2*x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) - (7*a^2*x*ArcTan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) - (15*
a*ArcTan[a*x]^4)/(32*c^3) + (3*a*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 - ((3*I)*a*ArcTan[a*x]*PolyLog[2, -
1 + 2/(1 - I*a*x)])/c^3 + (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3)

Rule 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int[
x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] &
& NeQ[p, -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 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 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 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 4992

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a + b*ArcT
an[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 4892

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

Rule 4930

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

Rule 261

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

Rule 4900

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*p*(d + e*x^2)^(q
+ 1)*(a + b*ArcTan[c*x])^(p - 1))/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(b^2*p*(p - 1))/(4*(q + 1)^2), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

Rule 4896

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*(d + e*x^2)^(q + 1))/(
4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x] - Si
mp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d
] && LtQ[q, -1] && NeQ[q, -3/2]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{1}{8} \left (3 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac{\left (3 a^2\right ) \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac{3 a}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}-\frac{7 a \tan ^{-1}(a x)^4}{32 c^3}+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx}{c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx}{c^2}+\frac{\left (9 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}+\frac{\left (9 a^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}+\frac{\left (3 a^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\\ &=\frac{3 a}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a^2 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}+\frac{9 a \tan ^{-1}(a x)^2}{128 c^3}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}+\frac{(3 a) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c^3}+\frac{\left (9 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}+\frac{\left (3 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}-\frac{\left (9 a^3\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{64 c}\\ &=\frac{3 a}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a}{128 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{93 a^2 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}+\frac{93 a \tan ^{-1}(a x)^2}{128 c^3}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^3}{c^3}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}+\frac{(3 i a) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}-\frac{\left (9 a^3\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-\frac{\left (3 a^3\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac{3 a}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{93 a}{128 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{93 a^2 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}+\frac{93 a \tan ^{-1}(a x)^2}{128 c^3}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^3}{c^3}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}+\frac{3 a \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{\left (6 a^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\\ &=\frac{3 a}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{93 a}{128 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{93 a^2 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}+\frac{93 a \tan ^{-1}(a x)^2}{128 c^3}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^3}{c^3}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}+\frac{3 a \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i a \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{\left (3 i a^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\\ &=\frac{3 a}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac{93 a}{128 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{93 a^2 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}+\frac{93 a \tan ^{-1}(a x)^2}{128 c^3}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}-\frac{i a \tan ^{-1}(a x)^3}{c^3}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}+\frac{3 a \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i a \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{3 a \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.546762, size = 232, normalized size = 0.7 \[ \frac{a \left (3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-\frac{a x \tan ^{-1}(a x)^3}{a^2 x^2+1}-\frac{15}{32} \tan ^{-1}(a x)^4-\frac{\tan ^{-1}(a x)^3}{a x}+i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-\frac{1}{32} \tan ^{-1}(a x)^3 \sin \left (4 \tan ^{-1}(a x)\right )+\frac{3}{4} \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+\frac{3}{256} \tan ^{-1}(a x) \sin \left (4 \tan ^{-1}(a x)\right )-\frac{3}{4} \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )-\frac{3}{128} \tan ^{-1}(a x)^2 \cos \left (4 \tan ^{-1}(a x)\right )+\frac{3}{8} \cos \left (2 \tan ^{-1}(a x)\right )+\frac{3 \cos \left (4 \tan ^{-1}(a x)\right )}{1024}-\frac{i \pi ^3}{8}\right )}{c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*((-I/8)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) - (a*x*ArcTan[a*x]^3)/(1 + a^2*x^2) - (15*ArcTan[a*x]^
4)/32 + (3*Cos[2*ArcTan[a*x]])/8 - (3*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/4 + (3*Cos[4*ArcTan[a*x]])/1024 - (3*A
rcTan[a*x]^2*Cos[4*ArcTan[a*x]])/128 + 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*Pol
yLog[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2 + (3*ArcTan[a*x]*Sin[2*ArcTan[a*x]]
)/4 + (3*ArcTan[a*x]*Sin[4*ArcTan[a*x]])/256 - (ArcTan[a*x]^3*Sin[4*ArcTan[a*x]])/32))/c^3

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Maple [C]  time = 3.306, size = 2315, normalized size = 7. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*I*a/c^3*Pi*arctan(a*x)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*
x^2+1)+1)^2)^2-3/2*I*a/c^3*Pi*arctan(a*x)^2*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)^2)^2+3/2*I*a/c^3*Pi*arctan(a*x)^2*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))+3/4*I*a/c^3*Pi*arctan(a*x)^2*csgn(I*((1+I*a*x)^2/(a^2*x
^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+3/512*I/c^3*arctan(a*x)/(a*x-I)^2*a^3*x^2+3/8*I/c^3*arctan(a
*x)/(a*x+I)*a^2*x-3/512*I/c^3*arctan(a*x)/(a*x+I)^2*a^3*x^2-3/8*I/c^3*arctan(a*x)/(a*x-I)*a^2*x+3/2*I*a/c^3*Pi
*arctan(a*x)^2*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3-3/4*I*a/c^3*Pi*arctan(a*x)^2*cs
gn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+3/4*I*a/c^3*Pi*arctan(a*x)^2*csgn(I*((1+I*a*x)^2
/(a^2*x^2+1)+1)^2)^3-3/2*I*a/c^3*Pi*arctan(a*x)^2*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1)
)^2-3/4*I*a/c^3*Pi*arctan(a*x)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+3/2*I*a/c^3*Pi*arctan(a*x)^2*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+6*a/c^3*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*a/c^3*pol
ylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3/2048*a/c^3/(a*x+I)^2-3/2048*a/c^3/(a*x-I)^2-9/8*a^2*x*arctan(a*x)^3/c^3
/(a^2*x^2+1)^2+3*a/c^3*arctan(a*x)^2*ln(2)+3*a/c^3*arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a/c^3*arc
tan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a/c^3*arctan(a*x)^2*ln(a*x)-3/2*a/c^3*arctan(a*x)^2*ln(a^2*x^2+
1)-3/16*I*a/c^3/(a*x-I)+3/16*I*a/c^3/(a*x+I)+3*a/c^3*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-3*a/c^3*arc
tan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-3/16/c^3/(a*x-I)*a^2*x+3/2048/c^3/(a*x+I)^2*a^3*x^2+3/2048/c^3/(a*x-I
)^2*a^3*x^2-3/16/c^3/(a*x+I)*a^2*x-arctan(a*x)^3/c^3/x-15/32*a*arctan(a*x)^4/c^3+93/128*a*arctan(a*x)^2/c^3+3/
8*a/c^3*arctan(a*x)/(a*x+I)+3/8*a/c^3*arctan(a*x)/(a*x-I)-3/4*I*a/c^3*Pi*arctan(a*x)^2*csgn(I/((1+I*a*x)^2/(a^
2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+3
/2*I*a/c^3*Pi*arctan(a*x)^2*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))-3/16*a*arctan(a*x)^2/c^3/(a^2*x^2+1)^2-21/16*a*arctan(a*
x)^2/c^3/(a^2*x^2+1)-I*a*arctan(a*x)^3/c^3-6*I*a/c^3*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3/2*I
*a/c^3*Pi*arctan(a*x)^2-3/512*I*a/c^3*arctan(a*x)/(a*x-I)^2+3/512*I*a/c^3*arctan(a*x)/(a*x+I)^2-3/1024*I/c^3/(
a*x+I)^2*a^2*x+3/1024*I/c^3/(a*x-I)^2*a^2*x-3/256/c^3*arctan(a*x)/(a*x-I)^2*a^2*x-3/256/c^3*arctan(a*x)/(a*x+I
)^2*a^2*x-7/8/c^3*arctan(a*x)^3*a^4*x^3/(a^2*x^2+1)^2-6*I*a/c^3*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1
/2))-3/2*I*a/c^3*Pi*arctan(a*x)^2*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))^2-3/2*I*a/c^3*Pi*arctan(a*x)^2*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+3/4*I*a/c^3*Pi*arctan(a*x)^2*csgn(I/(
(1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-3/4*I*a/c^3*Pi*a
rctan(a*x)^2*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))+3/2*I*a/c^3*Pi*arctan(a*x)^
2*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2-3/2*I*a/c^3*Pi*arctan(a*x)^2*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

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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(arctan(a*x)^3/x^2/(a^2*c*x^2+c)^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{\arctan \left (a x\right )^{3}}{a^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctan(a*x)^3/(a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}^{3}{\left (a x \right )}}{a^{6} x^{8} + 3 a^{4} x^{6} + 3 a^{2} x^{4} + x^{2}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atan(a*x)**3/(a**6*x**8 + 3*a**4*x**6 + 3*a**2*x**4 + x**2), x)/c**3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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