3.403 \(\int \frac{\tan ^{-1}(a x)^3}{x^4 (c+a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=332 \[ -\frac{7 a^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{7 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{3 a^3}{8 c^2 \left (a^2 x^2+1\right )}-\frac{a^3 \log \left (a^2 x^2+1\right )}{2 c^2}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (a^2 x^2+1\right )}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left (a^2 x^2+1\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (a^2 x^2+1\right )}+\frac{a^3 \log (x)}{c^2}+\frac{5 a^3 \tan ^{-1}(a x)^4}{8 c^2}+\frac{7 i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{7 a^3 \tan ^{-1}(a x)^2}{8 c^2}+\frac{2 a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{7 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3} \]

[Out]

(-3*a^3)/(8*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x])/(c^2*x) - (3*a^4*x*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (7*
a^3*ArcTan[a*x]^2)/(8*c^2) - (a*ArcTan[a*x]^2)/(2*c^2*x^2) + (3*a^3*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) + (((
7*I)/3)*a^3*ArcTan[a*x]^3)/c^2 - ArcTan[a*x]^3/(3*c^2*x^3) + (2*a^2*ArcTan[a*x]^3)/(c^2*x) + (a^4*x*ArcTan[a*x
]^3)/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^4)/(8*c^2) + (a^3*Log[x])/c^2 - (a^3*Log[1 + a^2*x^2])/(2*c^2)
- (7*a^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 + ((7*I)*a^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c
^2 - (7*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2)

________________________________________________________________________________________

Rubi [A]  time = 1.29004, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {4966, 4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610, 4892, 4930, 261} \[ -\frac{7 a^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{2 c^2}+\frac{7 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{3 a^3}{8 c^2 \left (a^2 x^2+1\right )}-\frac{a^3 \log \left (a^2 x^2+1\right )}{2 c^2}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (a^2 x^2+1\right )}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left (a^2 x^2+1\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (a^2 x^2+1\right )}+\frac{a^3 \log (x)}{c^2}+\frac{5 a^3 \tan ^{-1}(a x)^4}{8 c^2}+\frac{7 i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{7 a^3 \tan ^{-1}(a x)^2}{8 c^2}+\frac{2 a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{7 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^2),x]

[Out]

(-3*a^3)/(8*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x])/(c^2*x) - (3*a^4*x*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (7*
a^3*ArcTan[a*x]^2)/(8*c^2) - (a*ArcTan[a*x]^2)/(2*c^2*x^2) + (3*a^3*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) + (((
7*I)/3)*a^3*ArcTan[a*x]^3)/c^2 - ArcTan[a*x]^3/(3*c^2*x^3) + (2*a^2*ArcTan[a*x]^3)/(c^2*x) + (a^4*x*ArcTan[a*x
]^3)/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^4)/(8*c^2) + (a^3*Log[x])/c^2 - (a^3*Log[1 + a^2*x^2])/(2*c^2)
- (7*a^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 + ((7*I)*a^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c
^2 - (7*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2)

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=a^4 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^4} \, dx}{c^2}-2 \frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^4}{8 c^2}-\frac{1}{2} \left (3 a^5\right ) \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{a \int \frac{\tan ^{-1}(a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx}{c^2}-2 \left (\frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx}{c^2}-\frac{a^4 \int \frac{\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx}{c}\right )\\ &=\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^4}{8 c^2}-\frac{1}{2} \left (3 a^4\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{a \int \frac{\tan ^{-1}(a x)^2}{x^3} \, dx}{c^2}-\frac{a^3 \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c^2}-2 \left (-\frac{a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^4}{4 c^2}+\frac{\left (3 a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c^2}\right )\\ &=-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^2}{8 c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^4}{8 c^2}+\frac{1}{4} \left (3 a^5\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx+\frac{a^2 \int \frac{\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx}{c^2}-\frac{\left (i a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^3}{c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^4}{4 c^2}+\frac{\left (3 i a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^2}\right )\\ &=-\frac{3 a^3}{8 c^2 \left (1+a^2 x^2\right )}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^2}{8 c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^4}{8 c^2}-\frac{a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}+\frac{a^2 \int \frac{\tan ^{-1}(a x)}{x^2} \, dx}{c^2}-\frac{a^4 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{c^2}+\frac{\left (2 a^4\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^3}{c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^4}{4 c^2}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{\left (6 a^4\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right )\\ &=-\frac{3 a^3}{8 c^2 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{7 a^3 \tan ^{-1}(a x)^2}{8 c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^4}{8 c^2}-\frac{a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}+\frac{i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{a^3 \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx}{c^2}-\frac{\left (i a^4\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^3}{c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^4}{4 c^2}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{\left (3 i a^4\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right )\\ &=-\frac{3 a^3}{8 c^2 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{7 a^3 \tan ^{-1}(a x)^2}{8 c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^4}{8 c^2}-\frac{a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}+\frac{i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^3}{c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^4}{4 c^2}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{3 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}\right )+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^2}\\ &=-\frac{3 a^3}{8 c^2 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{7 a^3 \tan ^{-1}(a x)^2}{8 c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^4}{8 c^2}-\frac{a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}+\frac{i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^3}{c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^4}{4 c^2}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{3 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}\right )+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 c^2}-\frac{a^5 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^2}\\ &=-\frac{3 a^3}{8 c^2 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac{7 a^3 \tan ^{-1}(a x)^2}{8 c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac{i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^4}{8 c^2}+\frac{a^3 \log (x)}{c^2}-\frac{a^3 \log \left (1+a^2 x^2\right )}{2 c^2}-\frac{a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}+\frac{i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}-\frac{a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}-2 \left (-\frac{i a^3 \tan ^{-1}(a x)^3}{c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^3 \tan ^{-1}(a x)^4}{4 c^2}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^2}-\frac{3 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^2}+\frac{3 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.941906, size = 243, normalized size = 0.73 \[ \frac{a^3 \left (-7 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-\frac{7}{2} \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+\log \left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-\frac{\tan ^{-1}(a x)^3}{3 a^3 x^3}-\frac{\tan ^{-1}(a x)^2}{2 a^2 x^2}+\frac{5}{8} \tan ^{-1}(a x)^4+\frac{2 \tan ^{-1}(a x)^3}{a x}-\frac{7}{3} i \tan ^{-1}(a x)^3-\frac{1}{2} \tan ^{-1}(a x)^2-\frac{\tan ^{-1}(a x)}{a x}-7 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+\frac{1}{4} \tan ^{-1}(a x)^3 \sin \left (2 \tan ^{-1}(a x)\right )-\frac{3}{8} \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+\frac{3}{8} \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )-\frac{3}{16} \cos \left (2 \tan ^{-1}(a x)\right )+\frac{7 i \pi ^3}{24}\right )}{c^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^2),x]

[Out]

(a^3*(((7*I)/24)*Pi^3 - ArcTan[a*x]/(a*x) - ArcTan[a*x]^2/2 - ArcTan[a*x]^2/(2*a^2*x^2) - ((7*I)/3)*ArcTan[a*x
]^3 - ArcTan[a*x]^3/(3*a^3*x^3) + (2*ArcTan[a*x]^3)/(a*x) + (5*ArcTan[a*x]^4)/8 - (3*Cos[2*ArcTan[a*x]])/16 +
(3*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/8 - 7*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 +
a^2*x^2]] - (7*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (7*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2 -
(3*ArcTan[a*x]*Sin[2*ArcTan[a*x]])/8 + (ArcTan[a*x]^3*Sin[2*ArcTan[a*x]])/4))/c^2

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Maple [C]  time = 8.694, size = 5190, normalized size = 15.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/x^4/(a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{2} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)^3/x^4/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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