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

Optimal. Leaf size=317 \[ \frac{10 i a^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{3 c^3}-\frac{47 a^4 x}{64 c^3 \left (a^2 x^2+1\right )}-\frac{a^4 x}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{11 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (a^2 x^2+1\right )}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{11 a^3 \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^2}{3 c^3 x}+\frac{35 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac{10 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac{205 a^3 \tan ^{-1}(a x)}{192 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac{20 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c^3}-\frac{a \tan ^{-1}(a x)}{3 c^3 x^2}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3} \]

[Out]

-a^2/(3*c^3*x) - (a^4*x)/(32*c^3*(1 + a^2*x^2)^2) - (47*a^4*x)/(64*c^3*(1 + a^2*x^2)) - (205*a^3*ArcTan[a*x])/
(192*c^3) - (a*ArcTan[a*x])/(3*c^3*x^2) + (a^3*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) + (11*a^3*ArcTan[a*x])/(8*
c^3*(1 + a^2*x^2)) + (((10*I)/3)*a^3*ArcTan[a*x]^2)/c^3 - ArcTan[a*x]^2/(3*c^3*x^3) + (3*a^2*ArcTan[a*x]^2)/(c
^3*x) + (a^4*x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) + (35*a
^3*ArcTan[a*x]^3)/(24*c^3) - (20*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c^3) + (((10*I)/3)*a^3*PolyLog[2,
-1 + 2/(1 - I*a*x)])/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.52575, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 48, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205, 4900} \[ \frac{10 i a^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{3 c^3}-\frac{47 a^4 x}{64 c^3 \left (a^2 x^2+1\right )}-\frac{a^4 x}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{11 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (a^2 x^2+1\right )}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{11 a^3 \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^2}{3 c^3 x}+\frac{35 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac{10 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac{205 a^3 \tan ^{-1}(a x)}{192 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac{20 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c^3}-\frac{a \tan ^{-1}(a x)}{3 c^3 x^2}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^2/(3*c^3*x) - (a^4*x)/(32*c^3*(1 + a^2*x^2)^2) - (47*a^4*x)/(64*c^3*(1 + a^2*x^2)) - (205*a^3*ArcTan[a*x])/
(192*c^3) - (a*ArcTan[a*x])/(3*c^3*x^2) + (a^3*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) + (11*a^3*ArcTan[a*x])/(8*
c^3*(1 + a^2*x^2)) + (((10*I)/3)*a^3*ArcTan[a*x]^2)/c^3 - ArcTan[a*x]^2/(3*c^3*x^3) + (3*a^2*ArcTan[a*x]^2)/(c
^3*x) + (a^4*x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) + (35*a
^3*ArcTan[a*x]^3)/(24*c^3) - (20*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c^3) + (((10*I)/3)*a^3*PolyLog[2,
-1 + 2/(1 - I*a*x)])/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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{1}{8} a^4 \int \frac{1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^4} \, dx}{c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}+\frac{\left (3 a^4\right ) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac{a^4 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=-\frac{a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)^3}{8 c^3}+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx}{c^3}+\frac{a^4 \int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{c^2}-\frac{\left (3 a^4\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-\frac{\left (3 a^5\right ) \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (-\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 \tan ^{-1}(a x)^3}{6 c^3}+\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx}{c^3}-\frac{a^4 \int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{c^2}+\frac{a^5 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=-\frac{a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^2}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x^3} \, dx}{3 c^3}-\frac{\left (2 a^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac{\left (2 a^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac{\left (3 a^4\right ) \int \frac{1}{c+a^2 c x^2} \, dx}{64 c^2}-\frac{\left (3 a^4\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (-\frac{a^3 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 \tan ^{-1}(a x)^3}{2 c^3}+\frac{\left (2 a^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^3}+\frac{a^4 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=-\frac{a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)}{64 c^3}-\frac{a \tan ^{-1}(a x)}{3 c^3 x^2}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^2}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac{a^2 \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac{\left (2 i a^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{3 c^3}-\frac{\left (2 i a^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}-\frac{\left (3 a^4\right ) \int \frac{1}{c+a^2 c x^2} \, dx}{16 c^2}-2 \left (\frac{a^4 x}{4 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^2}{c^3}-\frac{a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 \tan ^{-1}(a x)^3}{2 c^3}+\frac{\left (2 i a^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}+\frac{a^4 \int \frac{1}{c+a^2 c x^2} \, dx}{4 c^2}\right )\\ &=-\frac{a^2}{3 c^3 x}-\frac{a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac{15 a^3 \tan ^{-1}(a x)}{64 c^3}-\frac{a \tan ^{-1}(a x)}{3 c^3 x^2}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^2}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^3}{24 c^3}-\frac{8 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{3 c^3}-\frac{a^4 \int \frac{1}{1+a^2 x^2} \, dx}{3 c^3}+\frac{\left (2 a^4\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{3 c^3}-2 \left (\frac{a^4 x}{4 c^3 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)}{4 c^3}-\frac{a^3 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^2}{c^3}-\frac{a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 \tan ^{-1}(a x)^3}{2 c^3}+\frac{2 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{\left (2 a^4\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\right )+\frac{\left (2 a^4\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\\ &=-\frac{a^2}{3 c^3 x}-\frac{a^4 x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 a^4 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac{109 a^3 \tan ^{-1}(a x)}{192 c^3}-\frac{a \tan ^{-1}(a x)}{3 c^3 x^2}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^2}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^3}{24 c^3}-\frac{8 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{3 c^3}+\frac{4 i a^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{3 c^3}-2 \left (\frac{a^4 x}{4 c^3 \left (1+a^2 x^2\right )}+\frac{a^3 \tan ^{-1}(a x)}{4 c^3}-\frac{a^3 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^2}{c^3}-\frac{a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 \tan ^{-1}(a x)^3}{2 c^3}+\frac{2 a^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{i a^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}\right )\\ \end{align*}

Mathematica [A]  time = 0.763991, size = 189, normalized size = 0.6 \[ \frac{a^3 \left (2560 i \left (\tan ^{-1}(a x)^2+\text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )\right )-\frac{256 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{a^3 x^3}-\frac{256 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{a^2 x^2}+1120 \tan ^{-1}(a x)^3+\frac{256 \left (10 \tan ^{-1}(a x)^2-1\right )}{a x}-5120 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+288 \left (2 \tan ^{-1}(a x)^2-1\right ) \sin \left (2 \tan ^{-1}(a x)\right )+3 \left (8 \tan ^{-1}(a x)^2-1\right ) \sin \left (4 \tan ^{-1}(a x)\right )+576 \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )+12 \tan ^{-1}(a x) \cos \left (4 \tan ^{-1}(a x)\right )\right )}{768 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^3*((-256*(1 + a^2*x^2)*ArcTan[a*x])/(a^2*x^2) - (256*(1 + a^2*x^2)*ArcTan[a*x]^2)/(a^3*x^3) + 1120*ArcTan[a
*x]^3 + (256*(-1 + 10*ArcTan[a*x]^2))/(a*x) + 576*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 12*ArcTan[a*x]*Cos[4*ArcTan
[a*x]] - 5120*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + (2560*I)*(ArcTan[a*x]^2 + PolyLog[2, E^((2*I)*ArcTa
n[a*x])]) + 288*(-1 + 2*ArcTan[a*x]^2)*Sin[2*ArcTan[a*x]] + 3*(-1 + 8*ArcTan[a*x]^2)*Sin[4*ArcTan[a*x]]))/(768
*c^3)

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Maple [A]  time = 0.116, size = 517, normalized size = 1.6 \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)^2/x^4/(a^2*c*x^2+c)^3,x)

[Out]

5/3*I*a^3/c^3*dilog(1/2*I*(a*x-I))-10/3*I*a^3/c^3*dilog(1+I*a*x)+10/3*I*a^3/c^3*dilog(1-I*a*x)-5/3*I*a^3/c^3*d
ilog(-1/2*I*(a*x+I))-47/64*a^6/c^3/(a^2*x^2+1)^2*x^3-20/3*a^3/c^3*arctan(a*x)*ln(a*x)+10/3*a^3/c^3*arctan(a*x)
*ln(a^2*x^2+1)-5/6*I*a^3/c^3*ln(a*x-I)^2+5/6*I*a^3/c^3*ln(a*x+I)^2+11/8*a^6/c^3*arctan(a*x)^2/(a^2*x^2+1)^2*x^
3+5/3*I*a^3/c^3*ln(a*x+I)*ln(1/2*I*(a*x-I))+10/3*I*a^3/c^3*ln(a*x)*ln(1-I*a*x)-10/3*I*a^3/c^3*ln(a*x)*ln(1+I*a
*x)+5/3*I*a^3/c^3*ln(a^2*x^2+1)*ln(a*x-I)-5/3*I*a^3/c^3*ln(a*x-I)*ln(-1/2*I*(a*x+I))-5/3*I*a^3/c^3*ln(a*x+I)*l
n(a^2*x^2+1)-49/64*a^4*x/c^3/(a^2*x^2+1)^2-1/3*a*arctan(a*x)/c^3/x^2+1/8*a^3*arctan(a*x)/c^3/(a^2*x^2+1)^2+11/
8*a^3*arctan(a*x)/c^3/(a^2*x^2+1)+3*a^2*arctan(a*x)^2/c^3/x+13/8*a^4*x*arctan(a*x)^2/c^3/(a^2*x^2+1)^2-1/3*a^2
/c^3/x-205/192*a^3*arctan(a*x)/c^3-1/3*arctan(a*x)^2/c^3/x^3+35/24*a^3*arctan(a*x)^3/c^3

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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)^2/x^4/(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 )^{2}}{a^{6} c^{3} x^{10} + 3 \, a^{4} c^{3} x^{8} + 3 \, a^{2} c^{3} x^{6} + c^{3} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctan(a*x)^2/(a^6*c^3*x^10 + 3*a^4*c^3*x^8 + 3*a^2*c^3*x^6 + c^3*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{a^{6} x^{10} + 3 a^{4} x^{8} + 3 a^{2} x^{6} + x^{4}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atan(a*x)**2/(a**6*x**10 + 3*a**4*x**8 + 3*a**2*x**6 + x**4), 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 )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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