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

Optimal. Leaf size=322 \[ -\frac {3 a^2 \text {Li}_3\left (\frac {2}{1-i a x}-1\right )}{2 c^3}+\frac {3 i a^2 \text {Li}_2\left (\frac {2}{1-i a x}-1\right ) \tan ^{-1}(a x)}{c^3}+\frac {19 a^2}{32 c^3 \left (a^2 x^2+1\right )}+\frac {a^2}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac {a^2 \log \left (a^2 x^2+1\right )}{2 c^3}-\frac {a^2 \tan ^{-1}(a x)^2}{c^3 \left (a^2 x^2+1\right )}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {a^2 \log (x)}{c^3}+\frac {i a^2 \tan ^{-1}(a x)^3}{c^3}+\frac {3 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac {3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3}+\frac {19 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (a^2 x^2+1\right )}+\frac {a^3 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac {\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac {a \tan ^{-1}(a x)}{c^3 x} \]

[Out]

1/32*a^2/c^3/(a^2*x^2+1)^2+19/32*a^2/c^3/(a^2*x^2+1)-a*arctan(a*x)/c^3/x+1/8*a^3*x*arctan(a*x)/c^3/(a^2*x^2+1)
^2+19/16*a^3*x*arctan(a*x)/c^3/(a^2*x^2+1)+3/32*a^2*arctan(a*x)^2/c^3-1/2*arctan(a*x)^2/c^3/x^2-1/4*a^2*arctan
(a*x)^2/c^3/(a^2*x^2+1)^2-a^2*arctan(a*x)^2/c^3/(a^2*x^2+1)+I*a^2*arctan(a*x)^3/c^3+a^2*ln(x)/c^3-1/2*a^2*ln(a
^2*x^2+1)/c^3-3*a^2*arctan(a*x)^2*ln(2-2/(1-I*a*x))/c^3+3*I*a^2*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^3-3/2*
a^2*polylog(3,-1+2/(1-I*a*x))/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.33, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 16, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {4966, 4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610, 4930, 4892, 261, 4896} \[ -\frac {3 a^2 \text {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 i a^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {19 a^2}{32 c^3 \left (a^2 x^2+1\right )}+\frac {a^2}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac {a^2 \log \left (a^2 x^2+1\right )}{2 c^3}+\frac {19 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (a^2 x^2+1\right )}+\frac {a^3 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )^2}-\frac {a^2 \tan ^{-1}(a x)^2}{c^3 \left (a^2 x^2+1\right )}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {a^2 \log (x)}{c^3}+\frac {i a^2 \tan ^{-1}(a x)^3}{c^3}+\frac {3 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac {3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3}-\frac {\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac {a \tan ^{-1}(a x)}{c^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2/(32*c^3*(1 + a^2*x^2)^2) + (19*a^2)/(32*c^3*(1 + a^2*x^2)) - (a*ArcTan[a*x])/(c^3*x) + (a^3*x*ArcTan[a*x])
/(8*c^3*(1 + a^2*x^2)^2) + (19*a^3*x*ArcTan[a*x])/(16*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x]^2)/(32*c^3) - Ar
cTan[a*x]^2/(2*c^3*x^2) - (a^2*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x]^2)/(c^3*(1 + a^2*x^2)
) + (I*a^2*ArcTan[a*x]^3)/c^3 + (a^2*Log[x])/c^3 - (a^2*Log[1 + a^2*x^2])/(2*c^3) - (3*a^2*ArcTan[a*x]^2*Log[2
- 2/(1 - I*a*x)])/c^3 + ((3*I)*a^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 - (3*a^2*PolyLog[3, -1 + 2
/(1 - I*a*x)])/(2*c^3)

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 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 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 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 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 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 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 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]

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{x^3 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {1}{2} a^3 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^3} \, dx}{c^3}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \left (\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^4 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=\frac {a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}-\frac {\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac {a \int \frac {\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac {\left (i a^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}+\frac {\left (3 a^3\right ) \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (\frac {a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac {\left (i a^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}-\frac {a^3 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=\frac {a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac {\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}-\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {a \int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c^3}-\frac {a^3 \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{c^3}+\frac {\left (2 a^3\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 {\left (3 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-2 \left (-\frac {a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {\left (2 a^3\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 {a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=\frac {a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {a \tan ^{-1}(a x)}{c^3 x}+\frac {a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac {\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}-\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {i a^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {a^2 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac {\left (i a^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-2 \left (-\frac {a^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i a^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {\left (i a^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\right )\\ &=\frac {a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {a \tan ^{-1}(a x)}{c^3 x}+\frac {a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac {\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}-\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {i a^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}-\frac {a^2 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-2 \left (-\frac {a^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i a^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {a^2 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}\right )+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac {a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {a \tan ^{-1}(a x)}{c^3 x}+\frac {a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac {\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}-\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {i a^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}-\frac {a^2 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-2 \left (-\frac {a^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i a^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {a^2 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}\right )+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^3}-\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac {a^2}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {a \tan ^{-1}(a x)}{c^3 x}+\frac {a^3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x \tan ^{-1}(a x)}{16 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac {\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac {a^2 \log (x)}{c^3}-\frac {a^2 \log \left (1+a^2 x^2\right )}{2 c^3}-\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {i a^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}-\frac {a^2 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-2 \left (-\frac {a^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^2}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^3}{3 c^3}+\frac {a^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i a^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {a^2 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.82, size = 226, normalized size = 0.70 \[ \frac {a^2 \left (\log \left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-\frac {\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{2 a^2 x^2}-3 i \tan ^{-1}(a x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )-\frac {3}{2} \text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )-i \tan ^{-1}(a x)^3-\frac {\tan ^{-1}(a x)}{a x}-3 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+\frac {5}{8} \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+\frac {1}{64} \tan ^{-1}(a x) \sin \left (4 \tan ^{-1}(a x)\right )-\frac {5}{8} \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )-\frac {1}{32} \tan ^{-1}(a x)^2 \cos \left (4 \tan ^{-1}(a x)\right )+\frac {5}{16} \cos \left (2 \tan ^{-1}(a x)\right )+\frac {1}{256} \cos \left (4 \tan ^{-1}(a x)\right )+\frac {i \pi ^3}{8}\right )}{c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*((I/8)*Pi^3 - ArcTan[a*x]/(a*x) - ((1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a^2*x^2) - I*ArcTan[a*x]^3 + (5*Cos[2*
ArcTan[a*x]])/16 - (5*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/8 + Cos[4*ArcTan[a*x]]/256 - (ArcTan[a*x]^2*Cos[4*ArcT
an[a*x]])/32 - 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - (3*I)*ArcTan[a
*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2 + (5*ArcTan[a*x]*Sin[2*ArcTa
n[a*x]])/8 + (ArcTan[a*x]*Sin[4*ArcTan[a*x]])/64))/c^3

________________________________________________________________________________________

fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (a x\right )^{2}}{a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctan(a*x)^2/(a^6*c^3*x^9 + 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 + c^3*x^3), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

maple [C]  time = 5.27, size = 2217, normalized size = 6.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/32*a^2*arctan(a*x)^2/c^3+3/4*I*a^2/c^3*arctan(a*x)^2*Pi*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^2/c^3*arctan(a*x)^2*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)^2)^2-3/4*I*a^2/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^2/c^3*Pi*arctan(a*x)^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csg
n(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-3/2*I*a^2/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^2/c^3*arctan(a*x)^2*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+3/2*I*a^2/c^3*arctan(a*x)^2*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))^2-3/4*I*a^2/c^
3*arctan(a*x)^2*Pi*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)^2-1/2*arctan(a*x)^2/c^3/x^2+I*a^2*arctan(a*x)^3/c^3+a^2/c^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-6*a^2/c^3*pol
ylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/256*a^2/c^3*cos(4*arctan(a*x))+a^2/c^3*ln(1+(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))+3/4*I*a^2/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^2/c^3*arctan(a*x)^2*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))-5/32*I*a^2/c^3/(a*x-I)-3*a^2/c^3*arctan(a*x)^2*ln(2)+1/64*a
^2/c^3*arctan(a*x)*sin(4*arctan(a*x))-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*arct
an(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+I*a*x)^2/(a^2*x^2+1)-1)-3*a^2/c^3*ar
ctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-5/32*a^3/c^3/(I+a*x)*x-5/32*a^3/c^3/(a*x-I)*x-3*a^2/c^3*arctan(a*x
)^2*ln(a*x)+5/2*a^2/c^3*arctan(a*x)/(8*I+8*a*x)+3/2*I*a^2/c^3*arctan(a*x)^2*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))^2-3/2*I*a^2/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))-a^2*arctan(a*x)^2/c^3/(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+5/2*I*a^3/c^3*arctan
(a*x)/(8*I+8*a*x)*x-5/2*I*a^3/c^3*arctan(a*x)/(8*a*x-8*I)*x+3/2*I*a^2/c^3*arctan(a*x)^2*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+3/4*I*a^2/c^3*arctan(a*x)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1
+I*a*x)^2/(a^2*x^2+1)+1)^2)^3-3/2*I*a^2/c^3*arctan(a*x)^2*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^
2*x^2+1)+1))^3-3/2*I*a^2/c^3*arctan(a*x)^2*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))^
3-3/4*I*a^2/c^3*arctan(a*x)^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+3/4*I*a^2/c^3*Pi*arctan(a*x)^2*csgn(I
*(1+I*a*x)^2/(a^2*x^2+1))^3+5/2*a^2/c^3*arctan(a*x)/(8*a*x-8*I)+3/2*a^2/c^3*arctan(a*x)^2*ln(a^2*x^2+1)-I*a^2/
c^3*arctan(a*x)+5/32*I*a^2/c^3/(I+a*x)-a*arctan(a*x)/c^3/x-1/4*a^2*arctan(a*x)^2/c^3/(a^2*x^2+1)^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^3\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{9} + 3 a^{4} x^{7} + 3 a^{2} x^{5} + x^{3}}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atan(a*x)**2/(a**6*x**9 + 3*a**4*x**7 + 3*a**2*x**5 + x**3), x)/c**3

________________________________________________________________________________________