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

Optimal. Leaf size=478 \[ -\frac {3 i a^2 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )}{2 c^3}-\frac {9 i a^2 \text {Li}_4\left (\frac {2}{1-i a x}-1\right )}{4 c^3}+\frac {9 i a^2 \text {Li}_2\left (\frac {2}{1-i a x}-1\right ) \tan ^{-1}(a x)^2}{2 c^3}-\frac {9 a^2 \text {Li}_3\left (\frac {2}{1-i a x}-1\right ) \tan ^{-1}(a x)}{2 c^3}-\frac {a^2 \tan ^{-1}(a x)^3}{c^3 \left (a^2 x^2+1\right )}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {57 a^2 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )}+\frac {3 a^2 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {3 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac {3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac {237 a^2 \tan ^{-1}(a x)}{256 c^3}-\frac {3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^3}{c^3}+\frac {3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^3}-\frac {237 a^3 x}{256 c^3 \left (a^2 x^2+1\right )}-\frac {3 a^3 x}{128 c^3 \left (a^2 x^2+1\right )^2}+\frac {57 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left (a^2 x^2+1\right )}+\frac {3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac {\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac {3 a \tan ^{-1}(a x)^2}{2 c^3 x} \]

[Out]

-3/128*a^3*x/c^3/(a^2*x^2+1)^2-237/256*a^3*x/c^3/(a^2*x^2+1)-237/256*a^2*arctan(a*x)/c^3+3/32*a^2*arctan(a*x)/
c^3/(a^2*x^2+1)^2+57/32*a^2*arctan(a*x)/c^3/(a^2*x^2+1)+3/4*I*a^2*arctan(a*x)^4/c^3-3/2*a*arctan(a*x)^2/c^3/x+
3/16*a^3*x*arctan(a*x)^2/c^3/(a^2*x^2+1)^2+57/32*a^3*x*arctan(a*x)^2/c^3/(a^2*x^2+1)+3/32*a^2*arctan(a*x)^3/c^
3-1/2*arctan(a*x)^3/c^3/x^2-1/4*a^2*arctan(a*x)^3/c^3/(a^2*x^2+1)^2-a^2*arctan(a*x)^3/c^3/(a^2*x^2+1)+9/2*I*a^
2*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c^3+3*a^2*arctan(a*x)*ln(2-2/(1-I*a*x))/c^3-3*a^2*arctan(a*x)^3*ln(2
-2/(1-I*a*x))/c^3-3/2*I*a^2*arctan(a*x)^2/c^3-3/2*I*a^2*polylog(2,-1+2/(1-I*a*x))/c^3-9/2*a^2*arctan(a*x)*poly
log(3,-1+2/(1-I*a*x))/c^3-9/4*I*a^2*polylog(4,-1+2/(1-I*a*x))/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.84, antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {4966, 4918, 4852, 4924, 4868, 2447, 4884, 4992, 4996, 6610, 4930, 4892, 199, 205, 4900} \[ -\frac {3 i a^2 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {9 i a^2 \text {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^3}+\frac {9 i a^2 \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {9 a^2 \tan ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {237 a^3 x}{256 c^3 \left (a^2 x^2+1\right )}-\frac {3 a^3 x}{128 c^3 \left (a^2 x^2+1\right )^2}-\frac {a^2 \tan ^{-1}(a x)^3}{c^3 \left (a^2 x^2+1\right )}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {57 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left (a^2 x^2+1\right )}+\frac {3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac {57 a^2 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )}+\frac {3 a^2 \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {3 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac {3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac {237 a^2 \tan ^{-1}(a x)}{256 c^3}-\frac {3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^3}{c^3}+\frac {3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)}{c^3}-\frac {\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac {3 a \tan ^{-1}(a x)^2}{2 c^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^3*x)/(128*c^3*(1 + a^2*x^2)^2) - (237*a^3*x)/(256*c^3*(1 + a^2*x^2)) - (237*a^2*ArcTan[a*x])/(256*c^3) +
(3*a^2*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (57*a^2*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)) - (((3*I)/2)*a^2*A
rcTan[a*x]^2)/c^3 - (3*a*ArcTan[a*x]^2)/(2*c^3*x) + (3*a^3*x*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) + (57*a^3
*x*ArcTan[a*x]^2)/(32*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x]^3)/(32*c^3) - ArcTan[a*x]^3/(2*c^3*x^2) - (a^2*A
rcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x]^3)/(c^3*(1 + a^2*x^2)) + (((3*I)/4)*a^2*ArcTan[a*x]^4
)/c^3 + (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 - (3*a^2*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^3 - ((
(3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + (((9*I)/2)*a^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)]
)/c^3 - (9*a^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3) - (((9*I)/4)*a^2*PolyLog[4, -1 + 2/(1 - I*a
*x)])/c^3

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 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 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 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 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 4996

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]

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)^3}{x^3 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac {x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac {a^2 \int \frac {\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {1}{4} \left (3 a^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^3}{x^3} \, dx}{c^3}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \left (\frac {a^2 \int \frac {\tan ^{-1}(a x)^3}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^4 \int \frac {x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=\frac {3 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}-\frac {1}{32} \left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {(3 a) \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx}{2 c^3}-\frac {\left (i a^2\right ) \int \frac {\tan ^{-1}(a x)^3}{x (i+a x)} \, dx}{c^3}+\frac {\left (9 a^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-2 \left (\frac {a^2 \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {\left (i a^2\right ) \int \frac {\tan ^{-1}(a x)^3}{x (i+a x)} \, dx}{c^3}-\frac {\left (3 a^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=-\frac {3 a^3 x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac {\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}-\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {(3 a) \int \frac {\tan ^{-1}(a x)^2}{x^2} \, dx}{2 c^3}-\frac {\left (3 a^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 c^3}+\frac {\left (3 a^3\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-\frac {\left (9 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{128 c}-\frac {\left (9 a^4\right ) \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-2 \left (-\frac {3 a^3 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {\left (3 a^3\right ) \int \frac {\tan ^{-1}(a x)^2 \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 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=-\frac {3 a^3 x}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 a^3 x}{256 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 a \tan ^{-1}(a x)^2}{2 c^3 x}+\frac {3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac {\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}-\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {3 i a^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {\left (3 a^2\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac {\left (3 i a^3\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}{c^3}-\frac {\left (9 a^3\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{256 c^2}-\frac {\left (9 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-2 \left (-\frac {3 a^2 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i a^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {\left (3 i a^3\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}{c^3}+\frac {\left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\right )\\ &=-\frac {3 a^3 x}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac {45 a^3 x}{256 c^3 \left (1+a^2 x^2\right )}-\frac {9 a^2 \tan ^{-1}(a x)}{256 c^3}+\frac {3 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac {3 a \tan ^{-1}(a x)^2}{2 c^3 x}+\frac {3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac {\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}-\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {3 i a^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {3 a^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {\left (3 i a^2\right ) \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}+\frac {\left (3 a^3\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c^3}-\frac {\left (9 a^3\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{64 c^2}-2 \left (\frac {3 a^3 x}{8 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^2 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i a^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 a^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {\left (3 a^3\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c^3}+\frac {\left (3 a^3\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{8 c^2}\right )\\ &=-\frac {3 a^3 x}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac {45 a^3 x}{256 c^3 \left (1+a^2 x^2\right )}-\frac {45 a^2 \tan ^{-1}(a x)}{256 c^3}+\frac {3 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac {3 a \tan ^{-1}(a x)^2}{2 c^3 x}+\frac {3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac {\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {3 a^2 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {3 i a^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {3 a^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {3 i a^2 \text {Li}_4\left (-1+\frac {2}{1-i a x}\right )}{4 c^3}-2 \left (\frac {3 a^3 x}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^2 \tan ^{-1}(a x)}{8 c^3}-\frac {3 a^2 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i a^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 a^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 i a^2 \text {Li}_4\left (-1+\frac {2}{1-i a x}\right )}{4 c^3}\right )-\frac {\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\\ &=-\frac {3 a^3 x}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac {45 a^3 x}{256 c^3 \left (1+a^2 x^2\right )}-\frac {45 a^2 \tan ^{-1}(a x)}{256 c^3}+\frac {3 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^2 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac {3 a \tan ^{-1}(a x)^2}{2 c^3 x}+\frac {3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac {\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {3 a^2 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i a^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 i a^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {3 a^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {3 i a^2 \text {Li}_4\left (-1+\frac {2}{1-i a x}\right )}{4 c^3}-2 \left (\frac {3 a^3 x}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^2 \tan ^{-1}(a x)}{8 c^3}-\frac {3 a^2 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \tan ^{-1}(a x)^3}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac {a^2 \tan ^{-1}(a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i a^2 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 a^2 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 i a^2 \text {Li}_4\left (-1+\frac {2}{1-i a x}\right )}{4 c^3}\right )\\ \end {align*}

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Mathematica [A]  time = 1.10, size = 295, normalized size = 0.62 \[ \frac {a^2 \left (-\frac {512 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3}{a^2 x^2}-4608 i \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )-4608 \tan ^{-1}(a x) \text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )-1536 i \text {Li}_2\left (e^{2 i \tan ^{-1}(a x)}\right )+2304 i \text {Li}_4\left (e^{-2 i \tan ^{-1}(a x)}\right )-768 i \tan ^{-1}(a x)^4-\frac {1536 \tan ^{-1}(a x)^2}{a x}-1536 i \tan ^{-1}(a x)^2-3072 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+3072 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+960 \tan ^{-1}(a x)^2 \sin \left (2 \tan ^{-1}(a x)\right )+24 \tan ^{-1}(a x)^2 \sin \left (4 \tan ^{-1}(a x)\right )-480 \sin \left (2 \tan ^{-1}(a x)\right )-3 \sin \left (4 \tan ^{-1}(a x)\right )-640 \tan ^{-1}(a x)^3 \cos \left (2 \tan ^{-1}(a x)\right )-32 \tan ^{-1}(a x)^3 \cos \left (4 \tan ^{-1}(a x)\right )+960 \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )+12 \tan ^{-1}(a x) \cos \left (4 \tan ^{-1}(a x)\right )+48 i \pi ^4\right )}{1024 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*((48*I)*Pi^4 - (1536*I)*ArcTan[a*x]^2 - (1536*ArcTan[a*x]^2)/(a*x) - (512*(1 + a^2*x^2)*ArcTan[a*x]^3)/(a
^2*x^2) - (768*I)*ArcTan[a*x]^4 + 960*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 640*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] +
12*ArcTan[a*x]*Cos[4*ArcTan[a*x]] - 32*ArcTan[a*x]^3*Cos[4*ArcTan[a*x]] - 3072*ArcTan[a*x]^3*Log[1 - E^((-2*I)
*ArcTan[a*x])] + 3072*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] - (4608*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)
*ArcTan[a*x])] - (1536*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] - 4608*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x
])] + (2304*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - 480*Sin[2*ArcTan[a*x]] + 960*ArcTan[a*x]^2*Sin[2*ArcTan[a*
x]] - 3*Sin[4*ArcTan[a*x]] + 24*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/(1024*c^3)

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (a x\right )^{3}}{a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctan(a*x)^3/(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)

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

[Out]

sage0*x

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maple [B]  time = 11.53, size = 891, normalized size = 1.86 \[ -\frac {\arctan \left (a x \right )^{3}}{2 c^{3} x^{2}}-\frac {a^{2} \arctan \left (a x \right )^{3}}{2 c^{3}}-\frac {3 a^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}-\frac {3 a^{2} \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}+\frac {3 a^{2} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}+\frac {3 a^{2} \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}+\frac {15 a^{2} \arctan \left (a x \right )^{2}}{32 c^{3} \left (a x +i\right )}+\frac {15 a^{2} \arctan \left (a x \right )^{2}}{32 c^{3} \left (a x -i\right )}-\frac {18 a^{2} \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}-\frac {18 a^{2} \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}-\frac {18 i a^{2} \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}-\frac {3 i a^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}-\frac {18 i a^{2} \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}-\frac {3 i a^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}+\frac {3 i a^{2} \arctan \left (a x \right )^{4}}{4 c^{3}}-\frac {3 i a^{2} \arctan \left (a x \right )^{2}}{2 c^{3}}+\frac {15 i a^{3} x}{64 c^{3} \left (a x -i\right )}+\frac {5 a^{3} \arctan \left (a x \right )^{3} x}{16 c^{3} \left (a x +i\right )}+\frac {9 i a^{2} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}+\frac {9 i a^{2} \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{3}}-\frac {15 a^{3} \arctan \left (a x \right ) x}{32 c^{3} \left (a x +i\right )}+\frac {5 a^{3} \arctan \left (a x \right )^{3} x}{16 c^{3} \left (a x -i\right )}-\frac {15 a^{3} \arctan \left (a x \right ) x}{32 c^{3} \left (a x -i\right )}-\frac {15 i a^{2} \arctan \left (a x \right )}{32 c^{3} \left (a x -i\right )}-\frac {5 i a^{2} \arctan \left (a x \right )^{3}}{16 c^{3} \left (a x +i\right )}+\frac {15 i a^{2} \arctan \left (a x \right )}{32 c^{3} \left (a x +i\right )}+\frac {5 i a^{2} \arctan \left (a x \right )^{3}}{16 c^{3} \left (a x -i\right )}-\frac {15 i a^{3} x}{64 c^{3} \left (a x +i\right )}+\frac {3 a^{2} \arctan \left (a x \right ) \cos \left (4 \arctan \left (a x \right )\right )}{256 c^{3}}-\frac {a^{2} \arctan \left (a x \right )^{3} \cos \left (4 \arctan \left (a x \right )\right )}{32 c^{3}}+\frac {3 a^{2} \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}}{128 c^{3}}-\frac {15 a^{2}}{64 c^{3} \left (a x +i\right )}-\frac {15 a^{2}}{64 c^{3} \left (a x -i\right )}-\frac {3 a^{2} \sin \left (4 \arctan \left (a x \right )\right )}{1024 c^{3}}+\frac {15 i a^{3} \arctan \left (a x \right )^{2} x}{32 c^{3} \left (a x +i\right )}-\frac {15 i a^{3} \arctan \left (a x \right )^{2} x}{32 c^{3} \left (a x -i\right )}-\frac {3 a \arctan \left (a x \right )^{2}}{2 c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(a*x)^3/c^3/x^2-1/2*a^2*arctan(a*x)^3/c^3-15/32*a^3/c^3/(I+a*x)*arctan(a*x)*x+5/16*a^3/c^3/(a*x-I)*
arctan(a*x)^3*x-15/32*a^3/c^3/(a*x-I)*arctan(a*x)*x-15/32*I*a^2/c^3/(a*x-I)*arctan(a*x)+9*I*a^2/c^3*arctan(a*x
)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+9*I*a^2/c^3*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-5
/16*I*a^2/c^3/(I+a*x)*arctan(a*x)^3+15/32*I*a^2/c^3/(I+a*x)*arctan(a*x)+5/16*I*a^2/c^3/(a*x-I)*arctan(a*x)^3-1
5/64*I*a^3/c^3/(I+a*x)*x+15/64*I*a^3/c^3/(a*x-I)*x+5/16*a^3/c^3/(I+a*x)*arctan(a*x)^3*x+3/256*a^2*arctan(a*x)/
c^3*cos(4*arctan(a*x))-3*a^2/c^3*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-18*a^2/c^3*arctan(a*x)*polylo
g(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*a^2/c^3*arctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-18*a^2/c^3*arctan(a
*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a^2/c^3*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*a^2/c^3*
arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/32*a^2*arctan(a*x)^3/c^3*cos(4*arctan(a*x))+3/128*a^2/c^3*sin(
4*arctan(a*x))*arctan(a*x)^2+15/32*a^2/c^3/(I+a*x)*arctan(a*x)^2+15/32*a^2/c^3/(a*x-I)*arctan(a*x)^2-18*I*a^2/
c^3*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*a^2/c^3*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-18*I*a^2/c^3*pol
ylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*a^2/c^3*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+15/32*I*a^3/c^3/(I+a*
x)*arctan(a*x)^2*x-15/32*I*a^3/c^3/(a*x-I)*arctan(a*x)^2*x-15/64*a^2/c^3/(I+a*x)-15/64*a^2/c^3/(a*x-I)-3/1024*
a^2/c^3*sin(4*arctan(a*x))+3/4*I*a^2*arctan(a*x)^4/c^3-3/2*a*arctan(a*x)^2/c^3/x-3/2*I*a^2*arctan(a*x)^2/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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