\(\int \frac {\arctan (a x)^3}{x^2 (c+a^2 c x^2)^3} \, dx\) [409]

Optimal result

Integrand size = 22, antiderivative size = 332 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {3 a}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac {93 a}{128 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^2 x \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {93 a^2 x \arctan (a x)}{64 c^3 \left (1+a^2 x^2\right )}+\frac {93 a \arctan (a x)^2}{128 c^3}-\frac {3 a \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {21 a \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^3}{c^3}-\frac {\arctan (a x)^3}{c^3 x}-\frac {a^2 x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {7 a^2 x \arctan (a x)^3}{8 c^3 \left (1+a^2 x^2\right )}-\frac {15 a \arctan (a x)^4}{32 c^3}+\frac {3 a \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i a \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^3}+\frac {3 a \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3} \]

[Out]

3/128*a/c^3/(a^2*x^2+1)^2+93/128*a/c^3/(a^2*x^2+1)+3/32*a^2*x*arctan(a*x)/c^3/(a^2*x^2+1)^2+93/64*a^2*x*arctan
(a*x)/c^3/(a^2*x^2+1)+93/128*a*arctan(a*x)^2/c^3-3/16*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-arctan(a*x)^3/c^3/x-1/4*a^2*x*arctan(a*x)^3/c^3/(a^2*x^2+1)^2-7/8*a^2*x*
arctan(a*x)^3/c^3/(a^2*x^2+1)-15/32*a*arctan(a*x)^4/c^3+3*a*arctan(a*x)^2*ln(2-2/(1-I*a*x))/c^3-3*I*a*arctan(a
*x)*polylog(2,-1+2/(1-I*a*x))/c^3+3/2*a*polylog(3,-1+2/(1-I*a*x))/c^3

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5086, 5038, 4946, 5044, 4988, 5004, 5112, 6745, 5012, 5050, 267, 5020, 5016} \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {7 a^2 x \arctan (a x)^3}{8 c^3 \left (a^2 x^2+1\right )}-\frac {a^2 x \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {21 a \arctan (a x)^2}{16 c^3 \left (a^2 x^2+1\right )}-\frac {3 a \arctan (a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac {93 a^2 x \arctan (a x)}{64 c^3 \left (a^2 x^2+1\right )}+\frac {3 a^2 x \arctan (a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {93 a}{128 c^3 \left (a^2 x^2+1\right )}+\frac {3 a}{128 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 i a \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c^3}-\frac {15 a \arctan (a x)^4}{32 c^3}-\frac {\arctan (a x)^3}{c^3 x}-\frac {i a \arctan (a x)^3}{c^3}+\frac {93 a \arctan (a x)^2}{128 c^3}+\frac {3 a \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {3 a \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 c^3} \]

[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 267

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 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4988

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 5004

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 5012

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 5016

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 5020

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 5038

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 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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 5050

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 5086

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 5112

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa
n[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 6745

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.70 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {a \left (-\frac {i \pi ^3}{8}+i \arctan (a x)^3-\frac {\arctan (a x)^3}{a x}-\frac {a x \arctan (a x)^3}{1+a^2 x^2}-\frac {15}{32} \arctan (a x)^4+\frac {3}{8} \cos (2 \arctan (a x))-\frac {3}{4} \arctan (a x)^2 \cos (2 \arctan (a x))+\frac {3 \cos (4 \arctan (a x))}{1024}-\frac {3}{128} \arctan (a x)^2 \cos (4 \arctan (a x))+3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+\frac {3}{4} \arctan (a x) \sin (2 \arctan (a x))+\frac {3}{256} \arctan (a x) \sin (4 \arctan (a x))-\frac {1}{32} \arctan (a x)^3 \sin (4 \arctan (a x))\right )}{c^3} \]

[In]

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

[Out]

(a*((-1/8*I)*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
*ArcTan[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]*P
olyLog[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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 82.41 (sec) , antiderivative size = 1799, normalized size of antiderivative = 5.42

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1799\)
default	\(\text {Expression too large to display}\)	\(1799\)
parts	\(\text {Expression too large to display}\)	\(1802\)

[In]

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

[Out]

a*(-7/8/c^3*arctan(a*x)^3/(a^2*x^2+1)^2*a^3*x^3-9/8/c^3*arctan(a*x)^3/(a^2*x^2+1)^2*a*x-15/8/c^3*arctan(a*x)^4
-1/c^3*arctan(a*x)^3/a/x-3/8/c^3*(-15/4*arctan(a*x)^4+1/2*arctan(a*x)^2/(a^2*x^2+1)^2+4*arctan(a*x)^2*ln(a^2*x
^2+1)+7/2*arctan(a*x)^2/(a^2*x^2+1)-8*arctan(a*x)^2*ln(a*x)-8*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+16
*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+8/3*I*arctan(a*x)^3-I*arctan(a*x)*(a*x-I)/(I+a*x)+1/2*(I
+a*x)/(a*x-I)+1/2*(a*x-I)/(I+a*x)+8*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-8*arctan(a*x)^2*ln((1+I*a*x)/(
a^2*x^2+1)^(1/2)+1)+16*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-16*polylog(3,-(1+I*a*x)/(a^2*x^2+
1)^(1/2))-8*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*arctan(a*x)*(I+a*x)/(a*x-I)-16*polylog(3,(1+I*a*
x)/(a^2*x^2+1)^(1/2))-1/16*(32*I*Pi*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
)-32*I*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))-64*I*Pi*csgn(((1+I*a*x)^2/(a^2
*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2-32*I*Pi*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)+32*I*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+64*I*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))^3-32*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3-64*I*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+64*I*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+32*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+32*I*Pi*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)^2-64*I*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))^2-32*I*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+64*I*Pi+64*I*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))+64*I*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))-
64*I*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))^2-64*I*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+
64*I*Pi*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+128*ln(2)+31)*arctan(a*x)^2-1/12
8*cos(4*arctan(a*x))-1/32*arctan(a*x)*sin(4*arctan(a*x))))

Fricas [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{8} + 3 a^{4} x^{6} + 3 a^{2} x^{4} + x^{2}}\, dx}{c^{3}} \]

[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

Maxima [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2}} \,d x } \]

[In]

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

[Out]

-1/16384*(2400*(a^5*x^5 + 2*a^3*x^3 + a*x)*arctan(a*x)^4 - 90*(a^5*x^5 + 2*a^3*x^3 + a*x)*log(a^2*x^2 + 1)^4 +
256*(15*a^4*x^4 + 25*a^2*x^2 + 8)*arctan(a*x)^3 - 48*(15*(a^5*x^5 + 2*a^3*x^3 + a*x)*arctan(a*x)^2 + 4*(15*a^
4*x^4 + 25*a^2*x^2 + 8)*arctan(a*x))*log(a^2*x^2 + 1)^2 - (a^4*c^3*x^5 + 2*a^2*c^3*x^3 + c^3*x)*(360*((8*a^2*x
^2 + 7)*a^2/(a^12*c^3*x^4 + 2*a^10*c^3*x^2 + a^8*c^3) + 2*(4*a^2*x^2 + 3)*log(a^2*x^2 + 1)/(a^10*c^3*x^4 + 2*a
^8*c^3*x^2 + a^6*c^3))*a^7 - 2949120*a^7*integrate(1/1024*x^7*arctan(a*x)^2*log(a^2*x^2 + 1)/(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) - 737280*a^7*integrate(1/1024*x^7*log(a^2*x^2 + 1)^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) + 360*(2*a^2*x^2 + 1)*a^5*log(a^2*x^2 + 1)^3/(a^8*c^3*x^4 + 2*a^
6*c^3*x^2 + a^4*c^3) + 5898240*a^6*integrate(1/1024*x^6*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) + 1474560*a^6*integrate(1/1024*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(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) - 11796480*a^6*integrate(1/1024*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(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) + 720*(2*a^2*x^2 + 1)*a^5*log(a^2*x^2 + 1)^2/(a^8*c^3*x^4 +
2*a^6*c^3*x^2 + a^4*c^3) + 270*(((16*a^2*x^2 + 15)*a^2/(a^14*c^3*x^4 + 2*a^12*c^3*x^2 + a^10*c^3) + 2*(8*a^2*
x^2 + 7)*log(a^2*x^2 + 1)/(a^12*c^3*x^4 + 2*a^10*c^3*x^2 + a^8*c^3))*a^4 + 2*(4*a^2*x^2 + 3)*a^2*log(a^2*x^2 +
1)^2/(a^10*c^3*x^4 + 2*a^8*c^3*x^2 + a^6*c^3))*a^5 + 600*a^5*(a^2/(a^10*c^3*x^4 + 2*a^8*c^3*x^2 + a^6*c^3) +
2*log(a^2*x^2 + 1)/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)) - 5898240*a^5*integrate(1/1024*x^5*arctan(a*x)^2*l
og(a^2*x^2 + 1)/(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) + 11796480*a^5*integrate(1/1024*x^
5*arctan(a*x)^2/(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) + 11796480*a^4*integrate(1/1024*x^
4*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) + 2949120*a^4*integrate(1/1024*x^4
*arctan(a*x)*log(a^2*x^2 + 1)^2/(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) - 19660800*a^4*int
egrate(1/1024*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(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) + 1
80*a^3*log(a^2*x^2 + 1)^3/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3) + 135*(a^4*(a^2/(a^12*c^3*x^4 + 2*a^10*c^3*x
^2 + a^8*c^3) + 2*log(a^2*x^2 + 1)/(a^10*c^3*x^4 + 2*a^8*c^3*x^2 + a^6*c^3)) + 2*a^2*log(a^2*x^2 + 1)^2/(a^8*c
^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3))*a^3 - 2949120*a^3*integrate(1/1024*x^3*arctan(a*x)^2*log(a^2*x^2 + 1)/(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) + 19660800*a^3*integrate(1/1024*x^3*arctan(a*x)^2/(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) + 1200*a^3*log(a^2*x^2 + 1)^2/(a^6*c^3*x^4 + 2*a^4*c^3*
x^2 + a^2*c^3) + 5898240*a^2*integrate(1/1024*x^2*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) + 1474560*a^2*integrate(1/1024*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(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) - 6291456*a^2*integrate(1/1024*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(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) + 6291456*a*integrate(1/1024*x*arctan(a*x)^2/(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) - 1572864*a*integrate(1/1024*x*log(a^2*x^2 + 1)^2/(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) + 14680064*integrate(1/1024*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) + 1572864*integrate(1/1024*arctan(a*x)*log(a^2*x^2 + 1)^2/(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)))/(a^4*c^3*x^5 + 2*a^2*c^3*x^3 + c^3*x)

Giac [F]

\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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