\(\int \frac {\arctan (a x)^3}{x^3 (c+a^2 c x^2)^2} \, dx\) [402]

Optimal result

Integrand size = 22, antiderivative size = 374 \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}-\frac {3 a^2 \arctan (a x)}{8 c^2}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^2 \arctan (a x)^4}{2 c^2}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {2 a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{2 c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5086, 5038, 4946, 5044, 4988, 2497, 5004, 5112, 5116, 6745, 5050, 5012, 205, 211} \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c^2}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{c^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (a^2 x^2+1\right )}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (a^2 x^2+1\right )}+\frac {i a^2 \arctan (a x)^4}{2 c^2}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a^2 \arctan (a x)}{8 c^2}-\frac {2 a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,\frac {2}{1-i a x}-1\right )}{2 c^2}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (a^2 x^2+1\right )}-\frac {3 a^3 x}{8 c^2 \left (a^2 x^2+1\right )}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x} \]

[In]

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

[Out]

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

Rule 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

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

Rule 2497

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 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 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 5116

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)*(PolyLo
g[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 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}{x \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = a^4 \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\arctan (a x)^3}{x^3} \, dx}{c^2}-2 \frac {a^2 \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = -\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {1}{2} \left (3 a^3\right ) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(3 a) \int \frac {\arctan (a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx}{2 c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {\left (i a^2\right ) \int \frac {\arctan (a x)^3}{x (i+a x)} \, dx}{c^2}\right ) \\ & = \frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {1}{2} \left (3 a^4\right ) \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(3 a) \int \frac {\arctan (a x)^2}{x^2} \, dx}{2 c^2}-\frac {\left (3 a^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{2 c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {\left (3 a^3\right ) \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right ) \\ & = \frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {1}{4} \left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\left (3 a^2\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {\left (3 i a^3\right ) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right ) \\ & = -\frac {3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\left (3 i a^2\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}-\frac {\left (3 a^3\right ) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c^2}\right )-\frac {\left (3 a^3\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{8 c} \\ & = -\frac {3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}-\frac {3 a^2 \arctan (a x)}{8 c^2}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^2}\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^2} \\ & = -\frac {3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}-\frac {3 a^2 \arctan (a x)}{8 c^2}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.65 \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2 \left (i \pi ^4-48 i \arctan (a x)^2-\frac {48 \arctan (a x)^2}{a x}-\frac {16 \left (1+a^2 x^2\right ) \arctan (a x)^3}{a^2 x^2}-16 i \arctan (a x)^4+12 \arctan (a x) \cos (2 \arctan (a x))-8 \arctan (a x)^3 \cos (2 \arctan (a x))-64 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+96 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )-96 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )-96 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+48 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-6 \sin (2 \arctan (a x))+12 \arctan (a x)^2 \sin (2 \arctan (a x))\right )}{32 c^2} \]

[In]

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

[Out]

(a^2*(I*Pi^4 - (48*I)*ArcTan[a*x]^2 - (48*ArcTan[a*x]^2)/(a*x) - (16*(1 + a^2*x^2)*ArcTan[a*x]^3)/(a^2*x^2) -
(16*I)*ArcTan[a*x]^4 + 12*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 8*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] - 64*ArcTan[a*x]
^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] - (96*I)*ArcTan[a*x]^2*Poly
Log[2, E^((-2*I)*ArcTan[a*x])] - (48*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]*PolyLog[3, E^((-2*I
)*ArcTan[a*x])] + (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - 6*Sin[2*ArcTan[a*x]] + 12*ArcTan[a*x]^2*Sin[2*Ar
cTan[a*x]]))/(32*c^2)

Maple [A] (verified)

Time = 96.56 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(a^{2} \left (\frac {6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {\left (6 i \arctan \left (a x \right )^{2}+4 \arctan \left (a x \right )^{3}-3 i-6 \arctan \left (a x \right )\right ) \left (a x -i\right )}{32 c^{2} \left (a x +i\right )}+\frac {\left (-6 i \arctan \left (a x \right )^{2}+4 \arctan \left (a x \right )^{3}+3 i-6 \arctan \left (a x \right )\right ) \left (a x +i\right )}{32 c^{2} \left (a x -i\right )}-\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 c^{2} a^{2} x^{2}}-\frac {3 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {3 i \arctan \left (a x \right )^{2}}{c^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c^{2}}+\frac {i \arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {2 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}\right )\)	\(531\)
default	\(a^{2} \left (\frac {6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {\left (6 i \arctan \left (a x \right )^{2}+4 \arctan \left (a x \right )^{3}-3 i-6 \arctan \left (a x \right )\right ) \left (a x -i\right )}{32 c^{2} \left (a x +i\right )}+\frac {\left (-6 i \arctan \left (a x \right )^{2}+4 \arctan \left (a x \right )^{3}+3 i-6 \arctan \left (a x \right )\right ) \left (a x +i\right )}{32 c^{2} \left (a x -i\right )}-\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 c^{2} a^{2} x^{2}}-\frac {3 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {3 i \arctan \left (a x \right )^{2}}{c^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c^{2}}+\frac {i \arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {2 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}\right )\)	\(531\)

[In]

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

[Out]

a^2*(6*I/c^2*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/32*(6*I*arctan(a*x)^2+4*arctan(a*x)^3-3*I-
6*arctan(a*x))*(a*x-I)/c^2/(I+a*x)+1/32*(-6*I*arctan(a*x)^2+4*arctan(a*x)^3+3*I-6*arctan(a*x))*(I+a*x)/c^2/(a*
x-I)-1/2/c^2*arctan(a*x)^2*(-I*arctan(a*x)-3*I*a*x+x*arctan(a*x)*a)*(I+a*x)/a^2/x^2-3*I/c^2*polylog(2,(1+I*a*x
)/(a^2*x^2+1)^(1/2))+3/c^2*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I/c^2*arctan(a*x)^2+3/c^2*arctan(a*
x)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-3*I/c^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*I/c^2*polylog(4,(1+I*a
*x)/(a^2*x^2+1)^(1/2))-2/c^2*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+1/2*I/c^2*arctan(a*x)^4-12/c^2*ar
ctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I/c^2*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2)
)-2/c^2*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*I/c^2*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-12/c^
2*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{7} + 2 a^{2} x^{5} + x^{3}}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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