\(\int \frac {\arctan (a x)^3}{x (c+a^2 c x^2)^2} \, dx\) [400]

Optimal result

Integrand size = 22, antiderivative size = 240 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {3 a x}{8 c^2 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)}{8 c^2}-\frac {3 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^3}{4 c^2}+\frac {\arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^2}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5086, 5044, 4988, 5004, 5112, 5116, 6745, 5050, 5012, 205, 211} \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {\arctan (a x)^3}{2 c^2 \left (a^2 x^2+1\right )}-\frac {3 a x \arctan (a x)^2}{4 c^2 \left (a^2 x^2+1\right )}-\frac {3 \arctan (a x)}{4 c^2 \left (a^2 x^2+1\right )}+\frac {3 a x}{8 c^2 \left (a^2 x^2+1\right )}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c^2}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 c^2}-\frac {i \arctan (a x)^4}{4 c^2}-\frac {\arctan (a x)^3}{4 c^2}+\frac {3 \arctan (a x)}{8 c^2}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {3 i \operatorname {PolyLog}\left (4,\frac {2}{1-i a x}-1\right )}{4 c^2} \]

[In]

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

[Out]

(3*a*x)/(8*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(8*c^2) - (3*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (3*a*x*ArcTa
n[a*x]^2)/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^3/(4*c^2) + ArcTan[a*x]^3/(2*c^2*(1 + a^2*x^2)) - ((I/4)*ArcTan[
a*x]^4)/c^2 + (ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^2 - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a
*x)])/c^2 + (3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2) + (((3*I)/4)*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 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 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 {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\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 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^2}-\frac {1}{2} (3 a) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {i \int \frac {\arctan (a x)^3}{x (i+a x)} \, dx}{c^2} \\ & = -\frac {3 a x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^3}{4 c^2}+\frac {\arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^2}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {1}{2} \left (3 a^2\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 \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2} \\ & = -\frac {3 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^3}{4 c^2}+\frac {\arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^2}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {1}{4} (3 a) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(3 i a) \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} \\ & = \frac {3 a x}{8 c^2 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^3}{4 c^2}+\frac {\arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^2}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}-\frac {(3 a) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c^2}+\frac {(3 a) \int \frac {1}{c+a^2 c x^2} \, dx}{8 c} \\ & = \frac {3 a x}{8 c^2 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)}{8 c^2}-\frac {3 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^3}{4 c^2}+\frac {\arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^2}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.65 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx=\frac {-i \pi ^4+16 i \arctan (a x)^4-24 \arctan (a x) \cos (2 \arctan (a x))+16 \arctan (a x)^3 \cos (2 \arctan (a x))+64 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+96 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+96 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )+12 \sin (2 \arctan (a x))-24 \arctan (a x)^2 \sin (2 \arctan (a x))}{64 c^2} \]

[In]

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

[Out]

((-I)*Pi^4 + (16*I)*ArcTan[a*x]^4 - 24*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 16*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] +
64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 9
6*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] + 12*Sin[2*ArcTan
[a*x]] - 24*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]])/(64*c^2)

Maple [C] (warning: unable to verify)

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

Time = 84.45 (sec) , antiderivative size = 1787, normalized size of antiderivative = 7.45

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1787\)
default	\(\text {Expression too large to display}\)	\(1787\)
parts	\(\text {Expression too large to display}\)	\(2218\)

[In]

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

[Out]

1/c^2*arctan(a*x)^3*ln(a*x)+1/2*arctan(a*x)^3/c^2/(a^2*x^2+1)-1/2/c^2*arctan(a*x)^3*ln(a^2*x^2+1)-3/2/c^2*(-2/
3*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+2*I*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*I*
arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-1/8*arctan(a*x)*(I+a*x)/(a*x-I)-I*arctan(a*x)^2*(I+a*x)/(
8*a*x-8*I)-I*(a*x-I)/(16*a*x+16*I)-1/8*arctan(a*x)*(a*x-I)/(I+a*x)+1/6*I*arctan(a*x)^4+2/3*arctan(a*x)^3*ln((1
+I*a*x)^2/(a^2*x^2+1)-1)-2/3*arctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-4*I*polylog(4,(1+I*a*x)/(a^2*x^2+
1)^(1/2))-4*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*(I+a*x)/(16*a*x-16*I)-2/3*arctan(a*x)^3*ln(1
-(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*I*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*arctan(a*x)*polylog(3,(1+I*a*x)/(a
^2*x^2+1)^(1/2))+I*arctan(a*x)^2*(a*x-I)/(8*a*x+8*I)-1/6*(2*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-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)-2*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-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3-I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csg
n(I*(1+I*a*x)^2/(a^2*x^2+1))-2*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))^2-2*I*Pi*csg
n(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+2*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))^3-2*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+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+I*Pi*csgn(I*((1+I*a*x)^2/(
a^2*x^2+1)+1)^2)^3-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+2*I*Pi+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+2*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
*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))+2*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))^3-2*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+I*Pi*csg
n(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)+4*ln(2)-1)*arctan(a*x)^3)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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