\(\int \frac {\arctan (a x)^3}{x^2 (c+a^2 c x^2)^2} \, dx\) [401]

Optimal result

Integrand size = 22, antiderivative size = 234 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {3 a}{8 c^2 \left (1+a^2 x^2\right )}+\frac {3 a^2 x \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}+\frac {3 a \arctan (a x)^2}{8 c^2}-\frac {3 a \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^3}{c^2}-\frac {\arctan (a x)^3}{c^2 x}-\frac {a^2 x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \arctan (a x)^4}{8 c^2}+\frac {3 a \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}+\frac {3 a \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5086, 5038, 4946, 5044, 4988, 5004, 5112, 6745, 5012, 5050, 267} \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a^2 x \arctan (a x)^3}{2 c^2 \left (a^2 x^2+1\right )}-\frac {3 a \arctan (a x)^2}{4 c^2 \left (a^2 x^2+1\right )}+\frac {3 a^2 x \arctan (a x)}{4 c^2 \left (a^2 x^2+1\right )}+\frac {3 a}{8 c^2 \left (a^2 x^2+1\right )}-\frac {3 i a \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c^2}-\frac {3 a \arctan (a x)^4}{8 c^2}-\frac {\arctan (a x)^3}{c^2 x}-\frac {i a \arctan (a x)^3}{c^2}+\frac {3 a \arctan (a x)^2}{8 c^2}+\frac {3 a \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {3 a \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 c^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a \left (-2 i \pi ^3+16 i \arctan (a x)^3-\frac {16 \arctan (a x)^3}{a x}-6 \arctan (a x)^4+3 \cos (2 \arctan (a x))-6 \arctan (a x)^2 \cos (2 \arctan (a x))+48 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+48 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+24 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+6 \arctan (a x) \sin (2 \arctan (a x))-4 \arctan (a x)^3 \sin (2 \arctan (a x))\right )}{16 c^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 77.72 (sec) , antiderivative size = 1731, normalized size of antiderivative = 7.40

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1731\)
default	\(\text {Expression too large to display}\)	\(1731\)
parts	\(\text {Expression too large to display}\)	\(1735\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/2048*(240*(a^3*x^3 + a*x)*arctan(a*x)^4 - 9*(a^3*x^3 + a*x)*log(a^2*x^2 + 1)^4 + 128*(3*a^2*x^2 + 2)*arctan
(a*x)^3 - 24*(3*(a^3*x^3 + a*x)*arctan(a*x)^2 + 4*(3*a^2*x^2 + 2)*arctan(a*x))*log(a^2*x^2 + 1)^2 - 4*(a^2*c^2
*x^3 + c^2*x)*(72*a^5*(a^2/(a^8*c^2*x^2 + a^6*c^2) + log(a^2*x^2 + 1)/(a^6*c^2*x^2 + a^4*c^2)) - 18432*a^5*int
egrate(1/256*x^5*arctan(a*x)^2*log(a^2*x^2 + 1)/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) - 4608*a^5*integra
te(1/256*x^5*log(a^2*x^2 + 1)^3/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 36864*a^4*integrate(1/256*x^4*ar
ctan(a*x)^3/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 9216*a^4*integrate(1/256*x^4*arctan(a*x)*log(a^2*x^2
+ 1)^2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) - 73728*a^4*integrate(1/256*x^4*arctan(a*x)*log(a^2*x^2 +
1)/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 9*a^3*log(a^2*x^2 + 1)^3/(a^4*c^2*x^2 + a^2*c^2) + 27*(2*a^4*
(a^2/(a^10*c^2*x^2 + a^8*c^2) + log(a^2*x^2 + 1)/(a^8*c^2*x^2 + a^6*c^2)) + a^2*log(a^2*x^2 + 1)^2/(a^6*c^2*x^
2 + a^4*c^2))*a^3 - 18432*a^3*integrate(1/256*x^3*arctan(a*x)^2*log(a^2*x^2 + 1)/(a^4*c^2*x^6 + 2*a^2*c^2*x^4
+ c^2*x^2), x) + 73728*a^3*integrate(1/256*x^3*arctan(a*x)^2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 36*
a^3*log(a^2*x^2 + 1)^2/(a^4*c^2*x^2 + a^2*c^2) + 36864*a^2*integrate(1/256*x^2*arctan(a*x)^3/(a^4*c^2*x^6 + 2*
a^2*c^2*x^4 + c^2*x^2), x) + 9216*a^2*integrate(1/256*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^4*c^2*x^6 + 2*a^2*
c^2*x^4 + c^2*x^2), x) - 49152*a^2*integrate(1/256*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^4*c^2*x^6 + 2*a^2*c^2*x
^4 + c^2*x^2), x) + 49152*a*integrate(1/256*x*arctan(a*x)^2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) - 1228
8*a*integrate(1/256*x*log(a^2*x^2 + 1)^2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 114688*integrate(1/256*
arctan(a*x)^3/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x) + 12288*integrate(1/256*arctan(a*x)*log(a^2*x^2 + 1)
^2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x)))/(a^2*c^2*x^3 + c^2*x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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