\(\int \frac {\arctan (a x)^2}{x^3 (c+a^2 c x^2)^2} \, dx\) [297]

Optimal result

Integrand size = 22, antiderivative size = 250 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)}{c^2 x}+\frac {a^3 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^2}{4 c^2}-\frac {\arctan (a x)^2}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {2 i a^2 \arctan (a x)^3}{3 c^2}+\frac {a^2 \log (x)}{c^2}-\frac {a^2 \log \left (1+a^2 x^2\right )}{2 c^2}-\frac {2 a^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {2 i a^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}-\frac {a^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {5086, 5038, 4946, 272, 36, 29, 31, 5004, 5044, 4988, 5112, 6745, 5050, 5012, 267} \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {2 i a^2 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c^2}-\frac {a^2 \arctan (a x)^2}{2 c^2 \left (a^2 x^2+1\right )}+\frac {2 i a^2 \arctan (a x)^3}{3 c^2}-\frac {a^2 \arctan (a x)^2}{4 c^2}-\frac {2 a^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {a^2 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{c^2}+\frac {a^2}{4 c^2 \left (a^2 x^2+1\right )}-\frac {a^2 \log \left (a^2 x^2+1\right )}{2 c^2}+\frac {a^2 \log (x)}{c^2}+\frac {a^3 x \arctan (a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac {\arctan (a x)^2}{2 c^2 x^2}-\frac {a \arctan (a x)}{c^2 x} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.64 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2 \left (-2 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+\frac {1}{24} \left (2 i \pi ^3-16 i \arctan (a x)^3+3 \cos (2 \arctan (a x))+6 \arctan (a x)^2 \left (-2-\frac {2}{a^2 x^2}-\cos (2 \arctan (a x))-8 \log \left (1-e^{-2 i \arctan (a x)}\right )\right )+24 \log (a x)-12 \log \left (1+a^2 x^2\right )+\frac {6 \arctan (a x) (-4+a x \sin (2 \arctan (a x)))}{a x}\right )\right )}{c^2} \]

[In]

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

[Out]

(a^2*((-2*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - PolyLog[3, E^((-2*I)*ArcTan[a*x])] + ((2*I)*Pi^3
- (16*I)*ArcTan[a*x]^3 + 3*Cos[2*ArcTan[a*x]] + 6*ArcTan[a*x]^2*(-2 - 2/(a^2*x^2) - Cos[2*ArcTan[a*x]] - 8*Lo
g[1 - E^((-2*I)*ArcTan[a*x])]) + 24*Log[a*x] - 12*Log[1 + a^2*x^2] + (6*ArcTan[a*x]*(-4 + a*x*Sin[2*ArcTan[a*x
]]))/(a*x))/24))/c^2

Maple [C] (warning: unable to verify)

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

Time = 85.27 (sec) , antiderivative size = 4110, normalized size of antiderivative = 16.44

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4110\)
default	\(\text {Expression too large to display}\)	\(4110\)
parts	\(\text {Expression too large to display}\)	\(5354\)

[In]

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

[Out]

a^2*(-1/2*arctan(a*x)^2/c^2/(a^2*x^2+1)+1/c^2*arctan(a*x)^2*ln(a^2*x^2+1)-1/2/c^2*arctan(a*x)^2/a^2/x^2-2/c^2*
arctan(a*x)^2*ln(a*x)-1/c^2*(2*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-2*arctan(a*x)^2*ln((1+I*a*x)^2/(a
^2*x^2+1)-1)-1/48/a/x/(I+a*x)*(-24*a^2*arctan(a*x)*x^2-24*I*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(
a^2*x^2+1)+I)^3*Pi*arctan(a*x)^2*a*x-3*a^3*x^3-48*I*arctan(a*x)*a*x-96*ln(2)*arctan(a*x)^2*a*x-96*ln(2)*arctan
(a*x)^2*a^3*x^3-48*arctan(a*x)+9*a*x-12*a^3*arctan(a*x)^2*x^3-12*a*arctan(a*x)^2*x+24*I*csgn(I*(1+I*a*x)^2/(a^
2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^
2)*arctan(a*x)^2*a^3*x^3-48*I*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)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)-I)*arctan(a*x)^2*a^3*x^3+24*I*
csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*Pi*csgn(I/((1+I*
a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2*a*x-48*I*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)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)-I)*arctan(a*
x)^2*a*x+32*I*arctan(a*x)^3*a^3*x^3+32*I*arctan(a*x)^3*a*x+48*I*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2
/(a^2*x^2+1)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-1/((
1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2*a^3*x^3-24*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^4
/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)*Pi*arctan(a*x)^2*a^3*x^3+48*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*
csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*Pi*arctan(a*x)^2*a^3*x^3-48*I*csgn(I/((1+I*a
*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)
+1)*(1+I*a*x)^2/(a^2*x^2+1)-1/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2*a^3*x^3-24*I*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*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*Pi*arctan(a*x)^2*a*x-24*I*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*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2*a*x
-48*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*a*x+24*I*csgn(I*(
1+I*a*x)^2/(a^2*x^2+1))*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*arctan(a*x)^2*a*x+48*I*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)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi*a
rctan(a*x)^2*a*x+48*I*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-I/((1+I*a*x)^2/(a^2*x^2+1)+1)
)^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)-I)*arctan(a*x)^2*a*x+48*I*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2
/(a^2*x^2+1)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-1/((
1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2*a*x-24*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^4/(a^
2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)*Pi*arctan(a*x)^2*a*x+48*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*csgn(I*(
1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*Pi*arctan(a*x)^2*a*x-48*I*csgn(I/((1+I*a*x)^2/(a^2*x
^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x
)^2/(a^2*x^2+1)-1/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2*a*x-24*I*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*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*Pi*arctan(a*x)^2*a^3*x^3-24*I*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*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2*a^3*x^3-48*I*cs
gn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*a^3*x^3+24*I*csgn(I*(1+I*
a*x)^2/(a^2*x^2+1))*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*arctan(a*x)^2*a^3*x^3+48*I*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)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi*a
rctan(a*x)^2*a^3*x^3+48*I*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-I/((1+I*a*x)^2/(a^2*x^2+1
)+1))^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)-I)*arctan(a*x)^2*a^3*x^3-48*I*arctan(a*x)*a^3*x^3-48*I*Pi*arctan(a*x
)^2*a^3*x^3-48*I*Pi*arctan(a*x)^2*a*x-48*I*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-I/((1+I*
a*x)^2/(a^2*x^2+1)+1))^3*Pi*arctan(a*x)^2*a*x-48*I*Pi*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+
1)-1/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*arctan(a*x)^2*a*x+48*I*Pi*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(
a^2*x^2+1)-1/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2*a*x+24*I*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*Pi*arctan(a*x)^2*a^3*x^3+24*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*Pi*arctan(a*x)^2*a^3*x^3-
48*I*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-I/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*Pi*arctan(a*x
)^2*a^3*x^3-24*I*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^3*Pi*arctan(a*x)^2*a^3*x^3-48
*I*Pi*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-1/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*arctan(a*x)^
2*a^3*x^3+48*I*Pi*csgn(1/((1+I*a*x)^2/(a^2*x^2+1)+1)*(1+I*a*x)^2/(a^2*x^2+1)-1/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*
arctan(a*x)^2*a^3*x^3+24*I*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*Pi*arctan(a*x)^2*a*
x+24*I*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*Pi*arctan(a*x)^2*a*x)/(a*x-I)-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-ln((1
+I*a*x)/(a^2*x^2+1)^(1/2)+1)+2*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-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))+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))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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