\(\int \frac {\arctan (a x)^2}{x^2 (c+a^2 c x^2)^2} \, dx\) [296]

Optimal result

Integrand size = 22, antiderivative size = 177 \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2 x}{4 c^2 \left (1+a^2 x^2\right )}+\frac {a \arctan (a x)}{4 c^2}-\frac {a \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^2}{c^2}-\frac {\arctan (a x)^2}{c^2 x}-\frac {a^2 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)^3}{2 c^2}+\frac {2 a \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i a \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 177, 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, 5038, 4946, 5044, 4988, 2497, 5004, 5012, 5050, 205, 211} \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a^2 x \arctan (a x)^2}{2 c^2 \left (a^2 x^2+1\right )}-\frac {a \arctan (a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac {a^2 x}{4 c^2 \left (a^2 x^2+1\right )}-\frac {a \arctan (a x)^3}{2 c^2}-\frac {\arctan (a x)^2}{c^2 x}-\frac {i a \arctan (a x)^2}{c^2}+\frac {a \arctan (a x)}{4 c^2}+\frac {2 a \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i a \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c^2} \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = -\frac {a^2 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)^3}{6 c^2}+a^3 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\arctan (a x)^2}{x^2} \, dx}{c^2}-\frac {a^2 \int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx}{c} \\ & = -\frac {a \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^2}{c^2 x}-\frac {a^2 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)^3}{2 c^2}+\frac {1}{2} a^2 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(2 a) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^2} \\ & = \frac {a^2 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^2}{c^2}-\frac {\arctan (a x)^2}{c^2 x}-\frac {a^2 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)^3}{2 c^2}+\frac {(2 i a) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^2}+\frac {a^2 \int \frac {1}{c+a^2 c x^2} \, dx}{4 c} \\ & = \frac {a^2 x}{4 c^2 \left (1+a^2 x^2\right )}+\frac {a \arctan (a x)}{4 c^2}-\frac {a \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^2}{c^2}-\frac {\arctan (a x)^2}{c^2 x}-\frac {a^2 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)^3}{2 c^2}+\frac {2 a \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {\left (2 a^2\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2} \\ & = \frac {a^2 x}{4 c^2 \left (1+a^2 x^2\right )}+\frac {a \arctan (a x)}{4 c^2}-\frac {a \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^2}{c^2}-\frac {\arctan (a x)^2}{c^2 x}-\frac {a^2 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)^3}{2 c^2}+\frac {2 a \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i a \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.62 \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {4 a x \arctan (a x)^3+2 a x \arctan (a x) \left (\cos (2 \arctan (a x))-8 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+8 i a x \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )-a x \sin (2 \arctan (a x))+2 \arctan (a x)^2 (4+4 i a x+a x \sin (2 \arctan (a x)))}{8 c^2 x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.77

method	result	size
derivativedivides	\(a \left (-\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\arctan \left (a x \right )^{2}}{c^{2} a x}-\frac {-\arctan \left (a x \right )^{3}+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}+2}+\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-2 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a x}{4 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4}-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{c^{2}}\right )\)	\(313\)
default	\(a \left (-\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\arctan \left (a x \right )^{2}}{c^{2} a x}-\frac {-\arctan \left (a x \right )^{3}+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}+2}+\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-2 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a x}{4 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4}-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{c^{2}}\right )\)	\(313\)
parts	\(-\frac {a^{2} x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 a \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\arctan \left (a x \right )^{2}}{c^{2} x}-\frac {2 \left (-\frac {a \arctan \left (a x \right )^{3}}{2}-\frac {a \left (-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+2 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}+i \ln \left (a x \right ) \ln \left (i a x +1\right )-i \ln \left (a x \right ) \ln \left (-i a x +1\right )+i \operatorname {dilog}\left (i a x +1\right )-i \operatorname {dilog}\left (-i a x +1\right )-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}\right )}{2}\right )}{c^{2}}\)	\(317\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/32*(6*a^3*x^3*arctan2(1, a*x) - 6*a^2*x^2 + 8*(a^3*x^3 + a*x)*arctan(a*x)^3 + 12*a*x*arctan(a*x) + 4*(3*a^2
*x^2 + 2)*arctan(a*x)^2 + 6*a*x*arctan2(1, a*x) - (3*a^2*x^2 + 2)*log(a^2*x^2 + 1)^2 + 192*(a^6*c^2*x^3 + a^4*
c^2*x)*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x) - 128*(a^2*c^2*x^3 + c^2*x)
*integrate(1/64*(4*(a^2*x^2 + 1)^(7/2)*a^2*arctan(a*x)*log(a^2*x^2 + 1)*sin(6*arctan(a*x)) - 24*(a^2*x^2 + 1)^
3*a^2*arctan(a*x)*log(a^2*x^2 + 1)*sin(5*arctan(a*x)) + 52*(a^2*x^2 + 1)^(5/2)*a^2*arctan(a*x)*log(a^2*x^2 + 1
)*sin(4*arctan(a*x)) - 48*(a^2*x^2 + 1)^2*a^2*arctan(a*x)*log(a^2*x^2 + 1)*sin(3*arctan(a*x)) + 16*(a^2*x^2 +
1)^(3/2)*a^2*arctan(a*x)*log(a^2*x^2 + 1)*sin(2*arctan(a*x)) - (4*(a^2*x^2 + 1)^(7/2)*a^2*arctan(a*x)^2 - (a^2
*x^2 + 1)^(7/2)*a^2*log(a^2*x^2 + 1)^2)*cos(6*arctan(a*x)) + 6*(4*(a^2*x^2 + 1)^3*a^2*arctan(a*x)^2 - (a^2*x^2
+ 1)^3*a^2*log(a^2*x^2 + 1)^2)*cos(5*arctan(a*x)) - 13*(4*(a^2*x^2 + 1)^(5/2)*a^2*arctan(a*x)^2 - (a^2*x^2 +
1)^(5/2)*a^2*log(a^2*x^2 + 1)^2)*cos(4*arctan(a*x)) + 12*(4*(a^2*x^2 + 1)^2*a^2*arctan(a*x)^2 - (a^2*x^2 + 1)^
2*a^2*log(a^2*x^2 + 1)^2)*cos(3*arctan(a*x)) - 4*(4*(a^2*x^2 + 1)^(3/2)*a^2*arctan(a*x)^2 - (a^2*x^2 + 1)^(3/2
)*a^2*log(a^2*x^2 + 1)^2)*cos(2*arctan(a*x)))*sqrt(a^2*x^2 + 1)/((a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^6*cos(6*arc
tan(a*x))^2 + (a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^6*sin(6*arctan(a*x))^2 + 36*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^
5*cos(5*arctan(a*x))^2 + 36*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^5*sin(5*arctan(a*x))^2 + 169*(a^2*c^2*x^2 + c^2)
*(a^2*x^2 + 1)^4*cos(4*arctan(a*x))^2 + 169*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^4*sin(4*arctan(a*x))^2 + 144*(a^
2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^3*cos(3*arctan(a*x))^2 + 144*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^3*sin(3*arctan(a
*x))^2 - 96*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^(5/2)*cos(3*arctan(a*x))*cos(2*arctan(a*x)) - 96*(a^2*c^2*x^2 +
c^2)*(a^2*x^2 + 1)^(5/2)*sin(3*arctan(a*x))*sin(2*arctan(a*x)) + 16*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^2*cos(2*
arctan(a*x))^2 + 16*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^2*sin(2*arctan(a*x))^2 - 2*(6*(a^2*c^2*x^2 + c^2)*(a^2*x
^2 + 1)^(11/2)*cos(5*arctan(a*x)) - 13*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^5*cos(4*arctan(a*x)) + 12*(a^2*c^2*x^
2 + c^2)*(a^2*x^2 + 1)^(9/2)*cos(3*arctan(a*x)) - 4*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^4*cos(2*arctan(a*x)))*co
s(6*arctan(a*x)) - 12*(13*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^(9/2)*cos(4*arctan(a*x)) - 12*(a^2*c^2*x^2 + c^2)*
(a^2*x^2 + 1)^4*cos(3*arctan(a*x)) + 4*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^(7/2)*cos(2*arctan(a*x)))*cos(5*arcta
n(a*x)) - 104*(3*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^(7/2)*cos(3*arctan(a*x)) - (a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1
)^3*cos(2*arctan(a*x)))*cos(4*arctan(a*x)) - 2*(6*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^(11/2)*sin(5*arctan(a*x))
- 13*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^5*sin(4*arctan(a*x)) + 12*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^(9/2)*sin(3
*arctan(a*x)) - 4*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^4*sin(2*arctan(a*x)))*sin(6*arctan(a*x)) - 12*(13*(a^2*c^2
*x^2 + c^2)*(a^2*x^2 + 1)^(9/2)*sin(4*arctan(a*x)) - 12*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^4*sin(3*arctan(a*x))
+ 4*(a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^(7/2)*sin(2*arctan(a*x)))*sin(5*arctan(a*x)) - 104*(3*(a^2*c^2*x^2 + c^
2)*(a^2*x^2 + 1)^(7/2)*sin(3*arctan(a*x)) - (a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^3*sin(2*arctan(a*x)))*sin(4*arct
an(a*x))), x) - 192*(a^6*c^2*x^3 + a^4*c^2*x)*integrate(1/64*(4*x^2*arctan(a*x)^2 + x^2*log(a^2*x^2 + 1)^2)/(a
^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x) - 256*(a^2*c^2*x^3 + c^2*x)*integrate(1/64*(4*arctan(a*x)^2 + 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) - 192*(a^4*c^2*x^3 + a^2*c^2*x)*integrate(1/64*(4*arctan
(a*x)^2 + log(a^2*x^2 + 1)^2)/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x) - 256*(a^3*c^2*x^3 + a*c^2*x)*integrate(
1/16*arctan(a*x)/(a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x), x) + 128*(a^4*c^2*x^3 + a^2*c^2*x)*integrate(1/16*log(
a^2*x^2 + 1)/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x))/(a^2*c^2*x^3 + c^2*x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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