\(\int \frac {x^2 \arctan (a x)^2}{(c+a^2 c x^2)^3} \, dx\) [300]

Optimal result

Integrand size = 22, antiderivative size = 181 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{64 a^3 c^3}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3} \]

[Out]

1/32*x/a^2/c^3/(a^2*x^2+1)^2-1/64*x/a^2/c^3/(a^2*x^2+1)-1/64*arctan(a*x)/a^3/c^3-1/8*arctan(a*x)/a^3/c^3/(a^2*
x^2+1)^2+1/8*arctan(a*x)/a^3/c^3/(a^2*x^2+1)-1/4*x*arctan(a*x)^2/a^2/c^3/(a^2*x^2+1)^2+1/8*x*arctan(a*x)^2/a^2
/c^3/(a^2*x^2+1)+1/24*arctan(a*x)^3/a^3/c^3

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5084, 5012, 5050, 205, 211, 5020} \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan (a x)^3}{24 a^3 c^3}-\frac {\arctan (a x)}{64 a^3 c^3}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac {x}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {x}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {\arctan (a x)}{8 a^3 c^3 \left (a^2 x^2+1\right )^2} \]

[In]

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

[Out]

x/(32*a^2*c^3*(1 + a^2*x^2)^2) - x/(64*a^2*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(64*a^3*c^3) - ArcTan[a*x]/(8*a^3*
c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]/(8*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x]^2)/(4*a^2*c^3*(1 + a^2*x^2)^2) +
(x*ArcTan[a*x]^2)/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^3/(24*a^3*c^3)

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 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 5020

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(d + e*x^2)^(q +
1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d*(q + 1)^2)), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q +
1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[b^2*p*((p - 1)/(4*(q + 1)^2)), Int[(d + e*x^2)^q*(a + b*ArcTan[c*x])^(
p - 2), x], x] - Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d*(q + 1))), x]) /; FreeQ[{a, b, c, d, e
}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]

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 5084

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/e, Int[
x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*Arc
Tan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m
, 1] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx}{a^2}+\frac {\int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2 c} \\ & = -\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{2 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a^3 c^3}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a^2}-\frac {3 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a c} \\ & = \frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{2 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^2 c}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2 c}+\frac {3 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a c} \\ & = \frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {13 x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{64 a^2 c^2}-\frac {\int \frac {1}{c+a^2 c x^2} \, dx}{4 a^2 c^2}+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a^2 c} \\ & = \frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {13 \arctan (a x)}{64 a^3 c^3}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3}+\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{16 a^2 c^2} \\ & = \frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{64 a^3 c^3}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 a x-3 a^3 x^3-3 \left (1-6 a^2 x^2+a^4 x^4\right ) \arctan (a x)+24 a x \left (-1+a^2 x^2\right ) \arctan (a x)^2+8 \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{192 a^3 c^3 \left (1+a^2 x^2\right )^2} \]

[In]

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

[Out]

(3*a*x - 3*a^3*x^3 - 3*(1 - 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x] + 24*a*x*(-1 + a^2*x^2)*ArcTan[a*x]^2 + 8*(1 + a^
2*x^2)^2*ArcTan[a*x]^3)/(192*a^3*c^3*(1 + a^2*x^2)^2)

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.68

method	result	size
parallelrisch	\(\frac {8 a^{4} \arctan \left (a x \right )^{3} x^{4}-3 \arctan \left (a x \right ) a^{4} x^{4}+24 a^{3} \arctan \left (a x \right )^{2} x^{3}+16 \arctan \left (a x \right )^{3} x^{2} a^{2}-3 a^{3} x^{3}+18 a^{2} \arctan \left (a x \right ) x^{2}-24 a \arctan \left (a x \right )^{2} x +8 \arctan \left (a x \right )^{3}+3 a x -3 \arctan \left (a x \right )}{192 c^{3} \left (a^{2} x^{2}+1\right )^{2} a^{3}}\)	\(123\)
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}+\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{4 c^{3}}}{a^{3}}\)	\(149\)
default	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}+\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{4 c^{3}}}{a^{3}}\)	\(149\)
parts	\(\frac {\arctan \left (a x \right )^{2} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {x \arctan \left (a x \right )^{2}}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 a^{3} c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3 a^{3}}+\frac {\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{a^{3}}}{4 c^{3}}\)	\(155\)
risch	\(\frac {i \ln \left (i a x +1\right )^{3}}{192 a^{3} c^{3}}-\frac {i \left (x^{4} \ln \left (-i a x +1\right ) a^{4}+2 a^{2} x^{2} \ln \left (-i a x +1\right )-2 i a^{3} x^{3}+\ln \left (-i a x +1\right )+2 i a x \right ) \ln \left (i a x +1\right )^{2}}{64 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {i \left (a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+2 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-4 i x^{3} \ln \left (-i a x +1\right ) a^{3}-4 a^{2} x^{2}+\ln \left (-i a x +1\right )^{2}+4 i a x \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{64 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}+\frac {i \left (-2 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}+12 i a^{3} x^{3} \ln \left (-i a x +1\right )^{2}+3 \ln \left (-i a x -1\right ) a^{4} x^{4}-3 \ln \left (i a x -1\right ) a^{4} x^{4}+6 i a^{3} x^{3}-4 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-12 i a x \ln \left (-i a x +1\right )^{2}+6 \ln \left (-i a x -1\right ) a^{2} x^{2}-6 \ln \left (i a x -1\right ) a^{2} x^{2}+24 a^{2} x^{2} \ln \left (-i a x +1\right )-6 i a x -2 \ln \left (-i a x +1\right )^{3}+3 \ln \left (-i a x -1\right )-3 \ln \left (i a x -1\right )\right )}{384 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}\)	\(444\)

[In]

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

[Out]

1/192*(8*a^4*arctan(a*x)^3*x^4-3*arctan(a*x)*a^4*x^4+24*a^3*arctan(a*x)^2*x^3+16*arctan(a*x)^3*x^2*a^2-3*a^3*x
^3+18*a^2*arctan(a*x)*x^2-24*a*arctan(a*x)^2*x+8*arctan(a*x)^3+3*a*x-3*arctan(a*x))/c^3/(a^2*x^2+1)^2/a^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{2} - 3 \, a x + 3 \, {\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{192 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]

[In]

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

[Out]

-1/192*(3*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 - 24*(a^3*x^3 - a*x)*arctan(a*x)^2 - 3*a*x + 3*(
a^4*x^4 - 6*a^2*x^2 + 1)*arctan(a*x))/(a^7*c^3*x^4 + 2*a^5*c^3*x^2 + a^3*c^3)

Sympy [F]

\[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(x**2*atan(a*x)**2/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.28 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{8} \, {\left (\frac {a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac {\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac {{\left (3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{192 \, {\left (a^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}\right )}} + \frac {{\left (a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )}{8 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]

[In]

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

[Out]

1/8*((a^2*x^3 - x)/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3) + arctan(a*x)/(a^3*c^3))*arctan(a*x)^2 - 1/192*(3*a
^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 - 3*a*x + 3*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x))*a^2/(a^9
*c^3*x^4 + 2*a^7*c^3*x^2 + a^5*c^3) + 1/8*(a^2*x^2 - (a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2)*a*arctan(a*x)/(a
^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {x}{8\,a^2}-\frac {x^3}{8}}{8\,a^4\,c^3\,x^4+16\,a^2\,c^3\,x^2+8\,c^3}-\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {x}{8\,a^4\,c^3}-\frac {x^3}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {\mathrm {atan}\left (a\,x\right )}{64\,a^3\,c^3}+\frac {{\mathrm {atan}\left (a\,x\right )}^3}{24\,a^3\,c^3}+\frac {x^2\,\mathrm {atan}\left (a\,x\right )}{8\,a^3\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )} \]

[In]

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

[Out]

(x/(8*a^2) - x^3/8)/(8*c^3 + 16*a^2*c^3*x^2 + 8*a^4*c^3*x^4) - (atan(a*x)^2*(x/(8*a^4*c^3) - x^3/(8*a^2*c^3)))
/(1/a^2 + 2*x^2 + a^2*x^4) - atan(a*x)/(64*a^3*c^3) + atan(a*x)^3/(24*a^3*c^3) + (x^2*atan(a*x))/(8*a^3*c^3*(1
/a^2 + 2*x^2 + a^2*x^4))