\(\int \frac {x^2 \arctan (a x)^3}{(c+a^2 c x^2)^3} \, dx\) [405]

Optimal result

Integrand size = 22, antiderivative size = 237 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \arctan (a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^2}{128 a^3 c^3}-\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{32 a^3 c^3} \]

[Out]

3/128/a^3/c^3/(a^2*x^2+1)^2-3/128/a^3/c^3/(a^2*x^2+1)+3/32*x*arctan(a*x)/a^2/c^3/(a^2*x^2+1)^2-3/64*x*arctan(a
*x)/a^2/c^3/(a^2*x^2+1)-3/128*arctan(a*x)^2/a^3/c^3-3/16*arctan(a*x)^2/a^3/c^3/(a^2*x^2+1)^2+3/16*arctan(a*x)^
2/a^3/c^3/(a^2*x^2+1)-1/4*x*arctan(a*x)^3/a^2/c^3/(a^2*x^2+1)^2+1/8*x*arctan(a*x)^3/a^2/c^3/(a^2*x^2+1)+1/32*a
rctan(a*x)^4/a^3/c^3

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 237, 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, 267, 5020, 5016} \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan (a x)^4}{32 a^3 c^3}-\frac {3 \arctan (a x)^2}{128 a^3 c^3}+\frac {x \arctan (a x)^3}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac {x \arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 x \arctan (a x)}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {3 x \arctan (a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )^2}-\frac {3}{128 a^3 c^3 \left (a^2 x^2+1\right )}+\frac {3}{128 a^3 c^3 \left (a^2 x^2+1\right )^2} \]

[In]

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

[Out]

3/(128*a^3*c^3*(1 + a^2*x^2)^2) - 3/(128*a^3*c^3*(1 + a^2*x^2)) + (3*x*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)^
2) - (3*x*ArcTan[a*x])/(64*a^2*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(128*a^3*c^3) - (3*ArcTan[a*x]^2)/(16*a^
3*c^3*(1 + a^2*x^2)^2) + (3*ArcTan[a*x]^2)/(16*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x]^3)/(4*a^2*c^3*(1 + a^2*
x^2)^2) + (x*ArcTan[a*x]^3)/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^4/(32*a^3*c^3)

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 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 5016

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

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)^3}{\left (c+a^2 c x^2\right )^3} \, dx}{a^2}+\frac {\int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2 c} \\ & = -\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {x \arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{2 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a^3 c^3}+\frac {3 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a^2}-\frac {3 \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a c} \\ & = \frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)^2}{4 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{32 a^3 c^3}+\frac {9 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^2 c}-\frac {3 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2 c}+\frac {9 \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a c} \\ & = \frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {39 x \arctan (a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {39 \arctan (a x)^2}{128 a^3 c^3}-\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{32 a^3 c^3}+\frac {9 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a^2 c}-\frac {9 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{64 a c}+\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a c} \\ & = \frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {39}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \arctan (a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^2}{128 a^3 c^3}-\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{32 a^3 c^3}-\frac {9 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c} \\ & = \frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \arctan (a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^2}{128 a^3 c^3}-\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{32 a^3 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.47 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-3 a^2 x^2+\left (6 a x-6 a^3 x^3\right ) \arctan (a x)-3 \left (1-6 a^2 x^2+a^4 x^4\right ) \arctan (a x)^2+16 a x \left (-1+a^2 x^2\right ) \arctan (a x)^3+4 \left (1+a^2 x^2\right )^2 \arctan (a x)^4}{128 a^3 c^3 \left (1+a^2 x^2\right )^2} \]

[In]

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

[Out]

(-3*a^2*x^2 + (6*a*x - 6*a^3*x^3)*ArcTan[a*x] - 3*(1 - 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x]^2 + 16*a*x*(-1 + a^2*x
^2)*ArcTan[a*x]^3 + 4*(1 + a^2*x^2)^2*ArcTan[a*x]^4)/(128*a^3*c^3*(1 + a^2*x^2)^2)

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.61

method	result	size
parallelrisch	\(\frac {4 a^{4} x^{4} \arctan \left (a x \right )^{4}-3 a^{4} \arctan \left (a x \right )^{2} x^{4}+16 \arctan \left (a x \right )^{3} a^{3} x^{3}+8 \arctan \left (a x \right )^{4} x^{2} a^{2}-6 \arctan \left (a x \right ) x^{3} a^{3}+18 x^{2} \arctan \left (a x \right )^{2} a^{2}-16 \arctan \left (a x \right )^{3} a x -3 a^{2} x^{2}+4 \arctan \left (a x \right )^{4}+6 x \arctan \left (a x \right ) a -3 \arctan \left (a x \right )^{2}}{128 c^{3} \left (a^{2} x^{2}+1\right )^{2} a^{3}}\)	\(145\)
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{3} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{3} a x}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{4}}{8 c^{3}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}+\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right ) a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{2}}{16}-\frac {1}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {1}{16 a^{2} x^{2}+16}\right )}{8 c^{3}}}{a^{3}}\)	\(197\)
default	\(\frac {\frac {\arctan \left (a x \right )^{3} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{3} a x}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{4}}{8 c^{3}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}+\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right ) a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{2}}{16}-\frac {1}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {1}{16 a^{2} x^{2}+16}\right )}{8 c^{3}}}{a^{3}}\)	\(197\)
parts	\(\frac {\arctan \left (a x \right )^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {x \arctan \left (a x \right )^{3}}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{4}}{8 a^{3} c^{3}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4 a^{3}}+\frac {\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right ) a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{2}}{16}-\frac {1}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {1}{16 a^{2} x^{2}+16}}{a^{3}}\right )}{8 c^{3}}\)	\(203\)
risch	\(\frac {\ln \left (i a x +1\right )^{4}}{512 a^{3} c^{3}}-\frac {\left (x^{4} \ln \left (-i a x +1\right ) a^{4}+2 a^{2} x^{2} \ln \left (-i a x +1\right )-2 i a^{3} x^{3}+\ln \left (-i a x +1\right )+2 i a x \right ) \ln \left (i a x +1\right )^{3}}{128 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}-8 i x^{3} \ln \left (-i a x +1\right ) a^{3}+a^{4} x^{4}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+8 i a x \ln \left (-i a x +1\right )-6 a^{2} x^{2}+2 \ln \left (-i a x +1\right )^{2}+1\right ) \ln \left (i a x +1\right )^{2}}{512 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}-\frac {\left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}-12 i x^{3} \ln \left (-i a x +1\right )^{2} a^{3}+3 x^{4} \ln \left (-i a x +1\right ) a^{4}-6 i a^{3} x^{3}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+12 i a x \ln \left (-i a x +1\right )^{2}-18 a^{2} x^{2} \ln \left (-i a x +1\right )+6 i a x +2 \ln \left (-i a x +1\right )^{3}+3 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{256 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}+\frac {a^{4} x^{4} \ln \left (-i a x +1\right )^{4}+3 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+2 a^{2} x^{2} \ln \left (-i a x +1\right )^{4}-8 i a^{3} x^{3} \ln \left (-i a x +1\right )^{3}-18 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-12 i x^{3} \ln \left (-i a x +1\right ) a^{3}-12 a^{2} x^{2}+\ln \left (-i a x +1\right )^{4}+8 i a x \ln \left (-i a x +1\right )^{3}+3 \ln \left (-i a x +1\right )^{2}+12 i a x \ln \left (-i a x +1\right )}{512 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}\)	\(593\)

[In]

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

[Out]

1/128*(4*a^4*x^4*arctan(a*x)^4-3*a^4*arctan(a*x)^2*x^4+16*arctan(a*x)^3*a^3*x^3+8*arctan(a*x)^4*x^2*a^2-6*arct
an(a*x)*x^3*a^3+18*x^2*arctan(a*x)^2*a^2-16*arctan(a*x)^3*a*x-3*a^2*x^2+4*arctan(a*x)^4+6*x*arctan(a*x)*a-3*ar
ctan(a*x)^2)/c^3/(a^2*x^2+1)^2/a^3

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.55 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {4 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 3 \, a^{2} x^{2} + 16 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{3} - 3 \, {\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )}{128 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]

[In]

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

[Out]

1/128*(4*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^4 - 3*a^2*x^2 + 16*(a^3*x^3 - a*x)*arctan(a*x)^3 - 3*(a^4*x^4 -
6*a^2*x^2 + 1)*arctan(a*x)^2 - 6*(a^3*x^3 - a*x)*arctan(a*x))/(a^7*c^3*x^4 + 2*a^5*c^3*x^2 + a^3*c^3)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*atan(a*x)**3/(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 = 0.35 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.41 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{8} \, {\left (\frac {a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac {\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{3} + \frac {3 \, {\left (a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )^{2}}{16 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} - \frac {1}{128} \, {\left (\frac {{\left (4 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a^{2}}{a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}} + \frac {2 \, {\left (3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}}\right )} a \]

[In]

integrate(x^2*arctan(a*x)^3/(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)^3 + 3/16*(a^2*
x^2 - (a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2)*a*arctan(a*x)^2/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) - 1/128
*((4*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^4 + 3*a^2*x^2 - 3*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2)*a^2/(a^1
0*c^3*x^4 + 2*a^8*c^3*x^2 + a^6*c^3) + 2*(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*arctan(a*x)/(a^9*c^3*x^4 + 2*a^7*c^3*x^2 + a^5*c^3))*a

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3\,x}{64\,a^4\,c^3}-\frac {3\,x^3}{64\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {3\,x^2}{2\,\left (64\,a^5\,c^3\,x^4+128\,a^3\,c^3\,x^2+64\,a\,c^3\right )}-{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {3}{128\,a^3\,c^3}-\frac {3\,x^2}{16\,a^3\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )}\right )-\frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {x}{8\,a^4\,c^3}-\frac {x^3}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}+\frac {{\mathrm {atan}\left (a\,x\right )}^4}{32\,a^3\,c^3} \]

[In]

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

[Out]

(atan(a*x)*((3*x)/(64*a^4*c^3) - (3*x^3)/(64*a^2*c^3)))/(1/a^2 + 2*x^2 + a^2*x^4) - (3*x^2)/(2*(64*a*c^3 + 128
*a^3*c^3*x^2 + 64*a^5*c^3*x^4)) - atan(a*x)^2*(3/(128*a^3*c^3) - (3*x^2)/(16*a^3*c^3*(1/a^2 + 2*x^2 + a^2*x^4)
)) - (atan(a*x)^3*(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)^4/(32*a^3*c^3)