\(\int \frac {x \arctan (a x)^3}{(c+a^2 c x^2)^3} \, dx\) [406]

Optimal result

Integrand size = 20, antiderivative size = 208 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a c^3 \left (1+a^2 x^2\right )}-\frac {45 \arctan (a x)}{256 a^2 c^3}+\frac {3 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \arctan (a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^3}{32 a^2 c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2} \]

[Out]

-3/128*x/a/c^3/(a^2*x^2+1)^2-45/256*x/a/c^3/(a^2*x^2+1)-45/256*arctan(a*x)/a^2/c^3+3/32*arctan(a*x)/a^2/c^3/(a
^2*x^2+1)^2+9/32*arctan(a*x)/a^2/c^3/(a^2*x^2+1)+3/16*x*arctan(a*x)^2/a/c^3/(a^2*x^2+1)^2+9/32*x*arctan(a*x)^2
/a/c^3/(a^2*x^2+1)+3/32*arctan(a*x)^3/a^2/c^3-1/4*arctan(a*x)^3/a^2/c^3/(a^2*x^2+1)^2

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5050, 5020, 5012, 205, 211} \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 x \arctan (a x)^2}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {3 x \arctan (a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \arctan (a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {3 \arctan (a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^3}{32 a^2 c^3}-\frac {45 \arctan (a x)}{256 a^2 c^3}-\frac {45 x}{256 a c^3 \left (a^2 x^2+1\right )}-\frac {3 x}{128 a c^3 \left (a^2 x^2+1\right )^2} \]

[In]

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

[Out]

(-3*x)/(128*a*c^3*(1 + a^2*x^2)^2) - (45*x)/(256*a*c^3*(1 + a^2*x^2)) - (45*ArcTan[a*x])/(256*a^2*c^3) + (3*Ar
cTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)) + (3*x*ArcTan[a*x]^2)/(16
*a*c^3*(1 + a^2*x^2)^2) + (9*x*ArcTan[a*x]^2)/(32*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^3)/(32*a^2*c^3) - ArcT
an[a*x]^3/(4*a^2*c^3*(1 + a^2*x^2)^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 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx}{4 a} \\ & = \frac {3 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx}{32 a}+\frac {9 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c} \\ & = -\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \arctan (a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^3}{32 a^2 c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-\frac {9 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{128 a c} \\ & = -\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {9 x}{256 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \arctan (a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^3}{32 a^2 c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \int \frac {1}{c+a^2 c x^2} \, dx}{256 a c^2}-\frac {9 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a c} \\ & = -\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a c^3 \left (1+a^2 x^2\right )}-\frac {9 \arctan (a x)}{256 a^2 c^3}+\frac {3 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \arctan (a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^3}{32 a^2 c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \int \frac {1}{c+a^2 c x^2} \, dx}{64 a c^2} \\ & = -\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a c^3 \left (1+a^2 x^2\right )}-\frac {45 \arctan (a x)}{256 a^2 c^3}+\frac {3 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \arctan (a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^3}{32 a^2 c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.50 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-3 a x \left (17+15 a^2 x^2\right )-3 \left (-17+6 a^2 x^2+15 a^4 x^4\right ) \arctan (a x)+24 a x \left (5+3 a^2 x^2\right ) \arctan (a x)^2+8 \left (-5+6 a^2 x^2+3 a^4 x^4\right ) \arctan (a x)^3}{256 c^3 \left (a+a^3 x^2\right )^2} \]

[In]

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

[Out]

(-3*a*x*(17 + 15*a^2*x^2) - 3*(-17 + 6*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x] + 24*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x]^
2 + 8*(-5 + 6*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x]^3)/(256*c^3*(a + a^3*x^2)^2)

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.59

method	result	size
parallelrisch	\(\frac {24 a^{4} \arctan \left (a x \right )^{3} x^{4}-45 \arctan \left (a x \right ) a^{4} x^{4}+72 a^{3} \arctan \left (a x \right )^{2} x^{3}+48 \arctan \left (a x \right )^{3} x^{2} a^{2}-45 a^{3} x^{3}-18 a^{2} \arctan \left (a x \right ) x^{2}+120 a \arctan \left (a x \right )^{2} x -40 \arctan \left (a x \right )^{3}-51 a x +51 \arctan \left (a x \right )}{256 c^{3} \left (a^{2} x^{2}+1\right )^{2} a^{2}}\)	\(123\)
derivativedivides	\(\frac {-\frac {\arctan \left (a x \right )^{3}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {3 \arctan \left (a x \right )^{2} a x}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {9 a \arctan \left (a x \right )^{2} x}{32 \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{3}}{32}+\frac {9 \arctan \left (a x \right )}{32 \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )}{32 \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \left (\frac {15}{8} a^{3} x^{3}+\frac {17}{8} a x \right )}{32 \left (a^{2} x^{2}+1\right )^{2}}-\frac {45 \arctan \left (a x \right )}{256}}{c^{3}}}{a^{2}}\)	\(150\)
default	\(\frac {-\frac {\arctan \left (a x \right )^{3}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {3 \arctan \left (a x \right )^{2} a x}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {9 a \arctan \left (a x \right )^{2} x}{32 \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{3}}{32}+\frac {9 \arctan \left (a x \right )}{32 \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )}{32 \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \left (\frac {15}{8} a^{3} x^{3}+\frac {17}{8} a x \right )}{32 \left (a^{2} x^{2}+1\right )^{2}}-\frac {45 \arctan \left (a x \right )}{256}}{c^{3}}}{a^{2}}\)	\(150\)
parts	\(-\frac {\arctan \left (a x \right )^{3}}{4 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {3 \arctan \left (a x \right )^{2} a x}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {9 a \arctan \left (a x \right )^{2} x}{32 \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{3}}{32}+\frac {9 \arctan \left (a x \right )}{32 \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )}{32 \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \left (\frac {15}{8} a^{3} x^{3}+\frac {17}{8} a x \right )}{32 \left (a^{2} x^{2}+1\right )^{2}}-\frac {45 \arctan \left (a x \right )}{256}}{a^{2} c^{3}}\)	\(152\)
risch	\(\frac {i \left (3 a^{4} x^{4}+6 a^{2} x^{2}-5\right ) \ln \left (i a x +1\right )^{3}}{256 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 i \left (-5 \ln \left (-i a x +1\right )+3 x^{4} \ln \left (-i a x +1\right ) a^{4}+6 a^{2} x^{2} \ln \left (-i a x +1\right )-6 i a^{3} x^{3}-10 i a x \right ) \ln \left (i a x +1\right )^{2}}{256 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{2}}+\frac {3 i \left (3 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+6 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-5 \ln \left (-i a x +1\right )^{2}-12 i x^{3} \ln \left (-i a x +1\right ) a^{3}-20 i a x \ln \left (-i a x +1\right )-12 a^{2} x^{2}-16\right ) \ln \left (i a x +1\right )}{256 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{2}}+\frac {i \left (72 a^{2} x^{2} \ln \left (-i a x +1\right )+96 \ln \left (-i a x +1\right )-6 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}-12 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+10 \ln \left (-i a x +1\right )^{3}+36 i x^{3} \ln \left (-i a x +1\right )^{2} a^{3}+60 i a x \ln \left (-i a x +1\right )^{2}+45 \ln \left (-a x +i\right ) a^{4} x^{4}+90 \ln \left (-a x +i\right ) a^{2} x^{2}+45 \ln \left (-a x +i\right )-45 \ln \left (a x +i\right ) a^{4} x^{4}-90 \ln \left (a x +i\right ) a^{2} x^{2}-45 \ln \left (a x +i\right )+90 i a^{3} x^{3}+102 i a x \right )}{512 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{2}}\)	\(492\)

[In]

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

[Out]

1/256*(24*a^4*arctan(a*x)^3*x^4-45*arctan(a*x)*a^4*x^4+72*a^3*arctan(a*x)^2*x^3+48*arctan(a*x)^3*x^2*a^2-45*a^
3*x^3-18*a^2*arctan(a*x)*x^2+120*a*arctan(a*x)^2*x-40*arctan(a*x)^3-51*a*x+51*arctan(a*x))/c^3/(a^2*x^2+1)^2/a
^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.56 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {45 \, a^{3} x^{3} - 8 \, {\left (3 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 5\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{2} + 51 \, a x + 3 \, {\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )}{256 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]

[In]

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

[Out]

-1/256*(45*a^3*x^3 - 8*(3*a^4*x^4 + 6*a^2*x^2 - 5)*arctan(a*x)^3 - 24*(3*a^3*x^3 + 5*a*x)*arctan(a*x)^2 + 51*a
*x + 3*(15*a^4*x^4 + 6*a^2*x^2 - 17)*arctan(a*x))/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3)

Sympy [F]

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

[In]

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

[Out]

Integral(x*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 = 0.33 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.31 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 \, {\left (\frac {3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}} + \frac {3 \, \arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{2}}{32 \, a c} - \frac {3 \, {\left (\frac {{\left (15 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{7} c^{2} x^{4} + 2 \, a^{5} c^{2} x^{2} + a^{3} c^{2}} - \frac {8 \, {\left (3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )}{a^{6} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{2} c^{2}}\right )}}{256 \, a c} - \frac {\arctan \left (a x\right )^{3}}{4 \, {\left (a^{2} c x^{2} + c\right )}^{2} a^{2} c} \]

[In]

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

[Out]

3/32*((3*a^2*x^3 + 5*x)/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2) + 3*arctan(a*x)/(a*c^2))*arctan(a*x)^2/(a*c) - 3/2
56*((15*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 + 17*a*x + 15*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x
))*a^2/(a^7*c^2*x^4 + 2*a^5*c^2*x^2 + a^3*c^2) - 8*(3*a^2*x^2 - 3*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2 + 4)
*a*arctan(a*x)/(a^6*c^2*x^4 + 2*a^4*c^2*x^2 + a^2*c^2))/(a*c) - 1/4*arctan(a*x)^3/((a^2*c*x^2 + c)^2*a^2*c)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.91 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx={\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {3}{32\,a^2\,c^3}-\frac {1}{4\,a^4\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )}\right )-\frac {\frac {45\,a^2\,x^3}{8}+\frac {51\,x}{8}}{32\,a^5\,c^3\,x^4+64\,a^3\,c^3\,x^2+32\,a\,c^3}+\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {15\,x}{32\,a^3\,c^3}+\frac {9\,x^3}{32\,a\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {45\,\mathrm {atan}\left (a\,x\right )}{256\,a^2\,c^3}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3}{8\,a^4\,c^3}+\frac {9\,x^2}{32\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4} \]

[In]

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

[Out]

atan(a*x)^3*(3/(32*a^2*c^3) - 1/(4*a^4*c^3*(1/a^2 + 2*x^2 + a^2*x^4))) - ((51*x)/8 + (45*a^2*x^3)/8)/(32*a*c^3
+ 64*a^3*c^3*x^2 + 32*a^5*c^3*x^4) + (atan(a*x)^2*((15*x)/(32*a^3*c^3) + (9*x^3)/(32*a*c^3)))/(1/a^2 + 2*x^2
+ a^2*x^4) - (45*atan(a*x))/(256*a^2*c^3) + (atan(a*x)*(3/(8*a^4*c^3) + (9*x^2)/(32*a^2*c^3)))/(1/a^2 + 2*x^2
+ a^2*x^4)