Integrand size = 20, antiderivative size = 208 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 x}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a c^3 \left (1+a^2 x^2\right )}-\frac {45 \arctan (a x)}{256 a^2 c^3}+\frac {3 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \arctan (a x)^2}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^3}{32 a^2 c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2} \]
-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
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.50 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-3 a x \left (17+15 a^2 x^2\right )-3 \left (-17+6 a^2 x^2+15 a^4 x^4\right ) \arctan (a x)+24 a x \left (5+3 a^2 x^2\right ) \arctan (a x)^2+8 \left (-5+6 a^2 x^2+3 a^4 x^4\right ) \arctan (a x)^3}{256 c^3 \left (a+a^3 x^2\right )^2} \]
(-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)*Arc Tan[a*x]^3)/(256*c^3*(a + a^3*x^2)^2)
Time = 0.62 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5465, 27, 5435, 215, 215, 216, 5427, 5465, 215, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x \arctan (a x)^3}{\left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {3 \int \frac {\arctan (a x)^2}{c^3 \left (a^2 x^2+1\right )^3}dx}{4 a}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^3}dx}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 5435 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx-\frac {1}{8} \int \frac {1}{\left (a^2 x^2+1\right )^3}dx+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}\right )}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {1}{8} \left (-\frac {3}{4} \int \frac {1}{\left (a^2 x^2+1\right )^2}dx-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}\right )}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}\right )}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )\right )}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )\right )}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )\right )}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {3 \left (\frac {3}{4} \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )\right )}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3 \left (\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )\right )}{4 a c^3}-\frac {\arctan (a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}\) |
-1/4*ArcTan[a*x]^3/(a^2*c^3*(1 + a^2*x^2)^2) + (3*(ArcTan[a*x]/(8*a*(1 + a ^2*x^2)^2) + (x*ArcTan[a*x]^2)/(4*(1 + a^2*x^2)^2) + (-1/4*x/(1 + a^2*x^2) ^2 - (3*(x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a)))/4)/8 + (3*((x*ArcTan[a* x]^2)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a) - a*(-1/2*ArcTan[a*x]/(a^2*( 1 + a^2*x^2)) + (x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a))/(2*a))))/4))/(4* a*c^3)
3.5.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sym bol] :> Simp[x*((a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b *ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S ymbol] :> Simp[b*p*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d* (q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d* (q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int[(d + e *x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] & & EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
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] - Simp[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]
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 | \(\frac {i \left (3 a^{4} x^{4}+6 a^{2} x^{2}-5\right ) \ln \left (i a x +1\right )^{3}}{256 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 i \left (-5 \ln \left (-i a x +1\right )+3 x^{4} \ln \left (-i a x +1\right ) a^{4}+6 a^{2} x^{2} \ln \left (-i a x +1\right )-6 i a^{3} x^{3}-10 i a x \right ) \ln \left (i a x +1\right )^{2}}{256 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{2}}+\frac {3 i \left (3 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+6 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-5 \ln \left (-i a x +1\right )^{2}-12 i x^{3} \ln \left (-i a x +1\right ) a^{3}-20 i a x \ln \left (-i a x +1\right )-12 a^{2} x^{2}-16\right ) \ln \left (i a x +1\right )}{256 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{2}}+\frac {i \left (72 a^{2} x^{2} \ln \left (-i a x +1\right )+96 \ln \left (-i a x +1\right )-6 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}-12 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+10 \ln \left (-i a x +1\right )^{3}+36 i x^{3} \ln \left (-i a x +1\right )^{2} a^{3}+60 i a x \ln \left (-i a x +1\right )^{2}+45 \ln \left (-a x +i\right ) a^{4} x^{4}+90 \ln \left (-a x +i\right ) a^{2} x^{2}+45 \ln \left (-a x +i\right )-45 \ln \left (a x +i\right ) a^{4} x^{4}-90 \ln \left (a x +i\right ) a^{2} x^{2}-45 \ln \left (a x +i\right )+90 i a^{3} x^{3}+102 i a x \right )}{512 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{2}}\) | \(492\) |
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*arc tan(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
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.56 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {45 \, a^{3} x^{3} - 8 \, {\left (3 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 5\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{2} + 51 \, a x + 3 \, {\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )}{256 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]
-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)*a rctan(a*x))/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3)
\[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Time = 0.33 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.31 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 \, {\left (\frac {3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}} + \frac {3 \, \arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{2}}{32 \, a c} - \frac {3 \, {\left (\frac {{\left (15 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{7} c^{2} x^{4} + 2 \, a^{5} c^{2} x^{2} + a^{3} c^{2}} - \frac {8 \, {\left (3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )}{a^{6} c^{2} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{2} c^{2}}\right )}}{256 \, a c} - \frac {\arctan \left (a x\right )^{3}}{4 \, {\left (a^{2} c x^{2} + c\right )}^{2} a^{2} c} \]
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) - 3/256*((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)
\[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
Time = 0.61 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.91 \[ \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx={\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {3}{32\,a^2\,c^3}-\frac {1}{4\,a^4\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )}\right )-\frac {\frac {45\,a^2\,x^3}{8}+\frac {51\,x}{8}}{32\,a^5\,c^3\,x^4+64\,a^3\,c^3\,x^2+32\,a\,c^3}+\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {15\,x}{32\,a^3\,c^3}+\frac {9\,x^3}{32\,a\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {45\,\mathrm {atan}\left (a\,x\right )}{256\,a^2\,c^3}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3}{8\,a^4\,c^3}+\frac {9\,x^2}{32\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4} \]
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)