Integrand size = 22, antiderivative size = 212 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {27 \arctan (a x)}{256 a^4 c^3}-\frac {3 x^4 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {3 x^3 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \arctan (a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^3}{32 a^4 c^3}+\frac {x^4 \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2} \]
-3/128*x^3/a/c^3/(a^2*x^2+1)^2-45/256*x/a^3/c^3/(a^2*x^2+1)-27/256*arctan( a*x)/a^4/c^3-3/32*x^4*arctan(a*x)/c^3/(a^2*x^2+1)^2+9/32*arctan(a*x)/a^4/c ^3/(a^2*x^2+1)+3/16*x^3*arctan(a*x)^2/a/c^3/(a^2*x^2+1)^2+9/32*x*arctan(a* x)^2/a^3/c^3/(a^2*x^2+1)-3/32*arctan(a*x)^3/a^4/c^3+1/4*x^4*arctan(a*x)^3/ c^3/(a^2*x^2+1)^2
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.50 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-3 a x \left (15+17 a^2 x^2\right )+\left (45+18 a^2 x^2-51 a^4 x^4\right ) \arctan (a x)+24 a x \left (3+5 a^2 x^2\right ) \arctan (a x)^2+8 \left (-3-6 a^2 x^2+5 a^4 x^4\right ) \arctan (a x)^3}{256 a^4 c^3 \left (1+a^2 x^2\right )^2} \]
(-3*a*x*(15 + 17*a^2*x^2) + (45 + 18*a^2*x^2 - 51*a^4*x^4)*ArcTan[a*x] + 2 4*a*x*(3 + 5*a^2*x^2)*ArcTan[a*x]^2 + 8*(-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTa n[a*x]^3)/(256*a^4*c^3*(1 + a^2*x^2)^2)
Time = 0.72 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5479, 27, 5475, 252, 252, 216, 5471, 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.
| \(\displaystyle \int \frac {x^3 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3}{4} a \int \frac {x^4 \arctan (a x)^2}{c^3 \left (a^2 x^2+1\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \int \frac {x^4 \arctan (a x)^2}{\left (a^2 x^2+1\right )^3}dx}{4 c^3}\) |
| \(\Big \downarrow \) 5475 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \left (\frac {3 \int \frac {x^2 \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{4 a^2}-\frac {1}{8} \int \frac {x^4}{\left (a^2 x^2+1\right )^3}dx+\frac {x^4 \arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{4 c^3}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \left (\frac {3 \int \frac {x^2 \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{4 a^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (a^2 x^2+1\right )^2}-\frac {3 \int \frac {x^2}{\left (a^2 x^2+1\right )^2}dx}{4 a^2}\right )+\frac {x^4 \arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{4 c^3}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \left (\frac {3 \int \frac {x^2 \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{4 a^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (a^2 x^2+1\right )^2}-\frac {3 \left (\frac {\int \frac {1}{a^2 x^2+1}dx}{2 a^2}-\frac {x}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )+\frac {x^4 \arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}\right )}{4 c^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \left (\frac {3 \int \frac {x^2 \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx}{4 a^2}+\frac {x^4 \arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (a^2 x^2+1\right )^2}-\frac {3 \left (\frac {\arctan (a x)}{2 a^3}-\frac {x}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )\right )}{4 c^3}\) |
| \(\Big \downarrow \) 5471 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \left (\frac {3 \left (\frac {\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}+\frac {\arctan (a x)^3}{6 a^3}-\frac {x \arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}+\frac {x^4 \arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (a^2 x^2+1\right )^2}-\frac {3 \left (\frac {\arctan (a x)}{2 a^3}-\frac {x}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )\right )}{4 c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \left (\frac {3 \left (\frac {\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a}+\frac {\arctan (a x)^3}{6 a^3}-\frac {x \arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}+\frac {x^4 \arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (a^2 x^2+1\right )^2}-\frac {3 \left (\frac {\arctan (a x)}{2 a^3}-\frac {x}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )\right )}{4 c^3}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \left (\frac {3 \left (\frac {\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a}+\frac {\arctan (a x)^3}{6 a^3}-\frac {x \arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}+\frac {x^4 \arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (a^2 x^2+1\right )^2}-\frac {3 \left (\frac {\arctan (a x)}{2 a^3}-\frac {x}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )\right )}{4 c^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {x^4 \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 a \left (\frac {x^4 \arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}-\frac {x^3 \arctan (a x)^2}{4 a^2 \left (a^2 x^2+1\right )^2}+\frac {3 \left (\frac {\arctan (a x)^3}{6 a^3}-\frac {x \arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}+\frac {\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a}\right )}{4 a^2}+\frac {1}{8} \left (\frac {x^3}{4 a^2 \left (a^2 x^2+1\right )^2}-\frac {3 \left (\frac {\arctan (a x)}{2 a^3}-\frac {x}{2 a^2 \left (a^2 x^2+1\right )}\right )}{4 a^2}\right )\right )}{4 c^3}\) |
(x^4*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) - (3*a*((x^4*ArcTan[a*x])/(8*a *(1 + a^2*x^2)^2) - (x^3*ArcTan[a*x]^2)/(4*a^2*(1 + a^2*x^2)^2) + (x^3/(4* a^2*(1 + a^2*x^2)^2) - (3*(-1/2*x/(a^2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a^3 )))/(4*a^2))/8 + (3*(-1/2*(x*ArcTan[a*x]^2)/(a^2*(1 + a^2*x^2)) + ArcTan[a *x]^3/(6*a^3) + (-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))/a))/(4*a^2)))/(4*c^3)
3.5.4.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^2)/((d_) + (e_.)*(x_)^2) ^2, x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(2*b*c^3*d^2*(p + 1)), x] + (-Simp[x*((a + b*ArcTan[c*x])^p/(2*c^2*d*(d + e*x^2))), x] + Simp[b*(p/( 2*c)) 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_)*((f_.)*(x_))^(m_)*((d_) + (e_.) *(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*Ar cTan[c*x])^(p - 1)/(c*d*m^2)), x] + (-Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + Simp[f^2*((m - 1)/(c^2*d*m)) Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[ b^2*p*((p - 1)/m^2) Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2) , x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2* q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1))) Int[(f*x) ^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Time = 1.39 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.58
| method | result | size |
| parallelrisch | \(\frac {40 a^{4} \arctan \left (a x \right )^{3} x^{4}-51 \arctan \left (a x \right ) a^{4} x^{4}+120 a^{3} \arctan \left (a x \right )^{2} x^{3}-48 \arctan \left (a x \right )^{3} x^{2} a^{2}-51 a^{3} x^{3}+18 a^{2} \arctan \left (a x \right ) x^{2}+72 a \arctan \left (a x \right )^{2} x -24 \arctan \left (a x \right )^{3}-45 a x +45 \arctan \left (a x \right )}{256 c^{3} \left (a^{2} x^{2}+1\right )^{2} a^{4}}\) | \(123\) |
| derivativedivides | \(\frac {-\frac {\arctan \left (a x \right )^{3}}{2 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \left (-\frac {5 \arctan \left (a x \right )^{2} a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \arctan \left (a x \right )^{2} a x}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {5 \arctan \left (a x \right )^{3}}{24}-\frac {5 \arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {17}{8} a^{3} x^{3}+\frac {15}{8} a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {17 \arctan \left (a x \right )}{64}\right )}{4 c^{3}}}{a^{4}}\) | \(176\) |
| default | \(\frac {-\frac {\arctan \left (a x \right )^{3}}{2 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \left (-\frac {5 \arctan \left (a x \right )^{2} a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \arctan \left (a x \right )^{2} a x}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {5 \arctan \left (a x \right )^{3}}{24}-\frac {5 \arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {17}{8} a^{3} x^{3}+\frac {15}{8} a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {17 \arctan \left (a x \right )}{64}\right )}{4 c^{3}}}{a^{4}}\) | \(176\) |
| parts | \(\frac {\arctan \left (a x \right )^{3}}{4 c^{3} a^{4} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{3}}{2 c^{3} a^{4} \left (a^{2} x^{2}+1\right )}-\frac {3 \left (-\frac {5 \arctan \left (a x \right )^{2} x^{3}}{8 a \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \arctan \left (a x \right )^{2} x}{8 a^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {5 \arctan \left (a x \right )^{3}}{8 a^{4}}+\frac {-\frac {5 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {17}{8} a^{3} x^{3}+\frac {15}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {17 \arctan \left (a x \right )}{16}+\frac {5 \arctan \left (a x \right )^{3}}{3}}{4 a^{4}}\right )}{4 c^{3}}\) | \(197\) |
| risch | \(\frac {i \left (5 a^{4} x^{4}-6 a^{2} x^{2}-3\right ) \ln \left (i a x +1\right )^{3}}{256 a^{4} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 i \left (-6 a^{2} x^{2} \ln \left (-i a x +1\right )-3 \ln \left (-i a x +1\right )+5 x^{4} \ln \left (-i a x +1\right ) a^{4}-10 i a^{3} x^{3}-6 i a x \right ) \ln \left (i a x +1\right )^{2}}{256 a^{4} \left (a x +i\right )^{2} \left (a x -i\right )^{2} c^{3}}+\frac {3 i \left (5 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}-6 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-3 \ln \left (-i a x +1\right )^{2}-20 i x^{3} \ln \left (-i a x +1\right ) a^{3}-12 i a x \ln \left (-i a x +1\right )-20 a^{2} x^{2}-16\right ) \ln \left (i a x +1\right )}{256 a^{4} \left (a x +i\right )^{2} \left (a x -i\right )^{2} c^{3}}-\frac {i \left (-120 a^{2} x^{2} \ln \left (-i a x +1\right )-96 \ln \left (-i a x +1\right )+10 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}-12 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-6 \ln \left (-i a x +1\right )^{3}-60 i x^{3} \ln \left (-i a x +1\right )^{2} a^{3}-36 i a x \ln \left (-i a x +1\right )^{2}+51 \ln \left (a x +i\right ) a^{4} x^{4}+102 \ln \left (a x +i\right ) a^{2} x^{2}+51 \ln \left (a x +i\right )-51 \ln \left (-a x +i\right ) a^{4} x^{4}-102 \ln \left (-a x +i\right ) a^{2} x^{2}-51 \ln \left (-a x +i\right )-102 i a^{3} x^{3}-90 i a x \right )}{512 a^{4} \left (a x +i\right )^{2} \left (a x -i\right )^{2} c^{3}}\) | \(492\) |
1/256*(40*a^4*arctan(a*x)^3*x^4-51*arctan(a*x)*a^4*x^4+120*a^3*arctan(a*x) ^2*x^3-48*arctan(a*x)^3*x^2*a^2-51*a^3*x^3+18*a^2*arctan(a*x)*x^2+72*a*arc tan(a*x)^2*x-24*arctan(a*x)^3-45*a*x+45*arctan(a*x))/c^3/(a^2*x^2+1)^2/a^4
Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.55 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {51 \, a^{3} x^{3} - 8 \, {\left (5 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 3\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (5 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{2} + 45 \, a x + 3 \, {\left (17 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 15\right )} \arctan \left (a x\right )}{256 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
-1/256*(51*a^3*x^3 - 8*(5*a^4*x^4 - 6*a^2*x^2 - 3)*arctan(a*x)^3 - 24*(5*a ^3*x^3 + 3*a*x)*arctan(a*x)^2 + 45*a*x + 3*(17*a^4*x^4 - 6*a^2*x^2 - 15)*a rctan(a*x))/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)
\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.36 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3}{32} \, a {\left (\frac {5 \, a^{2} x^{3} + 3 \, x}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac {5 \, \arctan \left (a x\right )}{a^{5} c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac {{\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3}}{4 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} - \frac {1}{256} \, {\left (\frac {{\left (51 \, a^{3} x^{3} - 40 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 45 \, a x + 51 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{11} c^{3} x^{4} + 2 \, a^{9} c^{3} x^{2} + a^{7} c^{3}} - \frac {24 \, {\left (5 \, a^{2} x^{2} - 5 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )}{a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}}\right )} a \]
3/32*a*((5*a^2*x^3 + 3*x)/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) + 5*arct an(a*x)/(a^5*c^3))*arctan(a*x)^2 - 1/4*(2*a^2*x^2 + 1)*arctan(a*x)^3/(a^8* c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3) - 1/256*((51*a^3*x^3 - 40*(a^4*x^4 + 2* a^2*x^2 + 1)*arctan(a*x)^3 + 45*a*x + 51*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan( a*x))*a^2/(a^11*c^3*x^4 + 2*a^9*c^3*x^2 + a^7*c^3) - 24*(5*a^2*x^2 - 5*(a^ 4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^2 + 4)*a*arctan(a*x)/(a^10*c^3*x^4 + 2* a^8*c^3*x^2 + a^6*c^3))*a
\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
Time = 0.66 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {9\,x}{32\,a^5\,c^3}+\frac {15\,x^3}{32\,a^3\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-{\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {\frac {1}{4\,a^6\,c^3}+\frac {x^2}{2\,a^4\,c^3}}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {5}{32\,a^4\,c^3}\right )-\frac {\frac {51\,a^2\,x^3}{8}+\frac {45\,x}{8}}{32\,a^7\,c^3\,x^4+64\,a^5\,c^3\,x^2+32\,a^3\,c^3}-\frac {51\,\mathrm {atan}\left (a\,x\right )}{256\,a^4\,c^3}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3}{8\,a^6\,c^3}+\frac {15\,x^2}{32\,a^4\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4} \]
(atan(a*x)^2*((9*x)/(32*a^5*c^3) + (15*x^3)/(32*a^3*c^3)))/(1/a^2 + 2*x^2 + a^2*x^4) - atan(a*x)^3*((1/(4*a^6*c^3) + x^2/(2*a^4*c^3))/(1/a^2 + 2*x^2 + a^2*x^4) - 5/(32*a^4*c^3)) - ((45*x)/8 + (51*a^2*x^3)/8)/(32*a^3*c^3 + 64*a^5*c^3*x^2 + 32*a^7*c^3*x^4) - (51*atan(a*x))/(256*a^4*c^3) + (atan(a* x)*(3/(8*a^6*c^3) + (15*x^2)/(32*a^4*c^3)))/(1/a^2 + 2*x^2 + a^2*x^4)