Integrand size = 22, antiderivative size = 237 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \arctan (a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \arctan (a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^2}{128 a^3 c^3}-\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{32 a^3 c^3} \]
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*arctan(a*x)^4/a^3/c^3
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.47 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {-3 a^2 x^2+\left (6 a x-6 a^3 x^3\right ) \arctan (a x)-3 \left (1-6 a^2 x^2+a^4 x^4\right ) \arctan (a x)^2+16 a x \left (-1+a^2 x^2\right ) \arctan (a x)^3+4 \left (1+a^2 x^2\right )^2 \arctan (a x)^4}{128 a^3 c^3 \left (1+a^2 x^2\right )^2} \]
(-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)
Time = 1.49 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.73, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5499, 27, 5427, 5435, 5427, 5431, 5427, 241, 5465, 5427, 241}
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.
| \(\displaystyle \int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^3} \, dx\) |
| \(\Big \downarrow \) 5499 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{c^2 \left (a^2 x^2+1\right )^2}dx}{a^2 c}-\frac {\int \frac {\arctan (a x)^3}{c^3 \left (a^2 x^2+1\right )^3}dx}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^3}-\frac {\int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^3}dx}{a^2 c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {\int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^3}dx}{a^2 c^3}\) |
| \(\Big \downarrow \) 5435 |
| \(\displaystyle \frac {-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {-\frac {3}{8} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^3}dx+\frac {3}{4} \int \frac {\arctan (a x)^3}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}}{a^2 c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {-\frac {3}{8} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^3}dx+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}}{a^2 c^3}\) |
| \(\Big \downarrow \) 5431 |
| \(\displaystyle \frac {-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {-\frac {3}{8} \left (\frac {3}{4} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}}{a^2 c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {-\frac {3}{8} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}}{a^2 c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{a^2 c^3}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {-\frac {3}{2} a \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{a^2 c^3}\) |
| \(\Big \downarrow \) 5427 |
| \(\displaystyle \frac {-\frac {3}{2} a \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^4}{8 a}\right )+\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{a^2 c^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}}{a^2 c^3}-\frac {\frac {x \arctan (a x)^3}{4 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^2}{16 a \left (a^2 x^2+1\right )^2}-\frac {3}{8} \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^3}{2 \left (a^2 x^2+1\right )}-\frac {3}{2} a \left (\frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a}-\frac {\arctan (a x)^2}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^4}{8 a}\right )}{a^2 c^3}\) |
((x*ArcTan[a*x]^3)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a) - (3*a*(-1/2*Ar cTan[a*x]^2/(a^2*(1 + a^2*x^2)) + (1/(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x]) /(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a))/a))/2)/(a^2*c^3) - ((3*ArcTan[a* x]^2)/(16*a*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^3)/(4*(1 + a^2*x^2)^2) - (3* (1/(16*a*(1 + a^2*x^2)^2) + (x*ArcTan[a*x])/(4*(1 + a^2*x^2)^2) + (3*(1/(4 *a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a )))/4))/8 + (3*((x*ArcTan[a*x]^3)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a) - (3*a*(-1/2*ArcTan[a*x]^2/(a^2*(1 + a^2*x^2)) + (1/(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a))/a))/2))/4)/(a^2*c ^3)
3.5.5.3.1 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[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol ] :> Simp[b*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x ^2)^(q + 1)*((a + b*ArcTan[c*x])/(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]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar cTan[c*x])^p, x], x] - Simp[d/e Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan [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]
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 | \(\frac {\ln \left (i a x +1\right )^{4}}{512 a^{3} c^{3}}-\frac {\left (x^{4} \ln \left (-i a x +1\right ) a^{4}+2 a^{2} x^{2} \ln \left (-i a x +1\right )-2 i a^{3} x^{3}+\ln \left (-i a x +1\right )+2 i a x \right ) \ln \left (i a x +1\right )^{3}}{128 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}-8 i x^{3} \ln \left (-i a x +1\right ) a^{3}+a^{4} x^{4}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+8 i a x \ln \left (-i a x +1\right )-6 a^{2} x^{2}+2 \ln \left (-i a x +1\right )^{2}+1\right ) \ln \left (i a x +1\right )^{2}}{512 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}-\frac {\left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}-12 i x^{3} \ln \left (-i a x +1\right )^{2} a^{3}+3 x^{4} \ln \left (-i a x +1\right ) a^{4}-6 i a^{3} x^{3}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+12 i a x \ln \left (-i a x +1\right )^{2}-18 a^{2} x^{2} \ln \left (-i a x +1\right )+6 i a x +2 \ln \left (-i a x +1\right )^{3}+3 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{256 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}+\frac {a^{4} x^{4} \ln \left (-i a x +1\right )^{4}+3 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+2 a^{2} x^{2} \ln \left (-i a x +1\right )^{4}-8 i a^{3} x^{3} \ln \left (-i a x +1\right )^{3}-18 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-12 i x^{3} \ln \left (-i a x +1\right ) a^{3}-12 a^{2} x^{2}+\ln \left (-i a x +1\right )^{4}+8 i a x \ln \left (-i a x +1\right )^{3}+3 \ln \left (-i a x +1\right )^{2}+12 i a x \ln \left (-i a x +1\right )}{512 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}\) | \(593\) |
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*arctan(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*arct an(a*x)^2)/c^3/(a^2*x^2+1)^2/a^3
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.55 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {4 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 3 \, a^{2} x^{2} + 16 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{3} - 3 \, {\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )}{128 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]
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)
\[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Time = 0.35 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.41 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{8} \, {\left (\frac {a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac {\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{3} + \frac {3 \, {\left (a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )^{2}}{16 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} - \frac {1}{128} \, {\left (\frac {{\left (4 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a^{2}}{a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}} + \frac {2 \, {\left (3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}}\right )} a \]
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^10*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
\[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
Time = 0.60 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3\,x}{64\,a^4\,c^3}-\frac {3\,x^3}{64\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {3\,x^2}{2\,\left (64\,a^5\,c^3\,x^4+128\,a^3\,c^3\,x^2+64\,a\,c^3\right )}-{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {3}{128\,a^3\,c^3}-\frac {3\,x^2}{16\,a^3\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )}\right )-\frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {x}{8\,a^4\,c^3}-\frac {x^3}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}+\frac {{\mathrm {atan}\left (a\,x\right )}^4}{32\,a^3\,c^3} \]
(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)