Integrand size = 22, antiderivative size = 106 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)}{4 a^3 c^2}-\frac {\arctan (a x)}{2 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a^3 c^2} \]
1/4*x/a^2/c^2/(a^2*x^2+1)+1/4*arctan(a*x)/a^3/c^2-1/2*arctan(a*x)/a^3/c^2/ (a^2*x^2+1)-1/2*x*arctan(a*x)^2/a^2/c^2/(a^2*x^2+1)+1/6*arctan(a*x)^3/a^3/ c^2
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.64 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {3 a x+3 \left (-1+a^2 x^2\right ) \arctan (a x)-6 a x \arctan (a x)^2+2 \left (1+a^2 x^2\right ) \arctan (a x)^3}{12 a^3 c^2 \left (1+a^2 x^2\right )} \]
(3*a*x + 3*(-1 + a^2*x^2)*ArcTan[a*x] - 6*a*x*ArcTan[a*x]^2 + 2*(1 + a^2*x ^2)*ArcTan[a*x]^3)/(12*a^3*c^2*(1 + a^2*x^2))
Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5471, 27, 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.
\(\displaystyle \int \frac {x^2 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^2} \, dx\) |
\(\Big \downarrow \) 5471 |
\(\displaystyle \frac {\int \frac {x \arctan (a x)}{c^2 \left (a^2 x^2+1\right )^2}dx}{a}+\frac {\arctan (a x)^3}{6 a^3 c^2}-\frac {x \arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a c^2}+\frac {\arctan (a x)^3}{6 a^3 c^2}-\frac {x \arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a c^2}+\frac {\arctan (a x)^3}{6 a^3 c^2}-\frac {x \arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a c^2}+\frac {\arctan (a x)^3}{6 a^3 c^2}-\frac {x \arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\arctan (a x)^3}{6 a^3 c^2}-\frac {x \arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a c^2}\) |
-1/2*(x*ArcTan[a*x]^2)/(a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a^3*c^2) + (-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*c^2)
3.3.92.3.1 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_.)*(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]
Time = 0.50 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71
method | result | size |
parallelrisch | \(\frac {2 \arctan \left (a x \right )^{3} x^{2} a^{2}+3 a^{2} \arctan \left (a x \right ) x^{2}-6 a \arctan \left (a x \right )^{2} x +2 \arctan \left (a x \right )^{3}+3 a x -3 \arctan \left (a x \right )}{12 c^{2} \left (a^{2} x^{2}+1\right ) a^{3}}\) | \(75\) |
derivativedivides | \(\frac {-\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}+2}-\frac {a x}{4 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4}}{c^{2}}}{a^{3}}\) | \(93\) |
default | \(\frac {-\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}+2}-\frac {a x}{4 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4}}{c^{2}}}{a^{3}}\) | \(93\) |
parts | \(-\frac {x \arctan \left (a x \right )^{2}}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 a^{3} c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3 a^{3}}-\frac {-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}}{a^{3}}}{c^{2}}\) | \(103\) |
risch | \(\frac {i \ln \left (i a x +1\right )^{3}}{48 c^{2} a^{3}}-\frac {i \left (a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )+2 i a x \right ) \ln \left (i a x +1\right )^{2}}{16 a^{3} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {i \left (a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+\ln \left (-i a x +1\right )^{2}+4 i a x \ln \left (-i a x +1\right )+4\right ) \ln \left (i a x +1\right )}{16 a^{3} c^{2} \left (a x +i\right ) \left (a x -i\right )}+\frac {i \left (-a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-6 i a x \ln \left (-i a x +1\right )^{2}+6 \ln \left (i a x -1\right ) a^{2} x^{2}-6 \ln \left (-i a x -1\right ) a^{2} x^{2}-12 i a x -\ln \left (-i a x +1\right )^{3}+6 \ln \left (i a x -1\right )-6 \ln \left (-i a x -1\right )-12 \ln \left (-i a x +1\right )\right )}{48 a^{3} c^{2} \left (a x +i\right ) \left (a x -i\right )}\) | \(293\) |
1/12*(2*arctan(a*x)^3*x^2*a^2+3*a^2*arctan(a*x)*x^2-6*a*arctan(a*x)^2*x+2* arctan(a*x)^3+3*a*x-3*arctan(a*x))/c^2/(a^2*x^2+1)/a^3
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.65 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {6 \, a x \arctan \left (a x\right )^{2} - 2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{12 \, {\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )}} \]
-1/12*(6*a*x*arctan(a*x)^2 - 2*(a^2*x^2 + 1)*arctan(a*x)^3 - 3*a*x - 3*(a^ 2*x^2 - 1)*arctan(a*x))/(a^5*c^2*x^2 + a^3*c^2)
\[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Time = 0.30 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.42 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {1}{2} \, {\left (\frac {x}{a^{4} c^{2} x^{2} + a^{2} c^{2}} - \frac {\arctan \left (a x\right )}{a^{3} c^{2}}\right )} \arctan \left (a x\right )^{2} + \frac {{\left (2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 3 \, a x + 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{12 \, {\left (a^{7} c^{2} x^{2} + a^{5} c^{2}\right )}} - \frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1\right )} a \arctan \left (a x\right )}{2 \, {\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} \]
-1/2*(x/(a^4*c^2*x^2 + a^2*c^2) - arctan(a*x)/(a^3*c^2))*arctan(a*x)^2 + 1 /12*(2*(a^2*x^2 + 1)*arctan(a*x)^3 + 3*a*x + 3*(a^2*x^2 + 1)*arctan(a*x))* a^2/(a^7*c^2*x^2 + a^5*c^2) - 1/2*((a^2*x^2 + 1)*arctan(a*x)^2 + 1)*a*arct an(a*x)/(a^6*c^2*x^2 + a^4*c^2)
\[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
Time = 0.46 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x}{2\,\left (2\,a^4\,c^2\,x^2+2\,a^2\,c^2\right )}+\frac {\mathrm {atan}\left (a\,x\right )}{4\,a^3\,c^2}+\frac {{\mathrm {atan}\left (a\,x\right )}^3}{6\,a^3\,c^2}-\frac {\mathrm {atan}\left (a\,x\right )}{2\,a^5\,c^2\,\left (\frac {1}{a^2}+x^2\right )}-\frac {x\,{\mathrm {atan}\left (a\,x\right )}^2}{2\,a^4\,c^2\,\left (\frac {1}{a^2}+x^2\right )} \]