Integrand size = 12, antiderivative size = 56 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=-\frac {x}{4 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)^2}{2 \left (1+x^2\right )}-\frac {1}{6} \cot ^{-1}(x)^3-\frac {\arctan (x)}{4} \]
-1/4*x/(x^2+1)-1/2*arccot(x)/(x^2+1)+1/2*x*arccot(x)^2/(x^2+1)-1/6*arccot( x)^3-1/4*arctan(x)
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=-\frac {6 \cot ^{-1}(x)-6 x \cot ^{-1}(x)^2+2 \left (1+x^2\right ) \cot ^{-1}(x)^3+3 \left (x+\left (1+x^2\right ) \arctan (x)\right )}{12 \left (1+x^2\right )} \]
-1/12*(6*ArcCot[x] - 6*x*ArcCot[x]^2 + 2*(1 + x^2)*ArcCot[x]^3 + 3*(x + (1 + x^2)*ArcTan[x]))/(1 + x^2)
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5428, 5466, 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.
\(\displaystyle \int \frac {\cot ^{-1}(x)^2}{\left (x^2+1\right )^2} \, dx\) |
\(\Big \downarrow \) 5428 |
\(\displaystyle \int \frac {x \cot ^{-1}(x)}{\left (x^2+1\right )^2}dx+\frac {x \cot ^{-1}(x)^2}{2 \left (x^2+1\right )}-\frac {1}{6} \cot ^{-1}(x)^3\) |
\(\Big \downarrow \) 5466 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{\left (x^2+1\right )^2}dx+\frac {x \cot ^{-1}(x)^2}{2 \left (x^2+1\right )}-\frac {\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{6} \cot ^{-1}(x)^3\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {1}{x^2+1}dx-\frac {x}{2 \left (x^2+1\right )}\right )+\frac {x \cot ^{-1}(x)^2}{2 \left (x^2+1\right )}-\frac {\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{6} \cot ^{-1}(x)^3\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{2} \left (-\frac {\arctan (x)}{2}-\frac {x}{2 \left (x^2+1\right )}\right )+\frac {x \cot ^{-1}(x)^2}{2 \left (x^2+1\right )}-\frac {\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{6} \cot ^{-1}(x)^3\) |
-1/2*ArcCot[x]/(1 + x^2) + (x*ArcCot[x]^2)/(2*(1 + x^2)) - ArcCot[x]^3/6 + (-1/2*x/(1 + x^2) - ArcTan[x]/2)/2
3.1.73.3.1 Defintions of rubi rules used
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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sym bol] :> Simp[x*((a + b*ArcCot[c*x])^p/(2*d*(d + e*x^2))), x] + (-Simp[(a + b*ArcCot[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] + Simp[b*c*(p/2) Int[x*((a + b*ArcCot[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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcCot[c*x])^p/(2*e*(q + 1))), x] + Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcCot[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 = 0.96 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {\operatorname {arccot}\left (x \right )^{2} x}{2 x^{2}+2}+\frac {\operatorname {arccot}\left (x \right )^{2} \arctan \left (x \right )}{2}-\frac {\pi \operatorname {arccot}\left (x \right )^{2}}{4}+\frac {\operatorname {arccot}\left (x \right )^{3}}{3}+\frac {x^{2} \operatorname {arccot}\left (x \right )}{2 x^{2}+2}-\frac {x}{4 \left (x^{2}+1\right )}-\frac {\operatorname {arccot}\left (x \right )}{4}\) | \(65\) |
parts | \(\frac {\operatorname {arccot}\left (x \right )^{2} x}{2 x^{2}+2}+\frac {\operatorname {arccot}\left (x \right )^{2} \arctan \left (x \right )}{2}-\frac {\pi \operatorname {arccot}\left (x \right )^{2}}{4}+\frac {\operatorname {arccot}\left (x \right )^{3}}{3}+\frac {x^{2} \operatorname {arccot}\left (x \right )}{2 x^{2}+2}-\frac {x}{4 \left (x^{2}+1\right )}-\frac {\operatorname {arccot}\left (x \right )}{4}\) | \(65\) |
risch | \(\frac {i \ln \left (i x +1\right )^{3}}{48}+\frac {\left (-i x^{2} \ln \left (-i x +1\right )+\pi \,x^{2}-i \ln \left (-i x +1\right )+\pi -2 x \right ) \ln \left (i x +1\right )^{2}}{16 x^{2}+16}-\frac {\left (-i x^{2} \ln \left (-i x +1\right )^{2}-i \ln \left (-i x +1\right )^{2}-4 \ln \left (-i x +1\right ) x +2 \ln \left (-i x +1\right ) \pi \,x^{2}+2 \pi \ln \left (-i x +1\right )-4 i \pi x +4 i\right ) \ln \left (i x +1\right )}{16 \left (i+x \right ) \left (x -i\right )}-\frac {i \left (-3 \ln \left (i+x \right ) \pi ^{2} x^{2}+3 \ln \left (x -i\right ) \pi ^{2} x^{2}+x^{2} \ln \left (-i x +1\right )^{3}+3 i \pi \ln \left (-i x +1\right )^{2}-3 \ln \left (i+x \right ) \pi ^{2}+6 \ln \left (i+x \right ) x^{2}+3 \ln \left (x -i\right ) \pi ^{2}-6 \ln \left (x -i\right ) x^{2}+12 \pi \ln \left (-i x +1\right ) x +\ln \left (-i x +1\right )^{3}-12 i \pi -12 i x +3 i \pi \,x^{2} \ln \left (-i x +1\right )^{2}+6 \ln \left (i+x \right )-6 \ln \left (x -i\right )-12 \ln \left (-i x +1\right )-6 i \ln \left (-i x +1\right )^{2} x +6 i \pi ^{2} x \right )}{48 \left (i+x \right ) \left (x -i\right )}\) | \(349\) |
1/2*x*arccot(x)^2/(x^2+1)+1/2*arccot(x)^2*arctan(x)-1/4*Pi*arccot(x)^2+1/3 *arccot(x)^3+1/2*x^2*arccot(x)/(x^2+1)-1/4*x/(x^2+1)-1/4*arccot(x)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.71 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=-\frac {2 \, {\left (x^{2} + 1\right )} \operatorname {arccot}\left (x\right )^{3} - 6 \, x \operatorname {arccot}\left (x\right )^{2} - 3 \, {\left (x^{2} - 1\right )} \operatorname {arccot}\left (x\right ) + 3 \, x}{12 \, {\left (x^{2} + 1\right )}} \]
\[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=\int \frac {\operatorname {acot}^{2}{\left (x \right )}}{\left (x^{2} + 1\right )^{2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.34 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{x^{2} + 1} + \arctan \left (x\right )\right )} \operatorname {arccot}\left (x\right )^{2} + \frac {{\left ({\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - 1\right )} \operatorname {arccot}\left (x\right )}{2 \, {\left (x^{2} + 1\right )}} + \frac {2 \, {\left (x^{2} + 1\right )} \arctan \left (x\right )^{3} - 3 \, {\left (x^{2} + 1\right )} \arctan \left (x\right ) - 3 \, x}{12 \, {\left (x^{2} + 1\right )}} \]
1/2*(x/(x^2 + 1) + arctan(x))*arccot(x)^2 + 1/2*((x^2 + 1)*arctan(x)^2 - 1 )*arccot(x)/(x^2 + 1) + 1/12*(2*(x^2 + 1)*arctan(x)^3 - 3*(x^2 + 1)*arctan (x) - 3*x)/(x^2 + 1)
\[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )^{2}}{{\left (x^{2} + 1\right )}^{2}} \,d x } \]
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^{-1}(x)^2}{\left (1+x^2\right )^2} \, dx=\frac {x\,{\mathrm {acot}\left (x\right )}^2}{2\,\left (x^2+1\right )}-\frac {{\mathrm {acot}\left (x\right )}^3}{6}-\frac {x}{4\,\left (x^2+1\right )}-\frac {\mathrm {acot}\left (x\right )}{2\,\left (x^2+1\right )}-\frac {\mathrm {atan}\left (x\right )}{4} \]