Integrand size = 29, antiderivative size = 123 \[ \int \frac {\sqrt {\frac {-2 x^2+x^4}{\left (-1+x^2\right )^2}}}{2+x^2} \, dx=-\frac {2 \left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \arctan \left (\frac {1}{2} \sqrt {-2+x^2}\right )}{3 x \sqrt {-2+x^2}}+\frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \arctan \left (\sqrt {-2+x^2}\right )}{3 x \sqrt {-2+x^2}} \]
-2/3*(-x^2+1)*arctan(1/2*(x^2-2)^(1/2))*((x^4-2*x^2)/(-x^2+1)^2)^(1/2)/x/( x^2-2)^(1/2)+1/3*(-x^2+1)*arctan((x^2-2)^(1/2))*((x^4-2*x^2)/(-x^2+1)^2)^( 1/2)/x/(x^2-2)^(1/2)
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {\frac {-2 x^2+x^4}{\left (-1+x^2\right )^2}}}{2+x^2} \, dx=\frac {\sqrt {\frac {x^2 \left (-2+x^2\right )}{\left (-1+x^2\right )^2}} \left (-1+x^2\right ) \left (2 \arctan \left (\frac {1}{2} \sqrt {-2+x^2}\right )-\arctan \left (\sqrt {-2+x^2}\right )\right )}{3 x \sqrt {-2+x^2}} \]
(Sqrt[(x^2*(-2 + x^2))/(-1 + x^2)^2]*(-1 + x^2)*(2*ArcTan[Sqrt[-2 + x^2]/2 ] - ArcTan[Sqrt[-2 + x^2]]))/(3*x*Sqrt[-2 + x^2])
Time = 0.46 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.67, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {7270, 25, 2467, 435, 94, 73, 216, 217}
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.
\(\displaystyle \int \frac {\sqrt {\frac {x^4-2 x^2}{\left (x^2-1\right )^2}}}{x^2+2} \, dx\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle -\frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \int -\frac {\sqrt {x^4-2 x^2}}{\left (1-x^2\right ) \left (x^2+2\right )}dx}{\sqrt {x^4-2 x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \int \frac {\sqrt {x^4-2 x^2}}{\left (1-x^2\right ) \left (x^2+2\right )}dx}{\sqrt {x^4-2 x^2}}\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \int \frac {x \sqrt {x^2-2}}{\left (1-x^2\right ) \left (x^2+2\right )}dx}{x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 435 |
\(\displaystyle \frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \int \frac {\sqrt {x^2-2}}{\left (1-x^2\right ) \left (x^2+2\right )}dx^2}{2 x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 94 |
\(\displaystyle \frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \left (-\frac {1}{3} \int \frac {1}{\left (1-x^2\right ) \sqrt {x^2-2}}dx^2-\frac {4}{3} \int \frac {1}{\sqrt {x^2-2} \left (x^2+2\right )}dx^2\right )}{2 x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \left (-\frac {2}{3} \int \frac {1}{-x^4-1}d\sqrt {x^2-2}-\frac {8}{3} \int \frac {1}{x^4+4}d\sqrt {x^2-2}\right )}{2 x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \left (-\frac {2}{3} \int \frac {1}{-x^4-1}d\sqrt {x^2-2}-\frac {4}{3} \arctan \left (\frac {\sqrt {x^2-2}}{2}\right )\right )}{2 x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\left (1-x^2\right ) \sqrt {-\frac {2 x^2-x^4}{\left (1-x^2\right )^2}} \left (\frac {2}{3} \arctan \left (\sqrt {x^2-2}\right )-\frac {4}{3} \arctan \left (\frac {\sqrt {x^2-2}}{2}\right )\right )}{2 x \sqrt {x^2-2}}\) |
((1 - x^2)*Sqrt[-((2*x^2 - x^4)/(1 - x^2)^2)]*((-4*ArcTan[Sqrt[-2 + x^2]/2 ])/3 + (2*ArcTan[Sqrt[-2 + x^2]])/3))/(2*x*Sqrt[-2 + x^2])
3.9.94.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[(b*e - a*f)/(b*c - a*d) Int[(e + f*x)^(p - 1)/(a + b*x), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[(e + f*x)^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]
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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Time = 1.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\sqrt {\frac {\left (x^{2}-2\right ) x^{2}}{\left (x^{2}-1\right )^{2}}}\, \left (x^{2}-1\right ) \left (\arctan \left (\frac {x +2}{\sqrt {x^{2}-2}}\right )-\arctan \left (\frac {x -2}{\sqrt {x^{2}-2}}\right )+4 \arctan \left (\frac {\sqrt {x^{2}-2}}{2}\right )\right )}{6 x \sqrt {x^{2}-2}}\) | \(75\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{7}-6 \sqrt {-\frac {-x^{4}+2 x^{2}}{x^{4}-2 x^{2}+1}}\, x^{6}-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+22 x^{4} \sqrt {-\frac {-x^{4}+2 x^{2}}{x^{4}-2 x^{2}+1}}+24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-24 x^{2} \sqrt {-\frac {-x^{4}+2 x^{2}}{x^{4}-2 x^{2}+1}}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +8 \sqrt {-\frac {-x^{4}+2 x^{2}}{x^{4}-2 x^{2}+1}}}{x \left (x^{2}+2\right )^{2} \left (x -1\right ) \left (x +1\right )}\right )}{6}\) | \(199\) |
1/6*((x^2-2)*x^2/(x^2-1)^2)^(1/2)*(x^2-1)*(arctan((x+2)/(x^2-2)^(1/2))-arc tan((x-2)/(x^2-2)^(1/2))+4*arctan(1/2*(x^2-2)^(1/2)))/x/(x^2-2)^(1/2)
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {\frac {-2 x^2+x^4}{\left (-1+x^2\right )^2}}}{2+x^2} \, dx=-\frac {1}{3} \, \arctan \left (\frac {{\left (x^{2} - 1\right )} \sqrt {\frac {x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}}}{x}\right ) + \frac {2}{3} \, \arctan \left (\frac {{\left (x^{2} - 1\right )} \sqrt {\frac {x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}}}{2 \, x}\right ) \]
-1/3*arctan((x^2 - 1)*sqrt((x^4 - 2*x^2)/(x^4 - 2*x^2 + 1))/x) + 2/3*arcta n(1/2*(x^2 - 1)*sqrt((x^4 - 2*x^2)/(x^4 - 2*x^2 + 1))/x)
\[ \int \frac {\sqrt {\frac {-2 x^2+x^4}{\left (-1+x^2\right )^2}}}{2+x^2} \, dx=\int \frac {\sqrt {\frac {x^{2} \left (x^{2} - 2\right )}{x^{4} - 2 x^{2} + 1}}}{x^{2} + 2}\, dx \]
\[ \int \frac {\sqrt {\frac {-2 x^2+x^4}{\left (-1+x^2\right )^2}}}{2+x^2} \, dx=\int { \frac {\sqrt {\frac {x^{4} - 2 \, x^{2}}{{\left (x^{2} - 1\right )}^{2}}}}{x^{2} + 2} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {\frac {-2 x^2+x^4}{\left (-1+x^2\right )^2}}}{2+x^2} \, dx=\frac {2}{3} \, \arctan \left (\frac {1}{2} \, \sqrt {x^{2} - 2}\right ) \mathrm {sgn}\left (x^{3} - x\right ) - \frac {1}{3} \, \arctan \left (\sqrt {x^{2} - 2}\right ) \mathrm {sgn}\left (x^{3} - x\right ) \]
Timed out. \[ \int \frac {\sqrt {\frac {-2 x^2+x^4}{\left (-1+x^2\right )^2}}}{2+x^2} \, dx=\int \frac {\sqrt {-\frac {2\,x^2-x^4}{{\left (x^2-1\right )}^2}}}{x^2+2} \,d x \]