Integrand size = 30, antiderivative size = 83 \[ \int \frac {\sqrt {-2 x^2+x^4}}{\left (1-x^2\right ) \left (2+x^2\right )} \, dx=-\frac {2 \sqrt {-2 x^2+x^4} \arctan \left (\frac {1}{2} \sqrt {-2+x^2}\right )}{3 x \sqrt {-2+x^2}}+\frac {\sqrt {-2 x^2+x^4} \arctan \left (\sqrt {-2+x^2}\right )}{3 x \sqrt {-2+x^2}} \] Output:
-2/3*(x^4-2*x^2)^(1/2)*arctan(1/2*(x^2-2)^(1/2))/x/(x^2-2)^(1/2)+1/3*(x^4- 2*x^2)^(1/2)*arctan((x^2-2)^(1/2))/x/(x^2-2)^(1/2)
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {-2 x^2+x^4}}{\left (1-x^2\right ) \left (2+x^2\right )} \, dx=-\frac {x \sqrt {-2+x^2} \left (2 \arctan \left (\frac {1}{2} \sqrt {-2+x^2}\right )-\arctan \left (\sqrt {-2+x^2}\right )\right )}{3 \sqrt {x^2 \left (-2+x^2\right )}} \] Input:
Integrate[Sqrt[-2*x^2 + x^4]/((1 - x^2)*(2 + x^2)),x]
Output:
-1/3*(x*Sqrt[-2 + x^2]*(2*ArcTan[Sqrt[-2 + x^2]/2] - ArcTan[Sqrt[-2 + x^2] ]))/Sqrt[x^2*(-2 + x^2)]
Time = 0.49 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.75, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {x^4-2 x^2}}{\left (1-x^2\right ) \left (x^2+2\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x^4-2 x^2} \int \frac {x \sqrt {x^2-2}}{\left (1-x^2\right ) \left (x^2+2\right )}dx}{x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 435 |
\(\displaystyle \frac {\sqrt {x^4-2 x^2} \int \frac {\sqrt {x^2-2}}{\left (1-x^2\right ) \left (x^2+2\right )}dx^2}{2 x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 94 |
\(\displaystyle \frac {\sqrt {x^4-2 x^2} \left (-\frac {1}{3} \int \frac {1}{\left (1-x^2\right ) \sqrt {x^2-2}}dx^2-\frac {4}{3} \int \frac {1}{\sqrt {x^2-2} \left (x^2+2\right )}dx^2\right )}{2 x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {x^4-2 x^2} \left (-\frac {2}{3} \int \frac {1}{-x^4-1}d\sqrt {x^2-2}-\frac {8}{3} \int \frac {1}{x^4+4}d\sqrt {x^2-2}\right )}{2 x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {x^4-2 x^2} \left (-\frac {2}{3} \int \frac {1}{-x^4-1}d\sqrt {x^2-2}-\frac {4}{3} \arctan \left (\frac {\sqrt {x^2-2}}{2}\right )\right )}{2 x \sqrt {x^2-2}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\sqrt {x^4-2 x^2} \left (\frac {2}{3} \arctan \left (\sqrt {x^2-2}\right )-\frac {4}{3} \arctan \left (\frac {\sqrt {x^2-2}}{2}\right )\right )}{2 x \sqrt {x^2-2}}\) |
Input:
Int[Sqrt[-2*x^2 + x^4]/((1 - x^2)*(2 + x^2)),x]
Output:
(Sqrt[-2*x^2 + x^4]*((-4*ArcTan[Sqrt[-2 + x^2]/2])/3 + (2*ArcTan[Sqrt[-2 + x^2]])/3))/(2*x*Sqrt[-2 + x^2])
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]
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {\arctan \left (\frac {1}{\sqrt {x^{2}-2}}\right )}{3}-\frac {i \operatorname {arctanh}\left (\frac {\left (i \sqrt {2}+x \right ) \sqrt {2}}{2 \sqrt {x^{2}-2}}\right )}{3}+\frac {\arctan \left (\frac {\sqrt {2}\, \left (\sqrt {2}+i x \right )}{2 \sqrt {x^{2}-2}}\right )}{3}\) | \(59\) |
default | \(\frac {\sqrt {x^{4}-2 x^{2}}\, \left (\arctan \left (\frac {x -2}{\sqrt {x^{2}-2}}\right )-\arctan \left (\frac {2+x}{\sqrt {x^{2}-2}}\right )-4 \arctan \left (\frac {\sqrt {x^{2}-2}}{2}\right )\right )}{6 x \sqrt {x^{2}-2}}\) | \(63\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{7}+15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{5}+6 \sqrt {x^{4}-2 x^{2}}\, x^{4}-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-16 \sqrt {x^{4}-2 x^{2}}\, x^{2}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +8 \sqrt {x^{4}-2 x^{2}}}{\left (x^{2}+2\right )^{2} \left (1+x \right ) \left (-1+x \right ) x}\right )}{6}\) | \(119\) |
Input:
int((x^4-2*x^2)^(1/2)/(-x^2+1)/(x^2+2),x,method=_RETURNVERBOSE)
Output:
-1/3*arctan(1/(x^2-2)^(1/2))-1/3*I*arctanh(1/2*(I*2^(1/2)+x)*2^(1/2)/(x^2- 2)^(1/2))+1/3*arctan(1/2*2^(1/2)/(x^2-2)^(1/2)*(2^(1/2)+I*x))
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {-2 x^2+x^4}}{\left (1-x^2\right ) \left (2+x^2\right )} \, dx=\frac {1}{3} \, \arctan \left (\frac {\sqrt {x^{4} - 2 \, x^{2}}}{x}\right ) - \frac {2}{3} \, \arctan \left (\frac {\sqrt {x^{4} - 2 \, x^{2}}}{2 \, x}\right ) \] Input:
integrate((x^4-2*x^2)^(1/2)/(-x^2+1)/(x^2+2),x, algorithm="fricas")
Output:
1/3*arctan(sqrt(x^4 - 2*x^2)/x) - 2/3*arctan(1/2*sqrt(x^4 - 2*x^2)/x)
\[ \int \frac {\sqrt {-2 x^2+x^4}}{\left (1-x^2\right ) \left (2+x^2\right )} \, dx=- \int \frac {\sqrt {x^{4} - 2 x^{2}}}{x^{4} + x^{2} - 2}\, dx \] Input:
integrate((x**4-2*x**2)**(1/2)/(-x**2+1)/(x**2+2),x)
Output:
-Integral(sqrt(x**4 - 2*x**2)/(x**4 + x**2 - 2), x)
\[ \int \frac {\sqrt {-2 x^2+x^4}}{\left (1-x^2\right ) \left (2+x^2\right )} \, dx=\int { -\frac {\sqrt {x^{4} - 2 \, x^{2}}}{{\left (x^{2} + 2\right )} {\left (x^{2} - 1\right )}} \,d x } \] Input:
integrate((x^4-2*x^2)^(1/2)/(-x^2+1)/(x^2+2),x, algorithm="maxima")
Output:
-integrate(sqrt(x^4 - 2*x^2)/((x^2 + 2)*(x^2 - 1)), x)
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {-2 x^2+x^4}}{\left (1-x^2\right ) \left (2+x^2\right )} \, dx=-\frac {1}{3} \, {\left (\arctan \left (i \, \sqrt {2}\right ) - 2 \, \arctan \left (\frac {1}{2} i \, \sqrt {2}\right )\right )} \mathrm {sgn}\left (x\right ) - \frac {2}{3} \, \arctan \left (\frac {1}{2} \, \sqrt {x^{2} - 2}\right ) \mathrm {sgn}\left (x\right ) + \frac {1}{3} \, \arctan \left (\sqrt {x^{2} - 2}\right ) \mathrm {sgn}\left (x\right ) \] Input:
integrate((x^4-2*x^2)^(1/2)/(-x^2+1)/(x^2+2),x, algorithm="giac")
Output:
-1/3*(arctan(I*sqrt(2)) - 2*arctan(1/2*I*sqrt(2)))*sgn(x) - 2/3*arctan(1/2 *sqrt(x^2 - 2))*sgn(x) + 1/3*arctan(sqrt(x^2 - 2))*sgn(x)
Timed out. \[ \int \frac {\sqrt {-2 x^2+x^4}}{\left (1-x^2\right ) \left (2+x^2\right )} \, dx=\int -\frac {\sqrt {x^4-2\,x^2}}{\left (x^2-1\right )\,\left (x^2+2\right )} \,d x \] Input:
int(-(x^4 - 2*x^2)^(1/2)/((x^2 - 1)*(x^2 + 2)),x)
Output:
int(-(x^4 - 2*x^2)^(1/2)/((x^2 - 1)*(x^2 + 2)), x)
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {-2 x^2+x^4}}{\left (1-x^2\right ) \left (2+x^2\right )} \, dx=\frac {\mathit {atan} \left (\frac {\sqrt {x^{2}-2}\, x +x^{2}-2}{\sqrt {x^{2}-2}+x}\right )}{3}-\frac {2 \mathit {atan} \left (\frac {\sqrt {x^{2}-2}\, x +x^{2}-2}{2 \sqrt {x^{2}-2}+2 x}\right )}{3} \] Input:
int((x^4-2*x^2)^(1/2)/(-x^2+1)/(x^2+2),x)
Output:
(atan((sqrt(x**2 - 2)*x + x**2 - 2)/(sqrt(x**2 - 2) + x)) - 2*atan((sqrt(x **2 - 2)*x + x**2 - 2)/(2*sqrt(x**2 - 2) + 2*x)))/3