Integrand size = 20, antiderivative size = 50 \[ \int \frac {x^2}{\sqrt {-8+6 x^2-x^4}} \, dx=\frac {2 \sqrt {-2+x^2} E\left (\arcsin \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{2}\right )}{\sqrt {2-x^2}}-2 \sqrt {2} \operatorname {EllipticF}\left (\arccos \left (\frac {x}{2}\right ),2\right ) \] Output:
2*(x^2-2)^(1/2)*EllipticE(1/2*x*2^(1/2),1/2*2^(1/2))/(-x^2+2)^(1/2)-2*2^(1 /2)*InverseJacobiAM(arccos(1/2*x),2^(1/2))
Time = 10.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.34 \[ \int \frac {x^2}{\sqrt {-8+6 x^2-x^4}} \, dx=-\frac {2 \sqrt {2-x^2} \sqrt {4-x^2} \left (E\left (\arcsin \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{2}\right )-\operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{2}\right )\right )}{\sqrt {-8+6 x^2-x^4}} \] Input:
Integrate[x^2/Sqrt[-8 + 6*x^2 - x^4],x]
Output:
(-2*Sqrt[2 - x^2]*Sqrt[4 - x^2]*(EllipticE[ArcSin[x/Sqrt[2]], 1/2] - Ellip ticF[ArcSin[x/Sqrt[2]], 1/2]))/Sqrt[-8 + 6*x^2 - x^4]
Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.62, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1452, 27, 389, 322, 328}
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.
\(\displaystyle \int \frac {x^2}{\sqrt {-x^4+6 x^2-8}} \, dx\) |
\(\Big \downarrow \) 1452 |
\(\displaystyle 2 \int \frac {x^2}{2 \sqrt {4-x^2} \sqrt {x^2-2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x^2}{\sqrt {4-x^2} \sqrt {x^2-2}}dx\) |
\(\Big \downarrow \) 389 |
\(\displaystyle 2 \int \frac {1}{\sqrt {4-x^2} \sqrt {x^2-2}}dx+\int \frac {\sqrt {x^2-2}}{\sqrt {4-x^2}}dx\) |
\(\Big \downarrow \) 322 |
\(\displaystyle \int \frac {\sqrt {x^2-2}}{\sqrt {4-x^2}}dx-\sqrt {2} \operatorname {EllipticF}\left (\arccos \left (\frac {x}{2}\right ),2\right )\) |
\(\Big \downarrow \) 328 |
\(\displaystyle -\sqrt {2} \operatorname {EllipticF}\left (\arccos \left (\frac {x}{2}\right ),2\right )-\sqrt {2} E\left (\left .\arccos \left (\frac {x}{2}\right )\right |2\right )\) |
Input:
Int[x^2/Sqrt[-8 + 6*x^2 - x^4],x]
Output:
-(Sqrt[2]*EllipticE[ArcCos[x/2], 2]) - Sqrt[2]*EllipticF[ArcCos[x/2], 2]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(-(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a - b*(c/d)])^(-1))*EllipticF[ArcCos[Rt[-d/ c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (-Sqrt[a - b*(c/d)]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcCos[Rt[-d/c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[1/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b Int [1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N eQ[b*c - a*d, 0] && !SimplerSqrtQ[-b/a, -d/c]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c] Int[x^2/(Sqrt[b + q + 2*c*x^2]*Sqrt [-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
Time = 1.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\sqrt {-x^{2}+4}\, \sqrt {-2 x^{2}+4}\, \left (\operatorname {EllipticF}\left (\frac {x}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {x}{2}, \sqrt {2}\right )\right )}{\sqrt {-x^{4}+6 x^{2}-8}}\) | \(51\) |
elliptic | \(\frac {\sqrt {-x^{2}+4}\, \sqrt {-2 x^{2}+4}\, \left (\operatorname {EllipticF}\left (\frac {x}{2}, \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {x}{2}, \sqrt {2}\right )\right )}{\sqrt {-x^{4}+6 x^{2}-8}}\) | \(51\) |
Input:
int(x^2/(-x^4+6*x^2-8)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-x^2+4)^(1/2)*(-2*x^2+4)^(1/2)/(-x^4+6*x^2-8)^(1/2)*(EllipticF(1/2*x,2^(1 /2))-EllipticE(1/2*x,2^(1/2)))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{\sqrt {-8+6 x^2-x^4}} \, dx=\frac {-8 i \, x E(\arcsin \left (\frac {2}{x}\right )\,|\,\frac {1}{2}) + 8 i \, x F(\arcsin \left (\frac {2}{x}\right )\,|\,\frac {1}{2}) - \sqrt {-x^{4} + 6 \, x^{2} - 8}}{x} \] Input:
integrate(x^2/(-x^4+6*x^2-8)^(1/2),x, algorithm="fricas")
Output:
(-8*I*x*elliptic_e(arcsin(2/x), 1/2) + 8*I*x*elliptic_f(arcsin(2/x), 1/2) - sqrt(-x^4 + 6*x^2 - 8))/x
\[ \int \frac {x^2}{\sqrt {-8+6 x^2-x^4}} \, dx=\int \frac {x^{2}}{\sqrt {- \left (x - 2\right ) \left (x + 2\right ) \left (x^{2} - 2\right )}}\, dx \] Input:
integrate(x**2/(-x**4+6*x**2-8)**(1/2),x)
Output:
Integral(x**2/sqrt(-(x - 2)*(x + 2)*(x**2 - 2)), x)
\[ \int \frac {x^2}{\sqrt {-8+6 x^2-x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {-x^{4} + 6 \, x^{2} - 8}} \,d x } \] Input:
integrate(x^2/(-x^4+6*x^2-8)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2/sqrt(-x^4 + 6*x^2 - 8), x)
\[ \int \frac {x^2}{\sqrt {-8+6 x^2-x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {-x^{4} + 6 \, x^{2} - 8}} \,d x } \] Input:
integrate(x^2/(-x^4+6*x^2-8)^(1/2),x, algorithm="giac")
Output:
integrate(x^2/sqrt(-x^4 + 6*x^2 - 8), x)
Timed out. \[ \int \frac {x^2}{\sqrt {-8+6 x^2-x^4}} \, dx=\int \frac {x^2}{\sqrt {-x^4+6\,x^2-8}} \,d x \] Input:
int(x^2/(6*x^2 - x^4 - 8)^(1/2),x)
Output:
int(x^2/(6*x^2 - x^4 - 8)^(1/2), x)
\[ \int \frac {x^2}{\sqrt {-8+6 x^2-x^4}} \, dx=-\left (\int \frac {\sqrt {-x^{4}+6 x^{2}-8}\, x^{2}}{x^{4}-6 x^{2}+8}d x \right ) \] Input:
int(x^2/(-x^4+6*x^2-8)^(1/2),x)
Output:
- int((sqrt( - x**4 + 6*x**2 - 8)*x**2)/(x**4 - 6*x**2 + 8),x)