Integrand size = 14, antiderivative size = 55 \[ \int \frac {1}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\frac {x \left (5-x^2\right )}{18 \sqrt {2+x^2-x^4}}+\frac {1}{18} E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {1}{6} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right ) \]
1/18*EllipticE(1/2*x*2^(1/2),I*2^(1/2))+1/6*EllipticF(1/2*x*2^(1/2),I*2^(1 /2))+1/18*x*(-x^2+5)/(-x^4+x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 5.77 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\frac {1}{18} \left (\frac {5 x}{\sqrt {2+x^2-x^4}}-\frac {x^3}{\sqrt {2+x^2-x^4}}+i \sqrt {2} E\left (i \text {arcsinh}(x)\left |-\frac {1}{2}\right .\right )-3 i \sqrt {2} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )\right ) \]
((5*x)/Sqrt[2 + x^2 - x^4] - x^3/Sqrt[2 + x^2 - x^4] + I*Sqrt[2]*EllipticE [I*ArcSinh[x], -1/2] - (3*I)*Sqrt[2]*EllipticF[I*ArcSinh[x], -1/2])/18
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1405, 25, 1494, 27, 399, 321, 327}
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.
\(\displaystyle \int \frac {1}{\left (-x^4+x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1405 |
\(\displaystyle \frac {x \left (5-x^2\right )}{18 \sqrt {-x^4+x^2+2}}-\frac {1}{18} \int -\frac {x^2+4}{\sqrt {-x^4+x^2+2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{18} \int \frac {x^2+4}{\sqrt {-x^4+x^2+2}}dx+\frac {x \left (5-x^2\right )}{18 \sqrt {-x^4+x^2+2}}\) |
\(\Big \downarrow \) 1494 |
\(\displaystyle \frac {1}{9} \int \frac {x^2+4}{2 \sqrt {2-x^2} \sqrt {x^2+1}}dx+\frac {x \left (5-x^2\right )}{18 \sqrt {-x^4+x^2+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{18} \int \frac {x^2+4}{\sqrt {2-x^2} \sqrt {x^2+1}}dx+\frac {x \left (5-x^2\right )}{18 \sqrt {-x^4+x^2+2}}\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {1}{18} \left (3 \int \frac {1}{\sqrt {2-x^2} \sqrt {x^2+1}}dx+\int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx\right )+\frac {x \left (5-x^2\right )}{18 \sqrt {-x^4+x^2+2}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {1}{18} \left (\int \frac {\sqrt {x^2+1}}{\sqrt {2-x^2}}dx+3 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )\right )+\frac {x \left (5-x^2\right )}{18 \sqrt {-x^4+x^2+2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {1}{18} \left (3 \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )+E\left (\left .\arcsin \left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\right )+\frac {x \left (5-x^2\right )}{18 \sqrt {-x^4+x^2+2}}\) |
(x*(5 - x^2))/(18*Sqrt[2 + x^2 - x^4]) + (EllipticE[ArcSin[x/Sqrt[2]], -2] + 3*EllipticF[ArcSin[x/Sqrt[2]], -2])/18
3.4.44.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c] Int[(d + e*x^2)/(Sqr t[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e }, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (53 ) = 106\).
Time = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.36
method | result | size |
risch | \(-\frac {x \left (x^{2}-5\right )}{18 \sqrt {-x^{4}+x^{2}+2}}+\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{9 \sqrt {-x^{4}+x^{2}+2}}-\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}\) | \(130\) |
default | \(\frac {\frac {5}{18} x -\frac {1}{18} x^{3}}{\sqrt {-x^{4}+x^{2}+2}}+\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{9 \sqrt {-x^{4}+x^{2}+2}}-\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}\) | \(133\) |
elliptic | \(\frac {\frac {5}{18} x -\frac {1}{18} x^{3}}{\sqrt {-x^{4}+x^{2}+2}}+\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{9 \sqrt {-x^{4}+x^{2}+2}}-\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )-E\left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\right )}{36 \sqrt {-x^{4}+x^{2}+2}}\) | \(133\) |
-1/18*x*(x^2-5)/(-x^4+x^2+2)^(1/2)+1/9*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1 /2)/(-x^4+x^2+2)^(1/2)*EllipticF(1/2*x*2^(1/2),I*2^(1/2))-1/36*2^(1/2)*(-2 *x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*(EllipticF(1/2*x*2^(1/2),I* 2^(1/2))-EllipticE(1/2*x*2^(1/2),I*2^(1/2)))
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\frac {{\left (x^{4} - x^{2} - 2\right )} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x\right )\,|\,-2) + 7 \, {\left (x^{4} - x^{2} - 2\right )} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x\right )\,|\,-2) + 2 \, \sqrt {-x^{4} + x^{2} + 2} {\left (x^{3} - 5 \, x\right )}}{36 \, {\left (x^{4} - x^{2} - 2\right )}} \]
1/36*((x^4 - x^2 - 2)*elliptic_e(arcsin(1/2*sqrt(2)*x), -2) + 7*(x^4 - x^2 - 2)*elliptic_f(arcsin(1/2*sqrt(2)*x), -2) + 2*sqrt(-x^4 + x^2 + 2)*(x^3 - 5*x))/(x^4 - x^2 - 2)
\[ \int \frac {1}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- x^{4} + x^{2} + 2\right )^{\frac {3}{2}}}\, dx \]
Timed out. \[ \int \frac {1}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (2+x^2-x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-x^4+x^2+2\right )}^{3/2}} \,d x \]