Integrand size = 16, antiderivative size = 57 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx=\frac {x \left (5+x^2\right )}{24 \sqrt {3-2 x^2-x^4}}-\frac {E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )}{8 \sqrt {3}}+\frac {\operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )}{4 \sqrt {3}} \] Output:
1/24*x*(x^2+5)/(-x^4-2*x^2+3)^(1/2)-1/24*EllipticE(x,1/3*I*3^(1/2))*3^(1/2 )+1/12*EllipticF(x,1/3*I*3^(1/2))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx=\frac {1}{24} \left (\frac {5 x}{\sqrt {3-2 x^2-x^4}}+\frac {x^3}{\sqrt {3-2 x^2-x^4}}-i E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-2 i \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right ),-3\right )\right ) \] Input:
Integrate[(3 - 2*x^2 - x^4)^(-3/2),x]
Output:
((5*x)/Sqrt[3 - 2*x^2 - x^4] + x^3/Sqrt[3 - 2*x^2 - x^4] - I*EllipticE[I*A rcSinh[x/Sqrt[3]], -3] - (2*I)*EllipticF[I*ArcSinh[x/Sqrt[3]], -3])/24
Time = 0.35 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1405, 27, 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.
| \(\displaystyle \int \frac {1}{\left (-x^4-2 x^2+3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (x^2+5\right )}{24 \sqrt {-x^4-2 x^2+3}}-\frac {1}{48} \int -\frac {2 \left (3-x^2\right )}{\sqrt {-x^4-2 x^2+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \int \frac {3-x^2}{\sqrt {-x^4-2 x^2+3}}dx+\frac {x \left (x^2+5\right )}{24 \sqrt {-x^4-2 x^2+3}}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {1}{12} \int \frac {3-x^2}{2 \sqrt {1-x^2} \sqrt {x^2+3}}dx+\frac {x \left (x^2+5\right )}{24 \sqrt {-x^4-2 x^2+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \int \frac {3-x^2}{\sqrt {1-x^2} \sqrt {x^2+3}}dx+\frac {x \left (x^2+5\right )}{24 \sqrt {-x^4-2 x^2+3}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {1}{24} \left (6 \int \frac {1}{\sqrt {1-x^2} \sqrt {x^2+3}}dx-\int \frac {\sqrt {x^2+3}}{\sqrt {1-x^2}}dx\right )+\frac {x \left (x^2+5\right )}{24 \sqrt {-x^4-2 x^2+3}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{24} \left (2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-\int \frac {\sqrt {x^2+3}}{\sqrt {1-x^2}}dx\right )+\frac {x \left (x^2+5\right )}{24 \sqrt {-x^4-2 x^2+3}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{24} \left (2 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )-\sqrt {3} E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )\right )+\frac {x \left (x^2+5\right )}{24 \sqrt {-x^4-2 x^2+3}}\) |
Input:
Int[(3 - 2*x^2 - x^4)^(-3/2),x]
Output:
(x*(5 + x^2))/(24*Sqrt[3 - 2*x^2 - x^4]) + (-(Sqrt[3]*EllipticE[ArcSin[x], -1/3]) + 2*Sqrt[3]*EllipticF[ArcSin[x], -1/3])/24
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. 118 vs. \(2 (49 ) = 98\).
Time = 1.89 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.09
| method | result | size |
| risch | \(\frac {x \left (x^{2}+5\right )}{24 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{24 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{24 \sqrt {-x^{4}-2 x^{2}+3}}\) | \(119\) |
| default | \(\frac {\frac {5}{24} x +\frac {1}{24} x^{3}}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{24 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{24 \sqrt {-x^{4}-2 x^{2}+3}}\) | \(122\) |
| elliptic | \(\frac {\frac {5}{24} x +\frac {1}{24} x^{3}}{\sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )}{24 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (\operatorname {EllipticF}\left (x , \frac {i \sqrt {3}}{3}\right )-\operatorname {EllipticE}\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{24 \sqrt {-x^{4}-2 x^{2}+3}}\) | \(122\) |
Input:
int(1/(-x^4-2*x^2+3)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/24*x*(x^2+5)/(-x^4-2*x^2+3)^(1/2)+1/24*(-x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(- x^4-2*x^2+3)^(1/2)*EllipticF(x,1/3*I*3^(1/2))+1/24*(-x^2+1)^(1/2)*(3*x^2+9 )^(1/2)/(-x^4-2*x^2+3)^(1/2)*(EllipticF(x,1/3*I*3^(1/2))-EllipticE(x,1/3*I *3^(1/2)))
Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx=-\frac {\sqrt {3} {\left (x^{4} + 2 \, x^{2} - 3\right )} E(\arcsin \left (x\right )\,|\,-\frac {1}{3}) - 2 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} - 3\right )} F(\arcsin \left (x\right )\,|\,-\frac {1}{3}) + \sqrt {-x^{4} - 2 \, x^{2} + 3} {\left (x^{3} + 5 \, x\right )}}{24 \, {\left (x^{4} + 2 \, x^{2} - 3\right )}} \] Input:
integrate(1/(-x^4-2*x^2+3)^(3/2),x, algorithm="fricas")
Output:
-1/24*(sqrt(3)*(x^4 + 2*x^2 - 3)*elliptic_e(arcsin(x), -1/3) - 2*sqrt(3)*( x^4 + 2*x^2 - 3)*elliptic_f(arcsin(x), -1/3) + sqrt(-x^4 - 2*x^2 + 3)*(x^3 + 5*x))/(x^4 + 2*x^2 - 3)
\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- x^{4} - 2 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(-x**4-2*x**2+3)**(3/2),x)
Output:
Integral((-x**4 - 2*x**2 + 3)**(-3/2), x)
\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} - 2 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-x^4-2*x^2+3)^(3/2),x, algorithm="maxima")
Output:
integrate((-x^4 - 2*x^2 + 3)^(-3/2), x)
\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-x^{4} - 2 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-x^4-2*x^2+3)^(3/2),x, algorithm="giac")
Output:
integrate((-x^4 - 2*x^2 + 3)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-x^4-2\,x^2+3\right )}^{3/2}} \,d x \] Input:
int(1/(3 - x^4 - 2*x^2)^(3/2),x)
Output:
int(1/(3 - x^4 - 2*x^2)^(3/2), x)
\[ \int \frac {1}{\left (3-2 x^2-x^4\right )^{3/2}} \, dx=\frac {-\sqrt {-x^{4}-2 x^{2}+3}\, x +3 \left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}}{x^{6}+x^{4}-5 x^{2}+3}d x \right ) x^{4}+6 \left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}}{x^{6}+x^{4}-5 x^{2}+3}d x \right ) x^{2}-9 \left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}}{x^{6}+x^{4}-5 x^{2}+3}d x \right )-\left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}\, x^{2}}{x^{6}+x^{4}-5 x^{2}+3}d x \right ) x^{4}-2 \left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}\, x^{2}}{x^{6}+x^{4}-5 x^{2}+3}d x \right ) x^{2}+3 \left (\int \frac {\sqrt {-x^{4}-2 x^{2}+3}\, x^{2}}{x^{6}+x^{4}-5 x^{2}+3}d x \right )}{12 x^{4}+24 x^{2}-36} \] Input:
int(1/(-x^4-2*x^2+3)^(3/2),x)
Output:
( - sqrt( - x**4 - 2*x**2 + 3)*x + 3*int(sqrt( - x**4 - 2*x**2 + 3)/(x**6 + x**4 - 5*x**2 + 3),x)*x**4 + 6*int(sqrt( - x**4 - 2*x**2 + 3)/(x**6 + x* *4 - 5*x**2 + 3),x)*x**2 - 9*int(sqrt( - x**4 - 2*x**2 + 3)/(x**6 + x**4 - 5*x**2 + 3),x) - int((sqrt( - x**4 - 2*x**2 + 3)*x**2)/(x**6 + x**4 - 5*x **2 + 3),x)*x**4 - 2*int((sqrt( - x**4 - 2*x**2 + 3)*x**2)/(x**6 + x**4 - 5*x**2 + 3),x)*x**2 + 3*int((sqrt( - x**4 - 2*x**2 + 3)*x**2)/(x**6 + x**4 - 5*x**2 + 3),x))/(12*(x**4 + 2*x**2 - 3))