\(\int \frac {1}{(-3+6 x^2-2 x^4)^{3/2}} \, dx\) [264]

Optimal result

Integrand size = 16, antiderivative size = 127 \[ \int \frac {1}{\left (-3+6 x^2-2 x^4\right )^{3/2}} \, dx=-\frac {x \left (2-x^2\right )}{3 \sqrt {-3+6 x^2-2 x^4}}+\frac {E\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{\sqrt {2} 3^{3/4}}+\frac {\left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{6 \sqrt {2} \sqrt [4]{3}} \] Output:

-1/3*x*(-x^2+2)/(-2*x^4+6*x^2-3)^(1/2)+1/6*EllipticE(1/3*(9-(9-3*3^(1/2))* 
x^2)^(1/2),1/2*(2+2*3^(1/2))^(1/2))*2^(1/2)*3^(1/4)+1/36*(1-3^(1/2))*Inver 
seJacobiAM(arccos(1/3*(9-3*3^(1/2))^(1/2)*x),1/2*(2+2*3^(1/2))^(1/2))*2^(1 
/2)*3^(3/4)
 

Mathematica [A] (warning: unable to verify)

Time = 6.50 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\left (-3+6 x^2-2 x^4\right )^{3/2}} \, dx=\frac {4 x \left (-2+x^2\right )+\sqrt {2} \left (1+\sqrt {3}\right ) \sqrt {3-\sqrt {3}-2 x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2} E\left (\arcsin \left (\sqrt {1+\frac {1}{\sqrt {3}}} x\right )|2-\sqrt {3}\right )-\frac {1}{3} \sqrt {2} \left (3+\sqrt {3}\right ) \sqrt {3-\sqrt {3}-2 x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {1+\frac {1}{\sqrt {3}}} x\right ),2-\sqrt {3}\right )}{12 \sqrt {-3+6 x^2-2 x^4}} \] Input:

Integrate[(-3 + 6*x^2 - 2*x^4)^(-3/2),x]
 

Output:

(4*x*(-2 + x^2) + Sqrt[2]*(1 + Sqrt[3])*Sqrt[3 - Sqrt[3] - 2*x^2]*Sqrt[3 + 
 (-3 + Sqrt[3])*x^2]*EllipticE[ArcSin[Sqrt[1 + 1/Sqrt[3]]*x], 2 - Sqrt[3]] 
 - (Sqrt[2]*(3 + Sqrt[3])*Sqrt[3 - Sqrt[3] - 2*x^2]*Sqrt[3 + (-3 + Sqrt[3] 
)*x^2]*EllipticF[ArcSin[Sqrt[1 + 1/Sqrt[3]]*x], 2 - Sqrt[3]])/3)/(12*Sqrt[ 
-3 + 6*x^2 - 2*x^4])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 130, 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, 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.

\(\displaystyle \int \frac {1}{\left (-2 x^4+6 x^2-3\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{36} \int \frac {12 \left (1-x^2\right )}{\sqrt {-2 x^4+6 x^2-3}}dx-\frac {x \left (2-x^2\right )}{3 \sqrt {-2 x^4+6 x^2-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {1-x^2}{\sqrt {-2 x^4+6 x^2-3}}dx-\frac {x \left (2-x^2\right )}{3 \sqrt {-2 x^4+6 x^2-3}}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {2}{3} \sqrt {2} \int \frac {1-x^2}{2 \sqrt {-2 x^2+\sqrt {3}+3} \sqrt {2 x^2+\sqrt {3}-3}}dx-\frac {x \left (2-x^2\right )}{3 \sqrt {-2 x^4+6 x^2-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \sqrt {2} \int \frac {1-x^2}{\sqrt {-2 x^2+\sqrt {3}+3} \sqrt {2 x^2+\sqrt {3}-3}}dx-\frac {x \left (2-x^2\right )}{3 \sqrt {-2 x^4+6 x^2-3}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {1}{3} \sqrt {2} \left (-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {-2 x^2+\sqrt {3}+3} \sqrt {2 x^2+\sqrt {3}-3}}dx-\frac {1}{2} \int \frac {\sqrt {2 x^2+\sqrt {3}-3}}{\sqrt {-2 x^2+\sqrt {3}+3}}dx\right )-\frac {x \left (2-x^2\right )}{3 \sqrt {-2 x^4+6 x^2-3}}\)

\(\Big \downarrow \) 322

\(\displaystyle \frac {1}{3} \sqrt {2} \left (\frac {\left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{4 \sqrt [4]{3}}-\frac {1}{2} \int \frac {\sqrt {2 x^2+\sqrt {3}-3}}{\sqrt {-2 x^2+\sqrt {3}+3}}dx\right )-\frac {x \left (2-x^2\right )}{3 \sqrt {-2 x^4+6 x^2-3}}\)

\(\Big \downarrow \) 328

\(\displaystyle \frac {1}{3} \sqrt {2} \left (\frac {\left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{4 \sqrt [4]{3}}+\frac {1}{2} \sqrt [4]{3} E\left (\arccos \left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|\frac {1}{2} \left (1+\sqrt {3}\right )\right )\right )-\frac {x \left (2-x^2\right )}{3 \sqrt {-2 x^4+6 x^2-3}}\)

Input:

Int[(-3 + 6*x^2 - 2*x^4)^(-3/2),x]
 

Output:

-1/3*(x*(2 - x^2))/Sqrt[-3 + 6*x^2 - 2*x^4] + (Sqrt[2]*((3^(1/4)*EllipticE 
[ArcCos[Sqrt[(3 - Sqrt[3])/3]*x], (1 + Sqrt[3])/2])/2 + ((1 - Sqrt[3])*Ell 
ipticF[ArcCos[Sqrt[(3 - Sqrt[3])/3]*x], (1 + Sqrt[3])/2])/(4*3^(1/4))))/3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 322

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]
 

rule 328

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]
 

rule 399

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])))))
 

rule 1405

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]
 

rule 1494

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(220\) vs. \(2(103)=206\).

Time = 2.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.74

method	result	size
risch	\(\frac {x \left (x^{2}-2\right )}{3 \sqrt {-2 x^{4}+6 x^{2}-3}}+\frac {\sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}}-\frac {6 \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}\, \left (6+2 \sqrt {3}\right )}\)	\(221\)
default	\(\frac {-\frac {2}{3} x +\frac {1}{3} x^{3}}{\sqrt {-2 x^{4}+6 x^{2}-3}}+\frac {\sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}}-\frac {6 \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}\, \left (6+2 \sqrt {3}\right )}\)	\(224\)
elliptic	\(\frac {-\frac {2}{3} x +\frac {1}{3} x^{3}}{\sqrt {-2 x^{4}+6 x^{2}-3}}+\frac {\sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}}-\frac {6 \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}\, \left (6+2 \sqrt {3}\right )}\)	\(224\)

Input:

int(1/(-2*x^4+6*x^2-3)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*x*(x^2-2)/(-2*x^4+6*x^2-3)^(1/2)+1/(9-3*3^(1/2))^(1/2)*(1-(1-1/3*3^(1/ 
2))*x^2)^(1/2)*(1-(1+1/3*3^(1/2))*x^2)^(1/2)/(-2*x^4+6*x^2-3)^(1/2)*Ellipt 
icF(1/3*(9-3*3^(1/2))^(1/2)*x,1/2*6^(1/2)+1/2*2^(1/2))-6/(9-3*3^(1/2))^(1/ 
2)*(1-(1-1/3*3^(1/2))*x^2)^(1/2)*(1-(1+1/3*3^(1/2))*x^2)^(1/2)/(-2*x^4+6*x 
^2-3)^(1/2)/(6+2*3^(1/2))*(EllipticF(1/3*(9-3*3^(1/2))^(1/2)*x,1/2*6^(1/2) 
+1/2*2^(1/2))-EllipticE(1/3*(9-3*3^(1/2))^(1/2)*x,1/2*6^(1/2)+1/2*2^(1/2)) 
)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (-3+6 x^2-2 x^4\right )^{3/2}} \, dx=\frac {2 \, \sqrt {3} \sqrt {-3} {\left (2 \, x^{4} - 6 \, x^{2} + 3\right )} \sqrt {\frac {1}{3} \, \sqrt {3} + 1} F(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} + 1}\right )\,|\,-\sqrt {3} + 2) - {\left (\sqrt {3} \sqrt {-3} {\left (2 \, x^{4} - 6 \, x^{2} + 3\right )} + 3 \, \sqrt {-3} {\left (2 \, x^{4} - 6 \, x^{2} + 3\right )}\right )} \sqrt {\frac {1}{3} \, \sqrt {3} + 1} E(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} + 1}\right )\,|\,-\sqrt {3} + 2) - 6 \, \sqrt {-2 \, x^{4} + 6 \, x^{2} - 3} {\left (x^{3} - 2 \, x\right )}}{18 \, {\left (2 \, x^{4} - 6 \, x^{2} + 3\right )}} \] Input:

integrate(1/(-2*x^4+6*x^2-3)^(3/2),x, algorithm="fricas")
 

Output:

1/18*(2*sqrt(3)*sqrt(-3)*(2*x^4 - 6*x^2 + 3)*sqrt(1/3*sqrt(3) + 1)*ellipti 
c_f(arcsin(x*sqrt(1/3*sqrt(3) + 1)), -sqrt(3) + 2) - (sqrt(3)*sqrt(-3)*(2* 
x^4 - 6*x^2 + 3) + 3*sqrt(-3)*(2*x^4 - 6*x^2 + 3))*sqrt(1/3*sqrt(3) + 1)*e 
lliptic_e(arcsin(x*sqrt(1/3*sqrt(3) + 1)), -sqrt(3) + 2) - 6*sqrt(-2*x^4 + 
 6*x^2 - 3)*(x^3 - 2*x))/(2*x^4 - 6*x^2 + 3)
 

Sympy [F]

\[ \int \frac {1}{\left (-3+6 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- 2 x^{4} + 6 x^{2} - 3\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(-2*x**4+6*x**2-3)**(3/2),x)
 

Output:

Integral((-2*x**4 + 6*x**2 - 3)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (-3+6 x^2-2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-2 \, x^{4} + 6 \, x^{2} - 3\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-2*x^4+6*x^2-3)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-2*x^4 + 6*x^2 - 3)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (-3+6 x^2-2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-2 \, x^{4} + 6 \, x^{2} - 3\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-2*x^4+6*x^2-3)^(3/2),x, algorithm="giac")
 

Output:

integrate((-2*x^4 + 6*x^2 - 3)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (-3+6 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-2\,x^4+6\,x^2-3\right )}^{3/2}} \,d x \] Input:

int(1/(6*x^2 - 2*x^4 - 3)^(3/2),x)
 

Output:

int(1/(6*x^2 - 2*x^4 - 3)^(3/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {1}{\left (-3+6 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-2 x^{4}+6 x^{2}-3}}{4 x^{8}-24 x^{6}+48 x^{4}-36 x^{2}+9}d x \] Input:

int(1/(-2*x^4+6*x^2-3)^(3/2),x)
 

Output:

int(sqrt( - 2*x**4 + 6*x**2 - 3)/(4*x**8 - 24*x**6 + 48*x**4 - 36*x**2 + 9 
),x)