\(\int \frac {1}{(-2+6 x^2-3 x^4)^{3/2}} \, dx\) [279]

Optimal result

Integrand size = 16, antiderivative size = 126 \[ \int \frac {1}{\left (-2+6 x^2-3 x^4\right )^{3/2}} \, dx=-\frac {x \left (4-3 x^2\right )}{4 \sqrt {-2+6 x^2-3 x^4}}+\frac {\sqrt [4]{3} E\left (\arccos \left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right )|\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{2 \sqrt {2}}+\frac {\left (1-\sqrt {3}\right ) \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{3+\sqrt {3}}} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{4 \sqrt {2} \sqrt [4]{3}} \] Output:

-1/4*x*(-3*x^2+4)/(-3*x^4+6*x^2-2)^(1/2)+1/4*3^(1/4)*EllipticE((1-3/(3+3^( 
1/2))*x^2)^(1/2),1/2*(2+2*3^(1/2))^(1/2))*2^(1/2)+1/24*(1-3^(1/2))*Inverse 
JacobiAM(arccos(3^(1/2)/(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 = 6.64 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (-2+6 x^2-3 x^4\right )^{3/2}} \, dx=\frac {6 x \left (-4+3 x^2\right )+3 \sqrt {2} \left (1+\sqrt {3}\right ) \sqrt {3-\sqrt {3}-3 x^2} \sqrt {2+\left (-3+\sqrt {3}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right )|2-\sqrt {3}\right )-\sqrt {2} \left (3+\sqrt {3}\right ) \sqrt {3-\sqrt {3}-3 x^2} \sqrt {2+\left (-3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right ),2-\sqrt {3}\right )}{24 \sqrt {-2+6 x^2-3 x^4}} \] Input:

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

Output:

(6*x*(-4 + 3*x^2) + 3*Sqrt[2]*(1 + Sqrt[3])*Sqrt[3 - Sqrt[3] - 3*x^2]*Sqrt 
[2 + (-3 + Sqrt[3])*x^2]*EllipticE[ArcSin[Sqrt[(3 + Sqrt[3])/2]*x], 2 - Sq 
rt[3]] - Sqrt[2]*(3 + Sqrt[3])*Sqrt[3 - Sqrt[3] - 3*x^2]*Sqrt[2 + (-3 + Sq 
rt[3])*x^2]*EllipticF[ArcSin[Sqrt[(3 + Sqrt[3])/2]*x], 2 - Sqrt[3]])/(24*S 
qrt[-2 + 6*x^2 - 3*x^4])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.03, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{24} \int \frac {6 \left (2-3 x^2\right )}{\sqrt {-3 x^4+6 x^2-2}}dx-\frac {x \left (4-3 x^2\right )}{4 \sqrt {-3 x^4+6 x^2-2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1494

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 322

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

\(\Big \downarrow \) 328

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

Input:

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

Output:

-1/4*(x*(4 - 3*x^2))/Sqrt[-2 + 6*x^2 - 3*x^4] + (Sqrt[3]*((Sqrt[2]*Ellipti 
cE[ArcCos[Sqrt[3/(3 + Sqrt[3])]*x], (1 + Sqrt[3])/2])/3^(1/4) + ((1 - Sqrt 
[3])*EllipticF[ArcCos[Sqrt[3/(3 + Sqrt[3])]*x], (1 + Sqrt[3])/2])/(Sqrt[2] 
*3^(3/4))))/4
 

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. \(222\) vs. \(2(101)=202\).

Time = 2.53 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.77

method	result	size
risch	\(\frac {x \left (3 x^{2}-4\right )}{4 \sqrt {-3 x^{4}+6 x^{2}-2}}+\frac {\sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {6-2 \sqrt {3}}\, \sqrt {-3 x^{4}+6 x^{2}-2}}-\frac {6 \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {6-2 \sqrt {3}}\, \sqrt {-3 x^{4}+6 x^{2}-2}\, \left (6+2 \sqrt {3}\right )}\)	\(223\)
default	\(\frac {\frac {3}{4} x^{3}-x}{\sqrt {-3 x^{4}+6 x^{2}-2}}+\frac {\sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {6-2 \sqrt {3}}\, \sqrt {-3 x^{4}+6 x^{2}-2}}-\frac {6 \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {6-2 \sqrt {3}}\, \sqrt {-3 x^{4}+6 x^{2}-2}\, \left (6+2 \sqrt {3}\right )}\)	\(224\)
elliptic	\(\frac {\frac {3}{4} x^{3}-x}{\sqrt {-3 x^{4}+6 x^{2}-2}}+\frac {\sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {6-2 \sqrt {3}}\, \sqrt {-3 x^{4}+6 x^{2}-2}}-\frac {6 \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6-2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {6-2 \sqrt {3}}\, \sqrt {-3 x^{4}+6 x^{2}-2}\, \left (6+2 \sqrt {3}\right )}\)	\(224\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\left (-2+6 x^2-3 x^4\right )^{3/2}} \, dx=-\frac {3 \, {\left (\sqrt {3} \sqrt {-2} {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )} + 3 \, \sqrt {-2} {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {3} + \frac {3}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {3} + \frac {3}{2}}\right )\,|\,-\sqrt {3} + 2) - {\left (5 \, \sqrt {3} \sqrt {-2} {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )} + 3 \, \sqrt {-2} {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {3} + \frac {3}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {3} + \frac {3}{2}}\right )\,|\,-\sqrt {3} + 2) + 6 \, \sqrt {-3 \, x^{4} + 6 \, x^{2} - 2} {\left (3 \, x^{3} - 4 \, x\right )}}{24 \, {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )}} \] Input:

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

Output:

-1/24*(3*(sqrt(3)*sqrt(-2)*(3*x^4 - 6*x^2 + 2) + 3*sqrt(-2)*(3*x^4 - 6*x^2 
 + 2))*sqrt(1/2*sqrt(3) + 3/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(3) + 3/2) 
), -sqrt(3) + 2) - (5*sqrt(3)*sqrt(-2)*(3*x^4 - 6*x^2 + 2) + 3*sqrt(-2)*(3 
*x^4 - 6*x^2 + 2))*sqrt(1/2*sqrt(3) + 3/2)*elliptic_f(arcsin(x*sqrt(1/2*sq 
rt(3) + 3/2)), -sqrt(3) + 2) + 6*sqrt(-3*x^4 + 6*x^2 - 2)*(3*x^3 - 4*x))/( 
3*x^4 - 6*x^2 + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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