\(\int \frac {1}{(-2+6 x^2+3 x^4)^{3/2}} \, dx\) [203]

Optimal result

Integrand size = 16, antiderivative size = 225 \[ \int \frac {1}{\left (-2+6 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {x \left (8+3 x^2\right )}{20 \sqrt {-2+6 x^2+3 x^4}}+\frac {\sqrt {\frac {3}{-3+\sqrt {15}}} \sqrt {2-\left (3-\sqrt {15}\right ) x^2} \sqrt {2-\left (3+\sqrt {15}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {15}\right )} x\right )|-4+\sqrt {15}\right )}{20 \sqrt {-2+6 x^2+3 x^4}}-\frac {\sqrt {2-\left (3-\sqrt {15}\right ) x^2} \sqrt {2-\left (3+\sqrt {15}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {15}\right )} x\right ),-4+\sqrt {15}\right )}{4 \sqrt {5 \left (-3+\sqrt {15}\right )} \sqrt {-2+6 x^2+3 x^4}} \] Output:

-1/20*x*(3*x^2+8)/(3*x^4+6*x^2-2)^(1/2)+1/20*3^(1/2)/(-3+15^(1/2))^(1/2)*( 
2-(3-15^(1/2))*x^2)^(1/2)*(2-(3+15^(1/2))*x^2)^(1/2)*EllipticE(1/2*(6+2*15 
^(1/2))^(1/2)*x,1/2*I*10^(1/2)-1/2*I*6^(1/2))/(3*x^4+6*x^2-2)^(1/2)-1/4*(2 
-(3-15^(1/2))*x^2)^(1/2)*(2-(3+15^(1/2))*x^2)^(1/2)*EllipticF(1/2*(6+2*15^ 
(1/2))^(1/2)*x,1/2*I*10^(1/2)-1/2*I*6^(1/2))/(-15+5*15^(1/2))^(1/2)/(3*x^4 
+6*x^2-2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.85 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (-2+6 x^2+3 x^4\right )^{3/2}} \, dx=\frac {-3 x \left (8+3 x^2\right )+3 i \sqrt {-3+\sqrt {15}} \sqrt {2-6 x^2-3 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{3+\sqrt {15}}} x\right )|-4-\sqrt {15}\right )-\frac {3 i \left (-5+\sqrt {15}\right ) \sqrt {2-6 x^2-3 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{3+\sqrt {15}}} x\right ),-4-\sqrt {15}\right )}{\sqrt {-3+\sqrt {15}}}}{60 \sqrt {-2+6 x^2+3 x^4}} \] Input:

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

Output:

(-3*x*(8 + 3*x^2) + (3*I)*Sqrt[-3 + Sqrt[15]]*Sqrt[2 - 6*x^2 - 3*x^4]*Elli 
pticE[I*ArcSinh[Sqrt[3/(3 + Sqrt[15])]*x], -4 - Sqrt[15]] - ((3*I)*(-5 + S 
qrt[15])*Sqrt[2 - 6*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[3/(3 + Sqrt[15]) 
]*x], -4 - Sqrt[15]])/Sqrt[-3 + Sqrt[15]])/(60*Sqrt[-2 + 6*x^2 + 3*x^4])
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.57, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1501, 27, 1411, 1498}

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

\(\displaystyle \frac {1}{120} \int -\frac {6 \left (2-3 x^2\right )}{\sqrt {3 x^4+6 x^2-2}}dx-\frac {x \left (3 x^2+8\right )}{20 \sqrt {3 x^4+6 x^2-2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{20} \int \frac {2-3 x^2}{\sqrt {3 x^4+6 x^2-2}}dx-\frac {x \left (3 x^2+8\right )}{20 \sqrt {3 x^4+6 x^2-2}}\)

\(\Big \downarrow \) 1501

\(\displaystyle \frac {1}{20} \left (\frac {1}{2} \int \frac {2 \left (3 x^2-\sqrt {15}+3\right )}{\sqrt {3 x^4+6 x^2-2}}dx-\left (5-\sqrt {15}\right ) \int \frac {1}{\sqrt {3 x^4+6 x^2-2}}dx\right )-\frac {x \left (3 x^2+8\right )}{20 \sqrt {3 x^4+6 x^2-2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{20} \left (\int \frac {3 x^2-\sqrt {15}+3}{\sqrt {3 x^4+6 x^2-2}}dx-\left (5-\sqrt {15}\right ) \int \frac {1}{\sqrt {3 x^4+6 x^2-2}}dx\right )-\frac {x \left (3 x^2+8\right )}{20 \sqrt {3 x^4+6 x^2-2}}\)

\(\Big \downarrow \) 1411

\(\displaystyle \frac {1}{20} \left (\int \frac {3 x^2-\sqrt {15}+3}{\sqrt {3 x^4+6 x^2-2}}dx-\frac {\left (5-\sqrt {15}\right ) \sqrt {\frac {2-\left (3-\sqrt {15}\right ) x^2}{2-\left (3+\sqrt {15}\right ) x^2}} \sqrt {\left (3+\sqrt {15}\right ) x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{15} x}{\sqrt {\left (3+\sqrt {15}\right ) x^2-2}}\right ),\frac {1}{10} \left (5+\sqrt {15}\right )\right )}{2 \sqrt [4]{15} \sqrt {\frac {1}{2-\left (3+\sqrt {15}\right ) x^2}} \sqrt {3 x^4+6 x^2-2}}\right )-\frac {x \left (3 x^2+8\right )}{20 \sqrt {3 x^4+6 x^2-2}}\)

\(\Big \downarrow \) 1498

\(\displaystyle \frac {1}{20} \left (-\frac {\left (5-\sqrt {15}\right ) \sqrt {\frac {2-\left (3-\sqrt {15}\right ) x^2}{2-\left (3+\sqrt {15}\right ) x^2}} \sqrt {\left (3+\sqrt {15}\right ) x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{15} x}{\sqrt {\left (3+\sqrt {15}\right ) x^2-2}}\right ),\frac {1}{10} \left (5+\sqrt {15}\right )\right )}{2 \sqrt [4]{15} \sqrt {\frac {1}{2-\left (3+\sqrt {15}\right ) x^2}} \sqrt {3 x^4+6 x^2-2}}-\frac {\sqrt [4]{15} \sqrt {\frac {2-\left (3-\sqrt {15}\right ) x^2}{2-\left (3+\sqrt {15}\right ) x^2}} \sqrt {\left (3+\sqrt {15}\right ) x^2-2} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{15} x}{\sqrt {\left (3+\sqrt {15}\right ) x^2-2}}\right )|\frac {1}{10} \left (5+\sqrt {15}\right )\right )}{\sqrt {\frac {1}{2-\left (3+\sqrt {15}\right ) x^2}} \sqrt {3 x^4+6 x^2-2}}+\frac {x \left (3 x^2+\sqrt {15}+3\right )}{\sqrt {3 x^4+6 x^2-2}}\right )-\frac {x \left (3 x^2+8\right )}{20 \sqrt {3 x^4+6 x^2-2}}\)

Input:

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

Output:

-1/20*(x*(8 + 3*x^2))/Sqrt[-2 + 6*x^2 + 3*x^4] + ((x*(3 + Sqrt[15] + 3*x^2 
))/Sqrt[-2 + 6*x^2 + 3*x^4] - (15^(1/4)*Sqrt[(2 - (3 - Sqrt[15])*x^2)/(2 - 
 (3 + Sqrt[15])*x^2)]*Sqrt[-2 + (3 + Sqrt[15])*x^2]*EllipticE[ArcSin[(Sqrt 
[2]*15^(1/4)*x)/Sqrt[-2 + (3 + Sqrt[15])*x^2]], (5 + Sqrt[15])/10])/(Sqrt[ 
(2 - (3 + Sqrt[15])*x^2)^(-1)]*Sqrt[-2 + 6*x^2 + 3*x^4]) - ((5 - Sqrt[15]) 
*Sqrt[(2 - (3 - Sqrt[15])*x^2)/(2 - (3 + Sqrt[15])*x^2)]*Sqrt[-2 + (3 + Sq 
rt[15])*x^2]*EllipticF[ArcSin[(Sqrt[2]*15^(1/4)*x)/Sqrt[-2 + (3 + Sqrt[15] 
)*x^2]], (5 + Sqrt[15])/10])/(2*15^(1/4)*Sqrt[(2 - (3 + Sqrt[15])*x^2)^(-1 
)]*Sqrt[-2 + 6*x^2 + 3*x^4]))/20
 

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 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 1411

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*(Sqrt[( 
2*a + (b + q)*x^2)/q]/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2) 
]))*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b + q)/(2*q)], x] 
] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
 

rule 1498

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[e*x*((b + q + 2*c*x^2)/(2*c*Sqrt[ 
a + b*x^2 + c*x^4])), x] - Simp[e*q*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q) 
*x^2)]*(Sqrt[(2*a + (b + q)*x^2)/q]/(2*c*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2* 
a + (b + q)*x^2)]))*EllipticE[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q)]], (b 
 + q)/(2*q)], x] /; EqQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] 
 && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
 

rule 1501

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*c*d - e*(b - q))/(2*c)   Int[1 
/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[e/(2*c)   Int[(b - q + 2*c*x^2)/Sqr 
t[a + b*x^2 + c*x^4], x], x] /; NeQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, 
c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]
 

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.02

method	result	size
risch	\(-\frac {x \left (3 x^{2}+8\right )}{20 \sqrt {3 x^{4}+6 x^{2}-2}}-\frac {\sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {15}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {15}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )}{5 \sqrt {6-2 \sqrt {15}}\, \sqrt {3 x^{4}+6 x^{2}-2}}+\frac {6 \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {15}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {15}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )\right )}{5 \sqrt {6-2 \sqrt {15}}\, \sqrt {3 x^{4}+6 x^{2}-2}\, \left (6+2 \sqrt {15}\right )}\)	\(230\)
default	\(-\frac {6 \left (\frac {1}{15} x +\frac {1}{40} x^{3}\right )}{\sqrt {3 x^{4}+6 x^{2}-2}}-\frac {\sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {15}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {15}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )}{5 \sqrt {6-2 \sqrt {15}}\, \sqrt {3 x^{4}+6 x^{2}-2}}+\frac {6 \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {15}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {15}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )\right )}{5 \sqrt {6-2 \sqrt {15}}\, \sqrt {3 x^{4}+6 x^{2}-2}\, \left (6+2 \sqrt {15}\right )}\)	\(231\)
elliptic	\(-\frac {6 \left (\frac {1}{15} x +\frac {1}{40} x^{3}\right )}{\sqrt {3 x^{4}+6 x^{2}-2}}-\frac {\sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {15}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {15}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )}{5 \sqrt {6-2 \sqrt {15}}\, \sqrt {3 x^{4}+6 x^{2}-2}}+\frac {6 \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {15}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {15}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {6-2 \sqrt {15}}}{2}, \frac {i \sqrt {6}}{2}+\frac {i \sqrt {10}}{2}\right )\right )}{5 \sqrt {6-2 \sqrt {15}}\, \sqrt {3 x^{4}+6 x^{2}-2}\, \left (6+2 \sqrt {15}\right )}\)	\(231\)

Input:

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

Output:

-1/20*x*(3*x^2+8)/(3*x^4+6*x^2-2)^(1/2)-1/5/(6-2*15^(1/2))^(1/2)*(1-(3/2-1 
/2*15^(1/2))*x^2)^(1/2)*(1-(3/2+1/2*15^(1/2))*x^2)^(1/2)/(3*x^4+6*x^2-2)^( 
1/2)*EllipticF(1/2*x*(6-2*15^(1/2))^(1/2),1/2*I*6^(1/2)+1/2*I*10^(1/2))+6/ 
5/(6-2*15^(1/2))^(1/2)*(1-(3/2-1/2*15^(1/2))*x^2)^(1/2)*(1-(3/2+1/2*15^(1/ 
2))*x^2)^(1/2)/(3*x^4+6*x^2-2)^(1/2)/(6+2*15^(1/2))*(EllipticF(1/2*x*(6-2* 
15^(1/2))^(1/2),1/2*I*6^(1/2)+1/2*I*10^(1/2))-EllipticE(1/2*x*(6-2*15^(1/2 
))^(1/2),1/2*I*6^(1/2)+1/2*I*10^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (-2+6 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {3 \, {\left (\sqrt {15} \sqrt {-2} {\left (3 \, x^{4} + 6 \, x^{2} - 2\right )} + 3 \, \sqrt {-2} {\left (3 \, x^{4} + 6 \, x^{2} - 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {15} + \frac {3}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {15} + \frac {3}{2}}\right )\,|\,\sqrt {15} - 4) - {\left (5 \, \sqrt {15} \sqrt {-2} {\left (3 \, x^{4} + 6 \, x^{2} - 2\right )} + 3 \, \sqrt {-2} {\left (3 \, x^{4} + 6 \, x^{2} - 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {15} + \frac {3}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {15} + \frac {3}{2}}\right )\,|\,\sqrt {15} - 4) + 6 \, \sqrt {3 \, x^{4} + 6 \, x^{2} - 2} {\left (3 \, x^{3} + 8 \, x\right )}}{120 \, {\left (3 \, x^{4} + 6 \, x^{2} - 2\right )}} \] Input:

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

Output:

-1/120*(3*(sqrt(15)*sqrt(-2)*(3*x^4 + 6*x^2 - 2) + 3*sqrt(-2)*(3*x^4 + 6*x 
^2 - 2))*sqrt(1/2*sqrt(15) + 3/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(15) + 
3/2)), sqrt(15) - 4) - (5*sqrt(15)*sqrt(-2)*(3*x^4 + 6*x^2 - 2) + 3*sqrt(- 
2)*(3*x^4 + 6*x^2 - 2))*sqrt(1/2*sqrt(15) + 3/2)*elliptic_f(arcsin(x*sqrt( 
1/2*sqrt(15) + 3/2)), sqrt(15) - 4) + 6*sqrt(3*x^4 + 6*x^2 - 2)*(3*x^3 + 8 
*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]

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