\(\int \frac {1}{(2-6 x^2+3 x^4)^{3/2}} \, dx\) [244]

Optimal result

Integrand size = 16, antiderivative size = 229 \[ \int \frac {1}{\left (2-6 x^2+3 x^4\right )^{3/2}} \, dx=\frac {x \left (4-3 x^2\right )}{4 \sqrt {2-6 x^2+3 x^4}}-\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} \sqrt {2-\left (3-\sqrt {3}\right ) x^2} \sqrt {2-\left (3+\sqrt {3}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right )|2-\sqrt {3}\right )}{4 \sqrt {2-6 x^2+3 x^4}}+\frac {\sqrt {2-\left (3-\sqrt {3}\right ) x^2} \sqrt {2-\left (3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right ),2-\sqrt {3}\right )}{4 \sqrt {3-\sqrt {3}} \sqrt {2-6 x^2+3 x^4}} \] Output:

1/4*x*(-3*x^2+4)/(3*x^4-6*x^2+2)^(1/2)-1/8*(6+2*3^(1/2))^(1/2)*(2-(3-3^(1/ 
2))*x^2)^(1/2)*(2-(3+3^(1/2))*x^2)^(1/2)*EllipticE(1/2*(6+2*3^(1/2))^(1/2) 
*x,1/2*6^(1/2)-1/2*2^(1/2))/(3*x^4-6*x^2+2)^(1/2)+1/4*(2-(3-3^(1/2))*x^2)^ 
(1/2)*(2-(3+3^(1/2))*x^2)^(1/2)*EllipticF(1/2*(6+2*3^(1/2))^(1/2)*x,1/2*6^ 
(1/2)-1/2*2^(1/2))/(3-3^(1/2))^(1/2)/(3*x^4-6*x^2+2)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 6.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (2-6 x^2+3 x^4\right )^{3/2}} \, dx=\frac {6 x \left (4-3 x^2\right )-3 \sqrt {2} \left (1+\sqrt {3}\right ) \sqrt {3-\sqrt {3}-3 x^2} \sqrt {2+\left (-3+\sqrt {3}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right )|2-\sqrt {3}\right )+\sqrt {2} \left (3+\sqrt {3}\right ) \sqrt {3-\sqrt {3}-3 x^2} \sqrt {2+\left (-3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right ),2-\sqrt {3}\right )}{24 \sqrt {2-6 x^2+3 x^4}} \] 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 - Sqr 
t[3]] + Sqrt[2]*(3 + Sqrt[3])*Sqrt[3 - Sqrt[3] - 3*x^2]*Sqrt[2 + (-3 + Sqr 
t[3])*x^2]*EllipticF[ArcSin[Sqrt[(3 + Sqrt[3])/2]*x], 2 - Sqrt[3]])/(24*Sq 
rt[2 - 6*x^2 + 3*x^4])
 

Rubi [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1497, 27, 1409, 1496}

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

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1497

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1409

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

\(\Big \downarrow \) 1496

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

Input:

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

Output:

(x*(4 - 3*x^2))/(4*Sqrt[2 - 6*x^2 + 3*x^4]) + (-(Sqrt[3/2]*((-2*x*Sqrt[2 - 
 6*x^2 + 3*x^4])/(2 + Sqrt[6]*x^2) + (2^(3/4)*(2 + Sqrt[6]*x^2)*Sqrt[(2 - 
6*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticE[2*ArcTan[(3/2)^(1/4)*x], (2 
+ Sqrt[6])/4])/(3^(1/4)*Sqrt[2 - 6*x^2 + 3*x^4]))) - ((2 - Sqrt[6])*(2 + S 
qrt[6]*x^2)*Sqrt[(2 - 6*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcT 
an[(3/2)^(1/4)*x], (2 + Sqrt[6])/4])/(2*6^(1/4)*Sqrt[2 - 6*x^2 + 3*x^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 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 1409

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

rule 1496

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 
- 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
 

rule 1497

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && GtQ 
[c/a, 0] && LtQ[b/a, 0]
 

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {x \left (3 x^{2}-4\right )}{4 \sqrt {3 x^{4}-6 x^{2}+2}}-\frac {\sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {6+2 \sqrt {3}}\, \sqrt {3 x^{4}-6 x^{2}+2}}-\frac {6 \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {6+2 \sqrt {3}}\, \sqrt {3 x^{4}-6 x^{2}+2}\, \left (-6+2 \sqrt {3}\right )}\)	\(224\)
default	\(-\frac {6 \left (\frac {1}{8} x^{3}-\frac {1}{6} x \right )}{\sqrt {3 x^{4}-6 x^{2}+2}}-\frac {\sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {6+2 \sqrt {3}}\, \sqrt {3 x^{4}-6 x^{2}+2}}-\frac {6 \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {6+2 \sqrt {3}}\, \sqrt {3 x^{4}-6 x^{2}+2}\, \left (-6+2 \sqrt {3}\right )}\)	\(225\)
elliptic	\(-\frac {6 \left (\frac {1}{8} x^{3}-\frac {1}{6} x \right )}{\sqrt {3 x^{4}-6 x^{2}+2}}-\frac {\sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {6+2 \sqrt {3}}\, \sqrt {3 x^{4}-6 x^{2}+2}}-\frac {6 \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {6+2 \sqrt {3}}\, \sqrt {3 x^{4}-6 x^{2}+2}\, \left (-6+2 \sqrt {3}\right )}\)	\(225\)

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)*E 
llipticF(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 = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (2-6 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {3 \, {\left (\sqrt {3} \sqrt {2} {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )} + 3 \, \sqrt {2} {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {3} + \frac {3}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {3} + \frac {3}{2}}\right )\,|\,-\sqrt {3} + 2) - {\left (5 \, \sqrt {3} \sqrt {2} {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )} + 3 \, \sqrt {2} {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {3} + \frac {3}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {3} + \frac {3}{2}}\right )\,|\,-\sqrt {3} + 2) + 6 \, \sqrt {3 \, x^{4} - 6 \, x^{2} + 2} {\left (3 \, x^{3} - 4 \, x\right )}}{24 \, {\left (3 \, x^{4} - 6 \, x^{2} + 2\right )}} \] 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*sqrt(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/(3*x^4 - 6*x^2 + 2)^(3/2),x)
 

Output:

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