\(\int \frac {1}{(3-6 x^2+2 x^4)^{3/2}} \, dx\) [260]

Optimal result

Integrand size = 16, antiderivative size = 227 \[ \int \frac {1}{\left (3-6 x^2+2 x^4\right )^{3/2}} \, dx=\frac {x \left (2-x^2\right )}{3 \sqrt {3-6 x^2+2 x^4}}-\frac {\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} \sqrt {3-\left (3-\sqrt {3}\right ) x^2} \sqrt {3-\left (3+\sqrt {3}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|2-\sqrt {3}\right )}{6 \sqrt {3-6 x^2+2 x^4}}+\frac {\sqrt {3+\sqrt {3}} \sqrt {3-\left (3-\sqrt {3}\right ) x^2} \sqrt {3-\left (3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right ),2-\sqrt {3}\right )}{18 \sqrt {3-6 x^2+2 x^4}} \] Output:

1/3*x*(-x^2+2)/(2*x^4-6*x^2+3)^(1/2)-1/18*(9+3*3^(1/2))^(1/2)*(3-(3-3^(1/2 
))*x^2)^(1/2)*(3-(3+3^(1/2))*x^2)^(1/2)*EllipticE(1/3*(9+3*3^(1/2))^(1/2)* 
x,1/2*6^(1/2)-1/2*2^(1/2))/(2*x^4-6*x^2+3)^(1/2)+1/18*(3+3^(1/2))^(1/2)*(3 
-(3-3^(1/2))*x^2)^(1/2)*(3-(3+3^(1/2))*x^2)^(1/2)*EllipticF(1/3*(9+3*3^(1/ 
2))^(1/2)*x,1/2*6^(1/2)-1/2*2^(1/2))/(2*x^4-6*x^2+3)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 6.13 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (3-6 x^2+2 x^4\right )^{3/2}} \, dx=\frac {-12 x \left (-2+x^2\right )-3 \sqrt {2} \left (1+\sqrt {3}\right ) \sqrt {3-\sqrt {3}-2 x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2} E\left (\arcsin \left (\sqrt {1+\frac {1}{\sqrt {3}}} x\right )|2-\sqrt {3}\right )+\sqrt {2} \left (3+\sqrt {3}\right ) \sqrt {3-\sqrt {3}-2 x^2} \sqrt {3+\left (-3+\sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {1+\frac {1}{\sqrt {3}}} x\right ),2-\sqrt {3}\right )}{36 \sqrt {3-6 x^2+2 x^4}} \] Input:

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

Output:

(-12*x*(-2 + x^2) - 3*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]])/(36*Sqrt[ 
3 - 6*x^2 + 2*x^4])
 

Rubi [A] (warning: unable to verify)

Time = 0.61 (sec) , antiderivative size = 265, 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.

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

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1497

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1409

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

\(\Big \downarrow \) 1496

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

Input:

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

Output:

(x*(2 - x^2))/(3*Sqrt[3 - 6*x^2 + 2*x^4]) + (-(((-3*x*Sqrt[3 - 6*x^2 + 2*x 
^4])/(3 + Sqrt[6]*x^2) + (3^(3/4)*(3 + Sqrt[6]*x^2)*Sqrt[(3 - 6*x^2 + 2*x^ 
4)/(3 + Sqrt[6]*x^2)^2]*EllipticE[2*ArcTan[(2/3)^(1/4)*x], (2 + Sqrt[6])/4 
])/(2^(1/4)*Sqrt[3 - 6*x^2 + 2*x^4]))/Sqrt[6]) - ((2 - Sqrt[6])*(3 + Sqrt[ 
6]*x^2)*Sqrt[(3 - 6*x^2 + 2*x^4)/(3 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTan[( 
2/3)^(1/4)*x], (2 + Sqrt[6])/4])/(4*6^(1/4)*Sqrt[3 - 6*x^2 + 2*x^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 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 = 1.76 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {x \left (x^{2}-2\right )}{3 \sqrt {2 x^{4}-6 x^{2}+3}}-\frac {\sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {9+3 \sqrt {3}}\, \sqrt {2 x^{4}-6 x^{2}+3}}-\frac {6 \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {9+3 \sqrt {3}}\, \sqrt {2 x^{4}-6 x^{2}+3}\, \left (-6+2 \sqrt {3}\right )}\)	\(222\)
default	\(-\frac {4 \left (-\frac {1}{6} x +\frac {1}{12} x^{3}\right )}{\sqrt {2 x^{4}-6 x^{2}+3}}-\frac {\sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {9+3 \sqrt {3}}\, \sqrt {2 x^{4}-6 x^{2}+3}}-\frac {6 \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {9+3 \sqrt {3}}\, \sqrt {2 x^{4}-6 x^{2}+3}\, \left (-6+2 \sqrt {3}\right )}\)	\(225\)
elliptic	\(-\frac {4 \left (-\frac {1}{6} x +\frac {1}{12} x^{3}\right )}{\sqrt {2 x^{4}-6 x^{2}+3}}-\frac {\sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {9+3 \sqrt {3}}\, \sqrt {2 x^{4}-6 x^{2}+3}}-\frac {6 \sqrt {1-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )\right )}{\sqrt {9+3 \sqrt {3}}\, \sqrt {2 x^{4}-6 x^{2}+3}\, \left (-6+2 \sqrt {3}\right )}\)	\(225\)

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)*Ellipti 
cF(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 = 141, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (3-6 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {{\left (2 \, x^{4} - 6 \, x^{2} + \sqrt {3} {\left (2 \, x^{4} - 6 \, x^{2} + 3\right )} + 3\right )} \sqrt {\frac {1}{3} \, \sqrt {3} + 1} E(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} + 1}\right )\,|\,-\sqrt {3} + 2) - 2 \, {\left (2 \, x^{4} - 6 \, x^{2} + 3\right )} \sqrt {\frac {1}{3} \, \sqrt {3} + 1} F(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {3} + 1}\right )\,|\,-\sqrt {3} + 2) + 2 \, \sqrt {2 \, x^{4} - 6 \, x^{2} + 3} {\left (x^{3} - 2 \, x\right )}}{6 \, {\left (2 \, x^{4} - 6 \, x^{2} + 3\right )}} \] Input:

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

Output:

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

Output:

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