\(\int \frac {1}{(-3-6 x^2+2 x^4)^{3/2}} \, dx\) [230]

Optimal result

Integrand size = 16, antiderivative size = 227 \[ \int \frac {1}{\left (-3-6 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {x \left (4-x^2\right )}{15 \sqrt {-3-6 x^2+2 x^4}}-\frac {\sqrt {\frac {1}{3} \left (-3+\sqrt {15}\right )} \sqrt {3+\left (3-\sqrt {15}\right ) x^2} \sqrt {3+\left (3+\sqrt {15}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{3} \left (-3+\sqrt {15}\right )} x\right )|-4-\sqrt {15}\right )}{30 \sqrt {-3-6 x^2+2 x^4}}-\frac {\sqrt {\frac {1}{5} \left (-3+\sqrt {15}\right )} \sqrt {3+\left (3-\sqrt {15}\right ) x^2} \sqrt {3+\left (3+\sqrt {15}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{3} \left (-3+\sqrt {15}\right )} x\right ),-4-\sqrt {15}\right )}{18 \sqrt {-3-6 x^2+2 x^4}} \] Output:

-1/15*x*(-x^2+4)/(2*x^4-6*x^2-3)^(1/2)-1/90*(-9+3*15^(1/2))^(1/2)*(3+(3-15 
^(1/2))*x^2)^(1/2)*(3+(3+15^(1/2))*x^2)^(1/2)*EllipticE(1/3*(-9+3*15^(1/2) 
)^(1/2)*x,1/2*I*6^(1/2)+1/2*I*10^(1/2))/(2*x^4-6*x^2-3)^(1/2)-1/90*(-15+5* 
15^(1/2))^(1/2)*(3+(3-15^(1/2))*x^2)^(1/2)*(3+(3+15^(1/2))*x^2)^(1/2)*Elli 
pticF(1/3*(-9+3*15^(1/2))^(1/2)*x,1/2*I*6^(1/2)+1/2*I*10^(1/2))/(2*x^4-6*x 
^2-3)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.91 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\left (-3-6 x^2+2 x^4\right )^{3/2}} \, dx=\frac {4 x \left (-4+x^2\right )-2 i \sqrt {3+\sqrt {15}} \sqrt {3+6 x^2-2 x^4} E\left (i \text {arcsinh}\left (\sqrt {1+\sqrt {\frac {5}{3}}} x\right )|-4+\sqrt {15}\right )+\frac {2 i \left (5+\sqrt {15}\right ) \sqrt {3+6 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1+\sqrt {\frac {5}{3}}} x\right ),-4+\sqrt {15}\right )}{\sqrt {3+\sqrt {15}}}}{60 \sqrt {-3-6 x^2+2 x^4}} \] Input:

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

Output:

(4*x*(-4 + x^2) - (2*I)*Sqrt[3 + Sqrt[15]]*Sqrt[3 + 6*x^2 - 2*x^4]*Ellipti 
cE[I*ArcSinh[Sqrt[1 + Sqrt[5/3]]*x], -4 + Sqrt[15]] + ((2*I)*(5 + Sqrt[15] 
)*Sqrt[3 + 6*x^2 - 2*x^4]*EllipticF[I*ArcSinh[Sqrt[1 + Sqrt[5/3]]*x], -4 + 
 Sqrt[15]])/Sqrt[3 + Sqrt[15]])/(60*Sqrt[-3 - 6*x^2 + 2*x^4])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.71, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{180} \int -\frac {12 \left (x^2+1\right )}{\sqrt {2 x^4-6 x^2-3}}dx-\frac {x \left (4-x^2\right )}{15 \sqrt {2 x^4-6 x^2-3}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1411

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

\(\Big \downarrow \) 1498

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

Input:

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

Output:

-1/15*(x*(4 - x^2))/Sqrt[-3 - 6*x^2 + 2*x^4] + (((x*(3 - Sqrt[15] - 2*x^2) 
)/Sqrt[-3 - 6*x^2 + 2*x^4] + ((5/3)^(1/4)*Sqrt[2]*Sqrt[-3 - (3 - Sqrt[15]) 
*x^2]*Sqrt[(3 + (3 + Sqrt[15])*x^2)/(3 + (3 - Sqrt[15])*x^2)]*EllipticE[Ar 
cSin[(Sqrt[2]*15^(1/4)*x)/Sqrt[-3 - (3 - Sqrt[15])*x^2]], (5 - Sqrt[15])/1 
0])/(Sqrt[(3 + (3 - Sqrt[15])*x^2)^(-1)]*Sqrt[-3 - 6*x^2 + 2*x^4]))/2 - (( 
5 + Sqrt[15])*Sqrt[-3 - (3 - Sqrt[15])*x^2]*Sqrt[(3 + (3 + Sqrt[15])*x^2)/ 
(3 + (3 - Sqrt[15])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*15^(1/4)*x)/Sqrt[-3 - 
(3 - Sqrt[15])*x^2]], (5 - Sqrt[15])/10])/(2*Sqrt[2]*3^(3/4)*5^(1/4)*Sqrt[ 
(3 + (3 - Sqrt[15])*x^2)^(-1)]*Sqrt[-3 - 6*x^2 + 2*x^4]))/15
 

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 = 2.00 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {x \left (x^{2}-4\right )}{15 \sqrt {2 x^{4}-6 x^{2}-3}}-\frac {\sqrt {1-\left (-1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )}{5 \sqrt {-9-3 \sqrt {15}}\, \sqrt {2 x^{4}-6 x^{2}-3}}-\frac {6 \sqrt {1-\left (-1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )\right )}{5 \sqrt {-9-3 \sqrt {15}}\, \sqrt {2 x^{4}-6 x^{2}-3}\, \left (-6+2 \sqrt {15}\right )}\)	\(228\)
default	\(-\frac {4 \left (\frac {1}{15} x -\frac {1}{60} x^{3}\right )}{\sqrt {2 x^{4}-6 x^{2}-3}}-\frac {\sqrt {1-\left (-1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )}{5 \sqrt {-9-3 \sqrt {15}}\, \sqrt {2 x^{4}-6 x^{2}-3}}-\frac {6 \sqrt {1-\left (-1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )\right )}{5 \sqrt {-9-3 \sqrt {15}}\, \sqrt {2 x^{4}-6 x^{2}-3}\, \left (-6+2 \sqrt {15}\right )}\)	\(231\)
elliptic	\(-\frac {4 \left (\frac {1}{15} x -\frac {1}{60} x^{3}\right )}{\sqrt {2 x^{4}-6 x^{2}-3}}-\frac {\sqrt {1-\left (-1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )}{5 \sqrt {-9-3 \sqrt {15}}\, \sqrt {2 x^{4}-6 x^{2}-3}}-\frac {6 \sqrt {1-\left (-1-\frac {\sqrt {15}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1+\frac {\sqrt {15}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-9-3 \sqrt {15}}\, x}{3}, \frac {i \sqrt {10}}{2}-\frac {i \sqrt {6}}{2}\right )\right )}{5 \sqrt {-9-3 \sqrt {15}}\, \sqrt {2 x^{4}-6 x^{2}-3}\, \left (-6+2 \sqrt {15}\right )}\)	\(231\)

Input:

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

Output:

1/15*x*(x^2-4)/(2*x^4-6*x^2-3)^(1/2)-1/5/(-9-3*15^(1/2))^(1/2)*(1-(-1-1/3* 
15^(1/2))*x^2)^(1/2)*(1-(-1+1/3*15^(1/2))*x^2)^(1/2)/(2*x^4-6*x^2-3)^(1/2) 
*EllipticF(1/3*(-9-3*15^(1/2))^(1/2)*x,1/2*I*10^(1/2)-1/2*I*6^(1/2))-6/5/( 
-9-3*15^(1/2))^(1/2)*(1-(-1-1/3*15^(1/2))*x^2)^(1/2)*(1-(-1+1/3*15^(1/2))* 
x^2)^(1/2)/(2*x^4-6*x^2-3)^(1/2)/(-6+2*15^(1/2))*(EllipticF(1/3*(-9-3*15^( 
1/2))^(1/2)*x,1/2*I*10^(1/2)-1/2*I*6^(1/2))-EllipticE(1/3*(-9-3*15^(1/2))^ 
(1/2)*x,1/2*I*10^(1/2)-1/2*I*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\left (-3-6 x^2+2 x^4\right )^{3/2}} \, dx=\frac {2 \, \sqrt {-3} {\left (2 \, x^{4} - 6 \, x^{2} - 3\right )} \sqrt {\sqrt {\frac {5}{3}} - 1} F(\arcsin \left (x \sqrt {\sqrt {\frac {5}{3}} - 1}\right )\,|\,-3 \, \sqrt {\frac {5}{3}} - 4) + {\left (\sqrt {\frac {5}{3}} \sqrt {-3} {\left (2 \, x^{4} - 6 \, x^{2} - 3\right )} - \sqrt {-3} {\left (2 \, x^{4} - 6 \, x^{2} - 3\right )}\right )} \sqrt {\sqrt {\frac {5}{3}} - 1} E(\arcsin \left (x \sqrt {\sqrt {\frac {5}{3}} - 1}\right )\,|\,-3 \, \sqrt {\frac {5}{3}} - 4) + 2 \, \sqrt {2 \, x^{4} - 6 \, x^{2} - 3} {\left (x^{3} - 4 \, x\right )}}{30 \, {\left (2 \, x^{4} - 6 \, x^{2} - 3\right )}} \] Input:

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

Output:

1/30*(2*sqrt(-3)*(2*x^4 - 6*x^2 - 3)*sqrt(sqrt(5/3) - 1)*elliptic_f(arcsin 
(x*sqrt(sqrt(5/3) - 1)), -3*sqrt(5/3) - 4) + (sqrt(5/3)*sqrt(-3)*(2*x^4 - 
6*x^2 - 3) - sqrt(-3)*(2*x^4 - 6*x^2 - 3))*sqrt(sqrt(5/3) - 1)*elliptic_e( 
arcsin(x*sqrt(sqrt(5/3) - 1)), -3*sqrt(5/3) - 4) + 2*sqrt(2*x^4 - 6*x^2 - 
3)*(x^3 - 4*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]

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