\(\int \frac {1}{(-2-3 x^2+3 x^4)^{3/2}} \, dx\) [212]

Optimal result

Integrand size = 16, antiderivative size = 229 \[ \int \frac {1}{\left (-2-3 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {x \left (7-3 x^2\right )}{22 \sqrt {-2-3 x^2+3 x^4}}-\frac {\sqrt {-3+\sqrt {33}} \sqrt {4+\left (3-\sqrt {33}\right ) x^2} \sqrt {4+\left (3+\sqrt {33}\right ) x^2} E\left (\arcsin \left (\frac {1}{2} \sqrt {-3+\sqrt {33}} x\right )|\frac {1}{4} \left (-7-\sqrt {33}\right )\right )}{88 \sqrt {-2-3 x^2+3 x^4}}-\frac {\sqrt {\frac {1}{33} \left (-3+\sqrt {33}\right )} \sqrt {4+\left (3-\sqrt {33}\right ) x^2} \sqrt {4+\left (3+\sqrt {33}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-3+\sqrt {33}} x\right ),\frac {1}{4} \left (-7-\sqrt {33}\right )\right )}{8 \sqrt {-2-3 x^2+3 x^4}} \] Output:

-1/22*x*(-3*x^2+7)/(3*x^4-3*x^2-2)^(1/2)-1/88*(-3+33^(1/2))^(1/2)*(4+(3-33 
^(1/2))*x^2)^(1/2)*(4+(3+33^(1/2))*x^2)^(1/2)*EllipticE(1/2*(-3+33^(1/2))^ 
(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2))/(3*x^4-3*x^2-2)^(1/2)-1/264*(-99+33* 
33^(1/2))^(1/2)*(4+(3-33^(1/2))*x^2)^(1/2)*(4+(3+33^(1/2))*x^2)^(1/2)*Elli 
pticF(1/2*(-3+33^(1/2))^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2))/(3*x^4-3*x^2 
-2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.80 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (-2-3 x^2+3 x^4\right )^{3/2}} \, dx=\frac {12 x \left (-7+3 x^2\right )-6 i \sqrt {3+\sqrt {33}} \sqrt {4+6 x^2-6 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )+\frac {6 i \left (11+\sqrt {33}\right ) \sqrt {4+6 x^2-6 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{\sqrt {3+\sqrt {33}}}}{264 \sqrt {-2-3 x^2+3 x^4}} \] Input:

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

Output:

(12*x*(-7 + 3*x^2) - (6*I)*Sqrt[3 + Sqrt[33]]*Sqrt[4 + 6*x^2 - 6*x^4]*Elli 
pticE[I*ArcSinh[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4] + ((6*I)*(1 
1 + Sqrt[33])*Sqrt[4 + 6*x^2 - 6*x^4]*EllipticF[I*ArcSinh[Sqrt[6/(-3 + Sqr 
t[33])]*x], (-7 + Sqrt[33])/4])/Sqrt[3 + Sqrt[33]])/(264*Sqrt[-2 - 3*x^2 + 
 3*x^4])
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.66, 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, 25, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{66} \int -\frac {3 \left (3 x^2+4\right )}{\sqrt {3 x^4-3 x^2-2}}dx-\frac {x \left (7-3 x^2\right )}{22 \sqrt {3 x^4-3 x^2-2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{22} \int \frac {3 x^2+4}{\sqrt {3 x^4-3 x^2-2}}dx-\frac {x \left (7-3 x^2\right )}{22 \sqrt {3 x^4-3 x^2-2}}\)

\(\Big \downarrow \) 1501

\(\displaystyle \frac {1}{22} \left (-\frac {1}{2} \left (11+\sqrt {33}\right ) \int \frac {1}{\sqrt {3 x^4-3 x^2-2}}dx-\frac {1}{2} \int -\frac {-6 x^2+\sqrt {33}+3}{\sqrt {3 x^4-3 x^2-2}}dx\right )-\frac {x \left (7-3 x^2\right )}{22 \sqrt {3 x^4-3 x^2-2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{22} \left (\frac {1}{2} \int \frac {-6 x^2+\sqrt {33}+3}{\sqrt {3 x^4-3 x^2-2}}dx-\frac {1}{2} \left (11+\sqrt {33}\right ) \int \frac {1}{\sqrt {3 x^4-3 x^2-2}}dx\right )-\frac {x \left (7-3 x^2\right )}{22 \sqrt {3 x^4-3 x^2-2}}\)

\(\Big \downarrow \) 1411

\(\displaystyle \frac {1}{22} \left (\frac {1}{2} \int \frac {-6 x^2+\sqrt {33}+3}{\sqrt {3 x^4-3 x^2-2}}dx-\frac {\left (11+\sqrt {33}\right ) \sqrt {-\left (\left (3-\sqrt {33}\right ) x^2\right )-4} \sqrt {\frac {\left (3+\sqrt {33}\right ) x^2+4}{\left (3-\sqrt {33}\right ) x^2+4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {-\left (\left (3-\sqrt {33}\right ) x^2\right )-4}}\right ),\frac {1}{22} \left (11-\sqrt {33}\right )\right )}{4 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{\left (3-\sqrt {33}\right ) x^2+4}} \sqrt {3 x^4-3 x^2-2}}\right )-\frac {x \left (7-3 x^2\right )}{22 \sqrt {3 x^4-3 x^2-2}}\)

\(\Big \downarrow \) 1498

\(\displaystyle \frac {1}{22} \left (\frac {1}{2} \left (\frac {\sqrt [4]{33} \sqrt {-\left (\left (3-\sqrt {33}\right ) x^2\right )-4} \sqrt {\frac {\left (3+\sqrt {33}\right ) x^2+4}{\left (3-\sqrt {33}\right ) x^2+4}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {-\left (\left (3-\sqrt {33}\right ) x^2\right )-4}}\right )|\frac {1}{22} \left (11-\sqrt {33}\right )\right )}{\sqrt {2} \sqrt {\frac {1}{\left (3-\sqrt {33}\right ) x^2+4}} \sqrt {3 x^4-3 x^2-2}}+\frac {x \left (-6 x^2-\sqrt {33}+3\right )}{\sqrt {3 x^4-3 x^2-2}}\right )-\frac {\left (11+\sqrt {33}\right ) \sqrt {-\left (\left (3-\sqrt {33}\right ) x^2\right )-4} \sqrt {\frac {\left (3+\sqrt {33}\right ) x^2+4}{\left (3-\sqrt {33}\right ) x^2+4}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {-\left (\left (3-\sqrt {33}\right ) x^2\right )-4}}\right ),\frac {1}{22} \left (11-\sqrt {33}\right )\right )}{4 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{\left (3-\sqrt {33}\right ) x^2+4}} \sqrt {3 x^4-3 x^2-2}}\right )-\frac {x \left (7-3 x^2\right )}{22 \sqrt {3 x^4-3 x^2-2}}\)

Input:

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

Output:

-1/22*(x*(7 - 3*x^2))/Sqrt[-2 - 3*x^2 + 3*x^4] + (((x*(3 - Sqrt[33] - 6*x^ 
2))/Sqrt[-2 - 3*x^2 + 3*x^4] + (33^(1/4)*Sqrt[-4 - (3 - Sqrt[33])*x^2]*Sqr 
t[(4 + (3 + Sqrt[33])*x^2)/(4 + (3 - Sqrt[33])*x^2)]*EllipticE[ArcSin[(Sqr 
t[2]*33^(1/4)*x)/Sqrt[-4 - (3 - Sqrt[33])*x^2]], (11 - Sqrt[33])/22])/(Sqr 
t[2]*Sqrt[(4 + (3 - Sqrt[33])*x^2)^(-1)]*Sqrt[-2 - 3*x^2 + 3*x^4]))/2 - (( 
11 + Sqrt[33])*Sqrt[-4 - (3 - Sqrt[33])*x^2]*Sqrt[(4 + (3 + Sqrt[33])*x^2) 
/(4 + (3 - Sqrt[33])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*33^(1/4)*x)/Sqrt[-4 - 
 (3 - Sqrt[33])*x^2]], (11 - Sqrt[33])/22])/(4*Sqrt[2]*33^(1/4)*Sqrt[(4 + 
(3 - Sqrt[33])*x^2)^(-1)]*Sqrt[-2 - 3*x^2 + 3*x^4]))/22
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 2.68 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {x \left (3 x^{2}-7\right )}{22 \sqrt {3 x^{4}-3 x^{2}-2}}-\frac {4 \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {-3-\sqrt {33}}\, \sqrt {3 x^{4}-3 x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {-3-\sqrt {33}}\, \sqrt {3 x^{4}-3 x^{2}-2}\, \left (-3+\sqrt {33}\right )}\)	\(228\)
default	\(-\frac {6 \left (\frac {7}{132} x -\frac {1}{44} x^{3}\right )}{\sqrt {3 x^{4}-3 x^{2}-2}}-\frac {4 \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {-3-\sqrt {33}}\, \sqrt {3 x^{4}-3 x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {-3-\sqrt {33}}\, \sqrt {3 x^{4}-3 x^{2}-2}\, \left (-3+\sqrt {33}\right )}\)	\(229\)
elliptic	\(-\frac {6 \left (\frac {7}{132} x -\frac {1}{44} x^{3}\right )}{\sqrt {3 x^{4}-3 x^{2}-2}}-\frac {4 \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {-3-\sqrt {33}}\, \sqrt {3 x^{4}-3 x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {-3-\sqrt {33}}\, \sqrt {3 x^{4}-3 x^{2}-2}\, \left (-3+\sqrt {33}\right )}\)	\(229\)

Input:

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

Output:

1/22*x*(3*x^2-7)/(3*x^4-3*x^2-2)^(1/2)-4/11/(-3-33^(1/2))^(1/2)*(1-(-3/4-1 
/4*33^(1/2))*x^2)^(1/2)*(1-(-3/4+1/4*33^(1/2))*x^2)^(1/2)/(3*x^4-3*x^2-2)^ 
(1/2)*EllipticF(1/2*(-3-33^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))-12 
/11/(-3-33^(1/2))^(1/2)*(1-(-3/4-1/4*33^(1/2))*x^2)^(1/2)*(1-(-3/4+1/4*33^ 
(1/2))*x^2)^(1/2)/(3*x^4-3*x^2-2)^(1/2)/(-3+33^(1/2))*(EllipticF(1/2*(-3-3 
3^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))-EllipticE(1/2*(-3-33^(1/2)) 
^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-2-3 x^2+3 x^4\right )^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {33} \sqrt {-2} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )} - 3 \, \sqrt {-2} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} - 3} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} - 3}\right )\,|\,-\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) + {\left (\sqrt {33} \sqrt {-2} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )} + 21 \, \sqrt {-2} {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} - 3} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} - 3}\right )\,|\,-\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) + 24 \, \sqrt {3 \, x^{4} - 3 \, x^{2} - 2} {\left (3 \, x^{3} - 7 \, x\right )}}{528 \, {\left (3 \, x^{4} - 3 \, x^{2} - 2\right )}} \] Input:

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

Output:

1/528*(3*(sqrt(33)*sqrt(-2)*(3*x^4 - 3*x^2 - 2) - 3*sqrt(-2)*(3*x^4 - 3*x^ 
2 - 2))*sqrt(sqrt(33) - 3)*elliptic_e(arcsin(1/2*x*sqrt(sqrt(33) - 3)), -1 
/4*sqrt(33) - 7/4) + (sqrt(33)*sqrt(-2)*(3*x^4 - 3*x^2 - 2) + 21*sqrt(-2)* 
(3*x^4 - 3*x^2 - 2))*sqrt(sqrt(33) - 3)*elliptic_f(arcsin(1/2*x*sqrt(sqrt( 
33) - 3)), -1/4*sqrt(33) - 7/4) + 24*sqrt(3*x^4 - 3*x^2 - 2)*(3*x^3 - 7*x) 
)/(3*x^4 - 3*x^2 - 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (-2-3 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {3 x^{4}-3 x^{2}-2}}{9 x^{8}-18 x^{6}-3 x^{4}+12 x^{2}+4}d x \] Input:

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

Output:

int(sqrt(3*x**4 - 3*x**2 - 2)/(9*x**8 - 18*x**6 - 3*x**4 + 12*x**2 + 4),x)