\(\int \frac {1}{(-3-2 x^2+2 x^4)^{3/2}} \, dx\) [226]

Optimal result

Integrand size = 16, antiderivative size = 233 \[ \int \frac {1}{\left (-3-2 x^2+2 x^4\right )^{3/2}} \, dx=-\frac {x \left (4-x^2\right )}{21 \sqrt {-3-2 x^2+2 x^4}}-\frac {\sqrt {\frac {1}{3} \left (-1+\sqrt {7}\right )} \sqrt {3+\left (1-\sqrt {7}\right ) x^2} \sqrt {3+\left (1+\sqrt {7}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{3} \left (-1+\sqrt {7}\right )} x\right )|\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{42 \sqrt {-3-2 x^2+2 x^4}}-\frac {\sqrt {3+\left (1-\sqrt {7}\right ) x^2} \sqrt {3+\left (1+\sqrt {7}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{3} \left (-1+\sqrt {7}\right )} x\right ),\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{3 \sqrt {14 \left (1+\sqrt {7}\right )} \sqrt {-3-2 x^2+2 x^4}} \] Output:

-1/21*x*(-x^2+4)/(2*x^4-2*x^2-3)^(1/2)-1/126*(-3+3*7^(1/2))^(1/2)*(3+(1-7^ 
(1/2))*x^2)^(1/2)*(3+(1+7^(1/2))*x^2)^(1/2)*EllipticE(1/3*(-3+3*7^(1/2))^( 
1/2)*x,1/6*I*6^(1/2)+1/6*I*42^(1/2))/(2*x^4-2*x^2-3)^(1/2)-1/3*(3+(1-7^(1/ 
2))*x^2)^(1/2)*(3+(1+7^(1/2))*x^2)^(1/2)*EllipticF(1/3*(-3+3*7^(1/2))^(1/2 
)*x,1/6*I*6^(1/2)+1/6*I*42^(1/2))/(14+14*7^(1/2))^(1/2)/(2*x^4-2*x^2-3)^(1 
/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.84 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (-3-2 x^2+2 x^4\right )^{3/2}} \, dx=\frac {4 x \left (-4+x^2\right )-2 i \sqrt {1+\sqrt {7}} \sqrt {3+2 x^2-2 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )+\frac {2 i \left (7+\sqrt {7}\right ) \sqrt {3+2 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}}}{84 \sqrt {-3-2 x^2+2 x^4}} \] Input:

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

Output:

(4*x*(-4 + x^2) - (2*I)*Sqrt[1 + Sqrt[7]]*Sqrt[3 + 2*x^2 - 2*x^4]*Elliptic 
E[I*ArcSinh[Sqrt[2/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3] + ((2*I)*(7 + Sqr 
t[7])*Sqrt[3 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[Sqrt[2/(-1 + Sqrt[7])]*x 
], (-4 + Sqrt[7])/3])/Sqrt[1 + Sqrt[7]])/(84*Sqrt[-3 - 2*x^2 + 2*x^4])
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.64, 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.

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

\(\Big \downarrow \) 1405

\(\displaystyle \frac {1}{84} \int -\frac {4 \left (x^2+3\right )}{\sqrt {2 x^4-2 x^2-3}}dx-\frac {x \left (4-x^2\right )}{21 \sqrt {2 x^4-2 x^2-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{21} \int \frac {x^2+3}{\sqrt {2 x^4-2 x^2-3}}dx-\frac {x \left (4-x^2\right )}{21 \sqrt {2 x^4-2 x^2-3}}\)

\(\Big \downarrow \) 1501

\(\displaystyle \frac {1}{21} \left (-\frac {1}{2} \left (7+\sqrt {7}\right ) \int \frac {1}{\sqrt {2 x^4-2 x^2-3}}dx-\frac {1}{4} \int -\frac {2 \left (-2 x^2+\sqrt {7}+1\right )}{\sqrt {2 x^4-2 x^2-3}}dx\right )-\frac {x \left (4-x^2\right )}{21 \sqrt {2 x^4-2 x^2-3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \left (\frac {1}{2} \int \frac {-2 x^2+\sqrt {7}+1}{\sqrt {2 x^4-2 x^2-3}}dx-\frac {1}{2} \left (7+\sqrt {7}\right ) \int \frac {1}{\sqrt {2 x^4-2 x^2-3}}dx\right )-\frac {x \left (4-x^2\right )}{21 \sqrt {2 x^4-2 x^2-3}}\)

\(\Big \downarrow \) 1411

\(\displaystyle \frac {1}{21} \left (\frac {1}{2} \int \frac {-2 x^2+\sqrt {7}+1}{\sqrt {2 x^4-2 x^2-3}}dx-\frac {\left (7+\sqrt {7}\right ) \sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-3} \sqrt {\frac {\left (1+\sqrt {7}\right ) x^2+3}{\left (1-\sqrt {7}\right ) x^2+3}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-3}}\right ),\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{2 \sqrt {6} \sqrt [4]{7} \sqrt {\frac {1}{\left (1-\sqrt {7}\right ) x^2+3}} \sqrt {2 x^4-2 x^2-3}}\right )-\frac {x \left (4-x^2\right )}{21 \sqrt {2 x^4-2 x^2-3}}\)

\(\Big \downarrow \) 1498

\(\displaystyle \frac {1}{21} \left (\frac {1}{2} \left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{7} \sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-3} \sqrt {\frac {\left (1+\sqrt {7}\right ) x^2+3}{\left (1-\sqrt {7}\right ) x^2+3}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-3}}\right )|\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{\sqrt {\frac {1}{\left (1-\sqrt {7}\right ) x^2+3}} \sqrt {2 x^4-2 x^2-3}}+\frac {x \left (-2 x^2-\sqrt {7}+1\right )}{\sqrt {2 x^4-2 x^2-3}}\right )-\frac {\left (7+\sqrt {7}\right ) \sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-3} \sqrt {\frac {\left (1+\sqrt {7}\right ) x^2+3}{\left (1-\sqrt {7}\right ) x^2+3}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-3}}\right ),\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{2 \sqrt {6} \sqrt [4]{7} \sqrt {\frac {1}{\left (1-\sqrt {7}\right ) x^2+3}} \sqrt {2 x^4-2 x^2-3}}\right )-\frac {x \left (4-x^2\right )}{21 \sqrt {2 x^4-2 x^2-3}}\)

Input:

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

Output:

-1/21*(x*(4 - x^2))/Sqrt[-3 - 2*x^2 + 2*x^4] + (((x*(1 - Sqrt[7] - 2*x^2)) 
/Sqrt[-3 - 2*x^2 + 2*x^4] + (Sqrt[2/3]*7^(1/4)*Sqrt[-3 - (1 - Sqrt[7])*x^2 
]*Sqrt[(3 + (1 + Sqrt[7])*x^2)/(3 + (1 - Sqrt[7])*x^2)]*EllipticE[ArcSin[( 
Sqrt[2]*7^(1/4)*x)/Sqrt[-3 - (1 - Sqrt[7])*x^2]], (7 - Sqrt[7])/14])/(Sqrt 
[(3 + (1 - Sqrt[7])*x^2)^(-1)]*Sqrt[-3 - 2*x^2 + 2*x^4]))/2 - ((7 + Sqrt[7 
])*Sqrt[-3 - (1 - Sqrt[7])*x^2]*Sqrt[(3 + (1 + Sqrt[7])*x^2)/(3 + (1 - Sqr 
t[7])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*7^(1/4)*x)/Sqrt[-3 - (1 - Sqrt[7])*x 
^2]], (7 - Sqrt[7])/14])/(2*Sqrt[6]*7^(1/4)*Sqrt[(3 + (1 - Sqrt[7])*x^2)^( 
-1)]*Sqrt[-3 - 2*x^2 + 2*x^4]))/21
 

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 = 2.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.98

method	result	size
risch	\(\frac {x \left (x^{2}-4\right )}{21 \sqrt {2 x^{4}-2 x^{2}-3}}-\frac {3 \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {-3-3 \sqrt {7}}\, \sqrt {2 x^{4}-2 x^{2}-3}}-\frac {6 \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {-3-3 \sqrt {7}}\, \sqrt {2 x^{4}-2 x^{2}-3}\, \left (-2+2 \sqrt {7}\right )}\)	\(228\)
default	\(-\frac {4 \left (\frac {1}{21} x -\frac {1}{84} x^{3}\right )}{\sqrt {2 x^{4}-2 x^{2}-3}}-\frac {3 \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {-3-3 \sqrt {7}}\, \sqrt {2 x^{4}-2 x^{2}-3}}-\frac {6 \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {-3-3 \sqrt {7}}\, \sqrt {2 x^{4}-2 x^{2}-3}\, \left (-2+2 \sqrt {7}\right )}\)	\(231\)
elliptic	\(-\frac {4 \left (\frac {1}{21} x -\frac {1}{84} x^{3}\right )}{\sqrt {2 x^{4}-2 x^{2}-3}}-\frac {3 \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {-3-3 \sqrt {7}}\, \sqrt {2 x^{4}-2 x^{2}-3}}-\frac {6 \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {-3-3 \sqrt {7}}\, \sqrt {2 x^{4}-2 x^{2}-3}\, \left (-2+2 \sqrt {7}\right )}\)	\(231\)

Input:

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

Output:

1/21*x*(x^2-4)/(2*x^4-2*x^2-3)^(1/2)-3/7/(-3-3*7^(1/2))^(1/2)*(1-(-1/3-1/3 
*7^(1/2))*x^2)^(1/2)*(1-(-1/3+1/3*7^(1/2))*x^2)^(1/2)/(2*x^4-2*x^2-3)^(1/2 
)*EllipticF(1/3*(-3-3*7^(1/2))^(1/2)*x,1/6*I*42^(1/2)-1/6*I*6^(1/2))-6/7/( 
-3-3*7^(1/2))^(1/2)*(1-(-1/3-1/3*7^(1/2))*x^2)^(1/2)*(1-(-1/3+1/3*7^(1/2)) 
*x^2)^(1/2)/(2*x^4-2*x^2-3)^(1/2)/(-2+2*7^(1/2))*(EllipticF(1/3*(-3-3*7^(1 
/2))^(1/2)*x,1/6*I*42^(1/2)-1/6*I*6^(1/2))-EllipticE(1/3*(-3-3*7^(1/2))^(1 
/2)*x,1/6*I*42^(1/2)-1/6*I*6^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-3-2 x^2+2 x^4\right )^{3/2}} \, dx=\frac {{\left (\sqrt {7} \sqrt {-3} {\left (2 \, x^{4} - 2 \, x^{2} - 3\right )} - \sqrt {-3} {\left (2 \, x^{4} - 2 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{3} \, \sqrt {7} - \frac {1}{3}} E(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {7} - \frac {1}{3}}\right )\,|\,-\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) + 2 \, {\left (\sqrt {7} \sqrt {-3} {\left (2 \, x^{4} - 2 \, x^{2} - 3\right )} + 2 \, \sqrt {-3} {\left (2 \, x^{4} - 2 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{3} \, \sqrt {7} - \frac {1}{3}} F(\arcsin \left (x \sqrt {\frac {1}{3} \, \sqrt {7} - \frac {1}{3}}\right )\,|\,-\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) + 6 \, \sqrt {2 \, x^{4} - 2 \, x^{2} - 3} {\left (x^{3} - 4 \, x\right )}}{126 \, {\left (2 \, x^{4} - 2 \, x^{2} - 3\right )}} \] Input:

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

Output:

1/126*((sqrt(7)*sqrt(-3)*(2*x^4 - 2*x^2 - 3) - sqrt(-3)*(2*x^4 - 2*x^2 - 3 
))*sqrt(1/3*sqrt(7) - 1/3)*elliptic_e(arcsin(x*sqrt(1/3*sqrt(7) - 1/3)), - 
1/3*sqrt(7) - 4/3) + 2*(sqrt(7)*sqrt(-3)*(2*x^4 - 2*x^2 - 3) + 2*sqrt(-3)* 
(2*x^4 - 2*x^2 - 3))*sqrt(1/3*sqrt(7) - 1/3)*elliptic_f(arcsin(x*sqrt(1/3* 
sqrt(7) - 1/3)), -1/3*sqrt(7) - 4/3) + 6*sqrt(2*x^4 - 2*x^2 - 3)*(x^3 - 4* 
x))/(2*x^4 - 2*x^2 - 3)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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