Integrand size = 16, antiderivative size = 235 \[ \int \frac {1}{\left (-2-2 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {x \left (8-3 x^2\right )}{28 \sqrt {-2-2 x^2+3 x^4}}-\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {7}\right )} \sqrt {2+\left (1-\sqrt {7}\right ) x^2} \sqrt {2+\left (1+\sqrt {7}\right ) x^2} E\left (\arcsin \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {7}\right )} x\right )|\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{28 \sqrt {-2-2 x^2+3 x^4}}-\frac {\sqrt {\frac {1}{14} \left (-1+\sqrt {7}\right )} \sqrt {2+\left (1-\sqrt {7}\right ) x^2} \sqrt {2+\left (1+\sqrt {7}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {7}\right )} x\right ),\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{4 \sqrt {-2-2 x^2+3 x^4}} \] Output:
-1/28*x*(-3*x^2+8)/(3*x^4-2*x^2-2)^(1/2)-1/56*(-2+2*7^(1/2))^(1/2)*(2+(1-7 ^(1/2))*x^2)^(1/2)*(2+(1+7^(1/2))*x^2)^(1/2)*EllipticE(1/2*(-2+2*7^(1/2))^ (1/2)*x,1/6*I*6^(1/2)+1/6*I*42^(1/2))/(3*x^4-2*x^2-2)^(1/2)-1/56*(-14+14*7 ^(1/2))^(1/2)*(2+(1-7^(1/2))*x^2)^(1/2)*(2+(1+7^(1/2))*x^2)^(1/2)*Elliptic F(1/2*(-2+2*7^(1/2))^(1/2)*x,1/6*I*6^(1/2)+1/6*I*42^(1/2))/(3*x^4-2*x^2-2) ^(1/2)
Result contains complex when optimal does not.
Time = 6.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (-2-2 x^2+3 x^4\right )^{3/2}} \, dx=\frac {3 x \left (-8+3 x^2\right )-3 i \sqrt {1+\sqrt {7}} \sqrt {2+2 x^2-3 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )+\frac {3 i \left (7+\sqrt {7}\right ) \sqrt {2+2 x^2-3 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}}}{84 \sqrt {-2-2 x^2+3 x^4}} \] Input:
Integrate[(-2 - 2*x^2 + 3*x^4)^(-3/2),x]
Output:
(3*x*(-8 + 3*x^2) - (3*I)*Sqrt[1 + Sqrt[7]]*Sqrt[2 + 2*x^2 - 3*x^4]*Ellipt icE[I*ArcSinh[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3] + ((3*I)*(7 + S qrt[7])*Sqrt[2 + 2*x^2 - 3*x^4]*EllipticF[I*ArcSinh[Sqrt[3/(-1 + Sqrt[7])] *x], (-4 + Sqrt[7])/3])/Sqrt[1 + Sqrt[7]])/(84*Sqrt[-2 - 2*x^2 + 3*x^4])
Time = 0.78 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.59, 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.
| \(\displaystyle \int \frac {1}{\left (3 x^4-2 x^2-2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {1}{56} \int -\frac {6 \left (x^2+2\right )}{\sqrt {3 x^4-2 x^2-2}}dx-\frac {x \left (8-3 x^2\right )}{28 \sqrt {3 x^4-2 x^2-2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{28} \int \frac {x^2+2}{\sqrt {3 x^4-2 x^2-2}}dx-\frac {x \left (8-3 x^2\right )}{28 \sqrt {3 x^4-2 x^2-2}}\) |
| \(\Big \downarrow \) 1501 |
| \(\displaystyle -\frac {3}{28} \left (\frac {1}{3} \left (7+\sqrt {7}\right ) \int \frac {1}{\sqrt {3 x^4-2 x^2-2}}dx+\frac {1}{6} \int -\frac {2 \left (-3 x^2+\sqrt {7}+1\right )}{\sqrt {3 x^4-2 x^2-2}}dx\right )-\frac {x \left (8-3 x^2\right )}{28 \sqrt {3 x^4-2 x^2-2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{28} \left (\frac {1}{3} \left (7+\sqrt {7}\right ) \int \frac {1}{\sqrt {3 x^4-2 x^2-2}}dx-\frac {1}{3} \int \frac {-3 x^2+\sqrt {7}+1}{\sqrt {3 x^4-2 x^2-2}}dx\right )-\frac {x \left (8-3 x^2\right )}{28 \sqrt {3 x^4-2 x^2-2}}\) |
| \(\Big \downarrow \) 1411 |
| \(\displaystyle -\frac {3}{28} \left (\frac {\left (7+\sqrt {7}\right ) \sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {7}\right ) x^2+2}{\left (1-\sqrt {7}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-2}}\right ),\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{6 \sqrt [4]{7} \sqrt {\frac {1}{\left (1-\sqrt {7}\right ) x^2+2}} \sqrt {3 x^4-2 x^2-2}}-\frac {1}{3} \int \frac {-3 x^2+\sqrt {7}+1}{\sqrt {3 x^4-2 x^2-2}}dx\right )-\frac {x \left (8-3 x^2\right )}{28 \sqrt {3 x^4-2 x^2-2}}\) |
| \(\Big \downarrow \) 1498 |
| \(\displaystyle -\frac {3}{28} \left (\frac {\left (7+\sqrt {7}\right ) \sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {7}\right ) x^2+2}{\left (1-\sqrt {7}\right ) x^2+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-2}}\right ),\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{6 \sqrt [4]{7} \sqrt {\frac {1}{\left (1-\sqrt {7}\right ) x^2+2}} \sqrt {3 x^4-2 x^2-2}}+\frac {1}{3} \left (-\frac {\sqrt [4]{7} \sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-2} \sqrt {\frac {\left (1+\sqrt {7}\right ) x^2+2}{\left (1-\sqrt {7}\right ) x^2+2}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-2}}\right )|\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{\sqrt {\frac {1}{\left (1-\sqrt {7}\right ) x^2+2}} \sqrt {3 x^4-2 x^2-2}}-\frac {x \left (-3 x^2-\sqrt {7}+1\right )}{\sqrt {3 x^4-2 x^2-2}}\right )\right )-\frac {x \left (8-3 x^2\right )}{28 \sqrt {3 x^4-2 x^2-2}}\) |
Input:
Int[(-2 - 2*x^2 + 3*x^4)^(-3/2),x]
Output:
-1/28*(x*(8 - 3*x^2))/Sqrt[-2 - 2*x^2 + 3*x^4] - (3*((-((x*(1 - Sqrt[7] - 3*x^2))/Sqrt[-2 - 2*x^2 + 3*x^4]) - (7^(1/4)*Sqrt[-2 - (1 - Sqrt[7])*x^2]* Sqrt[(2 + (1 + Sqrt[7])*x^2)/(2 + (1 - Sqrt[7])*x^2)]*EllipticE[ArcSin[(Sq rt[2]*7^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[7])*x^2]], (7 - Sqrt[7])/14])/(Sqrt[( 2 + (1 - Sqrt[7])*x^2)^(-1)]*Sqrt[-2 - 2*x^2 + 3*x^4]))/3 + ((7 + Sqrt[7]) *Sqrt[-2 - (1 - Sqrt[7])*x^2]*Sqrt[(2 + (1 + Sqrt[7])*x^2)/(2 + (1 - Sqrt[ 7])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*7^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[7])*x^2 ]], (7 - Sqrt[7])/14])/(6*7^(1/4)*Sqrt[(2 + (1 - Sqrt[7])*x^2)^(-1)]*Sqrt[ -2 - 2*x^2 + 3*x^4])))/28
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Time = 2.22 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(\frac {x \left (3 x^{2}-8\right )}{28 \sqrt {3 x^{4}-2 x^{2}-2}}-\frac {3 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {-2-2 \sqrt {7}}\, \sqrt {3 x^{4}-2 x^{2}-2}}-\frac {6 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {-2-2 \sqrt {7}}\, \sqrt {3 x^{4}-2 x^{2}-2}\, \left (-2+2 \sqrt {7}\right )}\) | \(230\) |
| default | \(-\frac {6 \left (\frac {1}{21} x -\frac {1}{56} x^{3}\right )}{\sqrt {3 x^{4}-2 x^{2}-2}}-\frac {3 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {-2-2 \sqrt {7}}\, \sqrt {3 x^{4}-2 x^{2}-2}}-\frac {6 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {-2-2 \sqrt {7}}\, \sqrt {3 x^{4}-2 x^{2}-2}\, \left (-2+2 \sqrt {7}\right )}\) | \(231\) |
| elliptic | \(-\frac {6 \left (\frac {1}{21} x -\frac {1}{56} x^{3}\right )}{\sqrt {3 x^{4}-2 x^{2}-2}}-\frac {3 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {-2-2 \sqrt {7}}\, \sqrt {3 x^{4}-2 x^{2}-2}}-\frac {6 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {-2-2 \sqrt {7}}\, \sqrt {3 x^{4}-2 x^{2}-2}\, \left (-2+2 \sqrt {7}\right )}\) | \(231\) |
Input:
int(1/(3*x^4-2*x^2-2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/28*x*(3*x^2-8)/(3*x^4-2*x^2-2)^(1/2)-3/7/(-2-2*7^(1/2))^(1/2)*(1-(-1/2-1 /2*7^(1/2))*x^2)^(1/2)*(1-(-1/2+1/2*7^(1/2))*x^2)^(1/2)/(3*x^4-2*x^2-2)^(1 /2)*EllipticF(1/2*(-2-2*7^(1/2))^(1/2)*x,1/6*I*42^(1/2)-1/6*I*6^(1/2))-6/7 /(-2-2*7^(1/2))^(1/2)*(1-(-1/2-1/2*7^(1/2))*x^2)^(1/2)*(1-(-1/2+1/2*7^(1/2 ))*x^2)^(1/2)/(3*x^4-2*x^2-2)^(1/2)/(-2+2*7^(1/2))*(EllipticF(1/2*(-2-2*7^ (1/2))^(1/2)*x,1/6*I*42^(1/2)-1/6*I*6^(1/2))-EllipticE(1/2*(-2-2*7^(1/2))^ (1/2)*x,1/6*I*42^(1/2)-1/6*I*6^(1/2)))
Time = 0.09 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-2-2 x^2+3 x^4\right )^{3/2}} \, dx=\frac {{\left (\sqrt {7} \sqrt {-2} {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )} - \sqrt {-2} {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {7} - \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {7} - \frac {1}{2}}\right )\,|\,-\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) + {\left (\sqrt {7} \sqrt {-2} {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )} + 3 \, \sqrt {-2} {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {7} - \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {7} - \frac {1}{2}}\right )\,|\,-\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) + 2 \, \sqrt {3 \, x^{4} - 2 \, x^{2} - 2} {\left (3 \, x^{3} - 8 \, x\right )}}{56 \, {\left (3 \, x^{4} - 2 \, x^{2} - 2\right )}} \] Input:
integrate(1/(3*x^4-2*x^2-2)^(3/2),x, algorithm="fricas")
Output:
1/56*((sqrt(7)*sqrt(-2)*(3*x^4 - 2*x^2 - 2) - sqrt(-2)*(3*x^4 - 2*x^2 - 2) )*sqrt(1/2*sqrt(7) - 1/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(7) - 1/2)), -1 /3*sqrt(7) - 4/3) + (sqrt(7)*sqrt(-2)*(3*x^4 - 2*x^2 - 2) + 3*sqrt(-2)*(3* x^4 - 2*x^2 - 2))*sqrt(1/2*sqrt(7) - 1/2)*elliptic_f(arcsin(x*sqrt(1/2*sqr t(7) - 1/2)), -1/3*sqrt(7) - 4/3) + 2*sqrt(3*x^4 - 2*x^2 - 2)*(3*x^3 - 8*x ))/(3*x^4 - 2*x^2 - 2)
\[ \int \frac {1}{\left (-2-2 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (3 x^{4} - 2 x^{2} - 2\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(3*x**4-2*x**2-2)**(3/2),x)
Output:
Integral((3*x**4 - 2*x**2 - 2)**(-3/2), x)
\[ \int \frac {1}{\left (-2-2 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} - 2 \, x^{2} - 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(3*x^4-2*x^2-2)^(3/2),x, algorithm="maxima")
Output:
integrate((3*x^4 - 2*x^2 - 2)^(-3/2), x)
\[ \int \frac {1}{\left (-2-2 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (3 \, x^{4} - 2 \, x^{2} - 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(3*x^4-2*x^2-2)^(3/2),x, algorithm="giac")
Output:
integrate((3*x^4 - 2*x^2 - 2)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (-2-2 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (3\,x^4-2\,x^2-2\right )}^{3/2}} \,d x \] Input:
int(1/(3*x^4 - 2*x^2 - 2)^(3/2),x)
Output:
int(1/(3*x^4 - 2*x^2 - 2)^(3/2), x)
\[ \int \frac {1}{\left (-2-2 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {3 x^{4}-2 x^{2}-2}}{9 x^{8}-12 x^{6}-8 x^{4}+8 x^{2}+4}d x \] Input:
int(1/(3*x^4-2*x^2-2)^(3/2),x)
Output:
int(sqrt(3*x**4 - 2*x**2 - 2)/(9*x**8 - 12*x**6 - 8*x**4 + 8*x**2 + 4),x)