Integrand size = 16, antiderivative size = 262 \[ \int \frac {1}{\left (-2-x^2-3 x^4\right )^{3/2}} \, dx=-\frac {x \left (11-3 x^2\right )}{46 \sqrt {-2-x^2-3 x^4}}+\frac {3 x \sqrt {-2-x^2-3 x^4}}{46 \left (\sqrt {6}+3 x^2\right )}+\frac {\sqrt [4]{3} \left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{23\ 2^{3/4} \sqrt {-2-x^2-3 x^4}}-\frac {\sqrt [4]{3} \left (1+2 \sqrt {6}\right ) \left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2+x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{46\ 2^{3/4} \sqrt {-2-x^2-3 x^4}} \] Output:
-1/46*x*(-3*x^2+11)/(-3*x^4-x^2-2)^(1/2)+3*x*(-3*x^4-x^2-2)^(1/2)/(46*6^(1 /2)+138*x^2)+1/46*3^(1/4)*(2+6^(1/2)*x^2)*((3*x^4+x^2+2)/(2+6^(1/2)*x^2)^2 )^(1/2)*EllipticE(sin(2*arctan(1/2*3^(1/4)*2^(3/4)*x)),1/12*(72-6*6^(1/2)) ^(1/2))*2^(1/4)/(-3*x^4-x^2-2)^(1/2)-1/92*3^(1/4)*(1+2*6^(1/2))*(2+6^(1/2) *x^2)*((3*x^4+x^2+2)/(2+6^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(1/2 *3^(1/4)*2^(3/4)*x),1/12*(72-6*6^(1/2))^(1/2))*2^(1/4)/(-3*x^4-x^2-2)^(1/2 )
Result contains complex when optimal does not.
Time = 6.60 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (-2-x^2-3 x^4\right )^{3/2}} \, dx=\frac {12 \sqrt {-\frac {i}{-i+\sqrt {23}}} x \left (-11+3 x^2\right )+\sqrt {6} \left (i+\sqrt {23}\right ) \sqrt {\frac {-i+\sqrt {23}-6 i x^2}{-i+\sqrt {23}}} \sqrt {\frac {i+\sqrt {23}+6 i x^2}{i+\sqrt {23}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {6 i}{-i+\sqrt {23}}} x\right )|\frac {i-\sqrt {23}}{i+\sqrt {23}}\right )-\sqrt {6} \left (-23 i+\sqrt {23}\right ) \sqrt {\frac {-i+\sqrt {23}-6 i x^2}{-i+\sqrt {23}}} \sqrt {\frac {i+\sqrt {23}+6 i x^2}{i+\sqrt {23}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {6 i}{-i+\sqrt {23}}} x\right ),\frac {i-\sqrt {23}}{i+\sqrt {23}}\right )}{552 \sqrt {-\frac {i}{-i+\sqrt {23}}} \sqrt {-2-x^2-3 x^4}} \] Input:
Integrate[(-2 - x^2 - 3*x^4)^(-3/2),x]
Output:
(12*Sqrt[(-I)/(-I + Sqrt[23])]*x*(-11 + 3*x^2) + Sqrt[6]*(I + Sqrt[23])*Sq rt[(-I + Sqrt[23] - (6*I)*x^2)/(-I + Sqrt[23])]*Sqrt[(I + Sqrt[23] + (6*I) *x^2)/(I + Sqrt[23])]*EllipticE[I*ArcSinh[Sqrt[(-6*I)/(-I + Sqrt[23])]*x], (I - Sqrt[23])/(I + Sqrt[23])] - Sqrt[6]*(-23*I + Sqrt[23])*Sqrt[(-I + Sq rt[23] - (6*I)*x^2)/(-I + Sqrt[23])]*Sqrt[(I + Sqrt[23] + (6*I)*x^2)/(I + Sqrt[23])]*EllipticF[I*ArcSinh[Sqrt[(-6*I)/(-I + Sqrt[23])]*x], (I - Sqrt[ 23])/(I + Sqrt[23])])/(552*Sqrt[(-I)/(-I + Sqrt[23])]*Sqrt[-2 - x^2 - 3*x^ 4])
Time = 0.61 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {1}{\left (-3 x^4-x^2-2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle -\frac {1}{46} \int \frac {3 \left (x^2+4\right )}{\sqrt {-3 x^4-x^2-2}}dx-\frac {x \left (11-3 x^2\right )}{46 \sqrt {-3 x^4-x^2-2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{46} \int \frac {x^2+4}{\sqrt {-3 x^4-x^2-2}}dx-\frac {x \left (11-3 x^2\right )}{46 \sqrt {-3 x^4-x^2-2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {3}{46} \left (\frac {1}{3} \left (12+\sqrt {6}\right ) \int \frac {1}{\sqrt {-3 x^4-x^2-2}}dx-\sqrt {\frac {2}{3}} \int \frac {2-\sqrt {6} x^2}{2 \sqrt {-3 x^4-x^2-2}}dx\right )-\frac {x \left (11-3 x^2\right )}{46 \sqrt {-3 x^4-x^2-2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{46} \left (\frac {1}{3} \left (12+\sqrt {6}\right ) \int \frac {1}{\sqrt {-3 x^4-x^2-2}}dx-\frac {\int \frac {2-\sqrt {6} x^2}{\sqrt {-3 x^4-x^2-2}}dx}{\sqrt {6}}\right )-\frac {x \left (11-3 x^2\right )}{46 \sqrt {-3 x^4-x^2-2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {3}{46} \left (\frac {\left (12+\sqrt {6}\right ) \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{6 \sqrt [4]{6} \sqrt {-3 x^4-x^2-2}}-\frac {\int \frac {2-\sqrt {6} x^2}{\sqrt {-3 x^4-x^2-2}}dx}{\sqrt {6}}\right )-\frac {x \left (11-3 x^2\right )}{46 \sqrt {-3 x^4-x^2-2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {3}{46} \left (\frac {\left (12+\sqrt {6}\right ) \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{6 \sqrt [4]{6} \sqrt {-3 x^4-x^2-2}}-\frac {\frac {2^{3/4} \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4+x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{24} \left (12-\sqrt {6}\right )\right )}{\sqrt [4]{3} \sqrt {-3 x^4-x^2-2}}+\frac {2 \sqrt {-3 x^4-x^2-2} x}{\sqrt {6} x^2+2}}{\sqrt {6}}\right )-\frac {x \left (11-3 x^2\right )}{46 \sqrt {-3 x^4-x^2-2}}\) |
Input:
Int[(-2 - x^2 - 3*x^4)^(-3/2),x]
Output:
-1/46*(x*(11 - 3*x^2))/Sqrt[-2 - x^2 - 3*x^4] - (3*(-(((2*x*Sqrt[-2 - x^2 - 3*x^4])/(2 + Sqrt[6]*x^2) + (2^(3/4)*(2 + Sqrt[6]*x^2)*Sqrt[(2 + x^2 + 3 *x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticE[2*ArcTan[(3/2)^(1/4)*x], (12 - Sqrt[6 ])/24])/(3^(1/4)*Sqrt[-2 - x^2 - 3*x^4]))/Sqrt[6]) + ((12 + Sqrt[6])*(2 + Sqrt[6]*x^2)*Sqrt[(2 + x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*ArcTa n[(3/2)^(1/4)*x], (12 - Sqrt[6])/24])/(6*6^(1/4)*Sqrt[-2 - x^2 - 3*x^4]))) /46
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[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] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
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] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
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] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Result contains complex when optimal does not.
Time = 2.17 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(\frac {x \left (3 x^{2}-11\right )}{46 \sqrt {-3 x^{4}-x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )}{23 \sqrt {-1-i \sqrt {23}}\, \sqrt {-3 x^{4}-x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )\right )}{23 \sqrt {-1-i \sqrt {23}}\, \sqrt {-3 x^{4}-x^{2}-2}\, \left (-1+i \sqrt {23}\right )}\) | \(237\) |
| default | \(\frac {-\frac {11}{46} x +\frac {3}{46} x^{3}}{\sqrt {-3 x^{4}-x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )}{23 \sqrt {-1-i \sqrt {23}}\, \sqrt {-3 x^{4}-x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )\right )}{23 \sqrt {-1-i \sqrt {23}}\, \sqrt {-3 x^{4}-x^{2}-2}\, \left (-1+i \sqrt {23}\right )}\) | \(238\) |
| elliptic | \(\frac {-\frac {11}{46} x +\frac {3}{46} x^{3}}{\sqrt {-3 x^{4}-x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )}{23 \sqrt {-1-i \sqrt {23}}\, \sqrt {-3 x^{4}-x^{2}-2}}-\frac {12 \sqrt {1-\left (-\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {-1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33-3 i \sqrt {23}}}{6}\right )\right )}{23 \sqrt {-1-i \sqrt {23}}\, \sqrt {-3 x^{4}-x^{2}-2}\, \left (-1+i \sqrt {23}\right )}\) | \(238\) |
Input:
int(1/(-3*x^4-x^2-2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/46*x*(3*x^2-11)/(-3*x^4-x^2-2)^(1/2)-12/23/(-1-I*23^(1/2))^(1/2)*(1-(-1/ 4-1/4*I*23^(1/2))*x^2)^(1/2)*(1-(-1/4+1/4*I*23^(1/2))*x^2)^(1/2)/(-3*x^4-x ^2-2)^(1/2)*EllipticF(1/2*(-1-I*23^(1/2))^(1/2)*x,1/6*(-33-3*I*23^(1/2))^( 1/2))-12/23/(-1-I*23^(1/2))^(1/2)*(1-(-1/4-1/4*I*23^(1/2))*x^2)^(1/2)*(1-( -1/4+1/4*I*23^(1/2))*x^2)^(1/2)/(-3*x^4-x^2-2)^(1/2)/(-1+I*23^(1/2))*(Elli pticF(1/2*(-1-I*23^(1/2))^(1/2)*x,1/6*(-33-3*I*23^(1/2))^(1/2))-EllipticE( 1/2*(-1-I*23^(1/2))^(1/2)*x,1/6*(-33-3*I*23^(1/2))^(1/2)))
Time = 0.07 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (-2-x^2-3 x^4\right )^{3/2}} \, dx=\frac {\sqrt {-2} {\left (3 \, x^{4} + x^{2} - \sqrt {-23} {\left (3 \, x^{4} + x^{2} + 2\right )} + 2\right )} \sqrt {\sqrt {-23} - 1} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {-23} - 1}\right )\,|\,\frac {1}{12} \, \sqrt {-23} - \frac {11}{12}) - \sqrt {-2} {\left (15 \, x^{4} + 5 \, x^{2} + 3 \, \sqrt {-23} {\left (3 \, x^{4} + x^{2} + 2\right )} + 10\right )} \sqrt {\sqrt {-23} - 1} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {-23} - 1}\right )\,|\,\frac {1}{12} \, \sqrt {-23} - \frac {11}{12}) - 8 \, \sqrt {-3 \, x^{4} - x^{2} - 2} {\left (3 \, x^{3} - 11 \, x\right )}}{368 \, {\left (3 \, x^{4} + x^{2} + 2\right )}} \] Input:
integrate(1/(-3*x^4-x^2-2)^(3/2),x, algorithm="fricas")
Output:
1/368*(sqrt(-2)*(3*x^4 + x^2 - sqrt(-23)*(3*x^4 + x^2 + 2) + 2)*sqrt(sqrt( -23) - 1)*elliptic_e(arcsin(1/2*x*sqrt(sqrt(-23) - 1)), 1/12*sqrt(-23) - 1 1/12) - sqrt(-2)*(15*x^4 + 5*x^2 + 3*sqrt(-23)*(3*x^4 + x^2 + 2) + 10)*sqr t(sqrt(-23) - 1)*elliptic_f(arcsin(1/2*x*sqrt(sqrt(-23) - 1)), 1/12*sqrt(- 23) - 11/12) - 8*sqrt(-3*x^4 - x^2 - 2)*(3*x^3 - 11*x))/(3*x^4 + x^2 + 2)
\[ \int \frac {1}{\left (-2-x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- 3 x^{4} - x^{2} - 2\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(-3*x**4-x**2-2)**(3/2),x)
Output:
Integral((-3*x**4 - x**2 - 2)**(-3/2), x)
\[ \int \frac {1}{\left (-2-x^2-3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-3*x^4-x^2-2)^(3/2),x, algorithm="maxima")
Output:
integrate((-3*x^4 - x^2 - 2)^(-3/2), x)
\[ \int \frac {1}{\left (-2-x^2-3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-3*x^4-x^2-2)^(3/2),x, algorithm="giac")
Output:
integrate((-3*x^4 - x^2 - 2)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (-2-x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-3\,x^4-x^2-2\right )}^{3/2}} \,d x \] Input:
int(1/(- x^2 - 3*x^4 - 2)^(3/2),x)
Output:
int(1/(- x^2 - 3*x^4 - 2)^(3/2), x)
\[ \int \frac {1}{\left (-2-x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-3 x^{4}-x^{2}-2}}{9 x^{8}+6 x^{6}+13 x^{4}+4 x^{2}+4}d x \] Input:
int(1/(-3*x^4-x^2-2)^(3/2),x)
Output:
int(sqrt( - 3*x**4 - x**2 - 2)/(9*x**8 + 6*x**6 + 13*x**4 + 4*x**2 + 4),x)