Integrand size = 16, antiderivative size = 121 \[ \int \frac {1}{\left (3-3 x^2-2 x^4\right )^{3/2}} \, dx=\frac {x \left (7+2 x^2\right )}{33 \sqrt {3-3 x^2-2 x^4}}-\frac {1}{33} \sqrt {\frac {1}{2} \left (3+\sqrt {33}\right )} E\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )+\frac {1}{3} \sqrt {\frac {1}{66} \left (3+\sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \] Output:
1/33*x*(2*x^2+7)/(-2*x^4-3*x^2+3)^(1/2)-1/66*(6+2*33^(1/2))^(1/2)*Elliptic E(2*x/(-3+33^(1/2))^(1/2),1/4*I*22^(1/2)-1/4*I*6^(1/2))+1/198*(198+66*33^( 1/2))^(1/2)*EllipticF(2*x/(-3+33^(1/2))^(1/2),1/4*I*22^(1/2)-1/4*I*6^(1/2) )
Result contains complex when optimal does not.
Time = 9.16 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\left (3-3 x^2-2 x^4\right )^{3/2}} \, dx=\frac {4 x \left (7+2 x^2\right )-2 i \sqrt {-3+\sqrt {33}} \sqrt {6-6 x^2-4 x^4} E\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {3+\sqrt {33}}}\right )|-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )+\frac {2 i \left (-11+\sqrt {33}\right ) \sqrt {6-6 x^2-4 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {3+\sqrt {33}}}\right ),-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )}{\sqrt {-3+\sqrt {33}}}}{132 \sqrt {3-3 x^2-2 x^4}} \] Input:
Integrate[(3 - 3*x^2 - 2*x^4)^(-3/2),x]
Output:
(4*x*(7 + 2*x^2) - (2*I)*Sqrt[-3 + Sqrt[33]]*Sqrt[6 - 6*x^2 - 4*x^4]*Ellip ticE[I*ArcSinh[(2*x)/Sqrt[3 + Sqrt[33]]], -7/4 - Sqrt[33]/4] + ((2*I)*(-11 + Sqrt[33])*Sqrt[6 - 6*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x)/Sqrt[3 + Sq rt[33]]], -7/4 - Sqrt[33]/4])/Sqrt[-3 + Sqrt[33]])/(132*Sqrt[3 - 3*x^2 - 2 *x^4])
Time = 0.49 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1494, 399, 321, 327}
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.
| \(\displaystyle \int \frac {1}{\left (-2 x^4-3 x^2+3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (2 x^2+7\right )}{33 \sqrt {-2 x^4-3 x^2+3}}-\frac {1}{99} \int -\frac {6 \left (2-x^2\right )}{\sqrt {-2 x^4-3 x^2+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{33} \int \frac {2-x^2}{\sqrt {-2 x^4-3 x^2+3}}dx+\frac {x \left (2 x^2+7\right )}{33 \sqrt {-2 x^4-3 x^2+3}}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {4}{33} \sqrt {2} \int \frac {2-x^2}{\sqrt {-4 x^2+\sqrt {33}-3} \sqrt {4 x^2+\sqrt {33}+3}}dx+\frac {x \left (2 x^2+7\right )}{33 \sqrt {-2 x^4-3 x^2+3}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {4}{33} \sqrt {2} \left (\frac {1}{4} \left (11+\sqrt {33}\right ) \int \frac {1}{\sqrt {-4 x^2+\sqrt {33}-3} \sqrt {4 x^2+\sqrt {33}+3}}dx-\frac {1}{4} \int \frac {\sqrt {4 x^2+\sqrt {33}+3}}{\sqrt {-4 x^2+\sqrt {33}-3}}dx\right )+\frac {x \left (2 x^2+7\right )}{33 \sqrt {-2 x^4-3 x^2+3}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {4}{33} \sqrt {2} \left (\frac {\left (11+\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{8 \sqrt {3+\sqrt {33}}}-\frac {1}{4} \int \frac {\sqrt {4 x^2+\sqrt {33}+3}}{\sqrt {-4 x^2+\sqrt {33}-3}}dx\right )+\frac {x \left (2 x^2+7\right )}{33 \sqrt {-2 x^4-3 x^2+3}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {4}{33} \sqrt {2} \left (\frac {\left (11+\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{8 \sqrt {3+\sqrt {33}}}-\frac {1}{8} \sqrt {3+\sqrt {33}} E\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )\right )+\frac {x \left (2 x^2+7\right )}{33 \sqrt {-2 x^4-3 x^2+3}}\) |
Input:
Int[(3 - 3*x^2 - 2*x^4)^(-3/2),x]
Output:
(x*(7 + 2*x^2))/(33*Sqrt[3 - 3*x^2 - 2*x^4]) + (4*Sqrt[2]*(-1/8*(Sqrt[3 + Sqrt[33]]*EllipticE[ArcSin[(2*x)/Sqrt[-3 + Sqrt[33]]], (-7 + Sqrt[33])/4]) + ((11 + Sqrt[33])*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[33]]], (-7 + Sqr t[33])/4])/(8*Sqrt[3 + Sqrt[33]])))/33
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
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[((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*Sqrt[-c] Int[(d + e*x^2)/(Sqr t[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e }, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (95 ) = 190\).
Time = 2.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.88
| method | result | size |
| risch | \(\frac {x \left (2 x^{2}+7\right )}{33 \sqrt {-2 x^{4}-3 x^{2}+3}}+\frac {8 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}}+\frac {24 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}\, \left (-3+\sqrt {33}\right )}\) | \(228\) |
| default | \(\frac {\frac {7}{33} x +\frac {2}{33} x^{3}}{\sqrt {-2 x^{4}-3 x^{2}+3}}+\frac {8 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}}+\frac {24 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}\, \left (-3+\sqrt {33}\right )}\) | \(229\) |
| elliptic | \(\frac {\frac {7}{33} x +\frac {2}{33} x^{3}}{\sqrt {-2 x^{4}-3 x^{2}+3}}+\frac {8 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}}+\frac {24 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {18+6 \sqrt {33}}\, x}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}\, \left (-3+\sqrt {33}\right )}\) | \(229\) |
Input:
int(1/(-2*x^4-3*x^2+3)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/33*x*(2*x^2+7)/(-2*x^4-3*x^2+3)^(1/2)+8/11/(18+6*33^(1/2))^(1/2)*(1-(1/2 +1/6*33^(1/2))*x^2)^(1/2)*(1-(1/2-1/6*33^(1/2))*x^2)^(1/2)/(-2*x^4-3*x^2+3 )^(1/2)*EllipticF(1/6*(18+6*33^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2) )+24/11/(18+6*33^(1/2))^(1/2)*(1-(1/2+1/6*33^(1/2))*x^2)^(1/2)*(1-(1/2-1/6 *33^(1/2))*x^2)^(1/2)/(-2*x^4-3*x^2+3)^(1/2)/(-3+33^(1/2))*(EllipticF(1/6* (18+6*33^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))-EllipticE(1/6*(18+6* 33^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2)))
Time = 0.09 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\left (3-3 x^2-2 x^4\right )^{3/2}} \, dx=-\frac {{\left (\sqrt {\frac {11}{3}} \sqrt {3} {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )} + \sqrt {3} {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {\frac {11}{3}} + \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {\frac {11}{3}} + \frac {1}{2}}\right )\,|\,\frac {3}{4} \, \sqrt {\frac {11}{3}} - \frac {7}{4}) - {\left (3 \, \sqrt {\frac {11}{3}} \sqrt {3} {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )} - \sqrt {3} {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {\frac {11}{3}} + \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {\frac {11}{3}} + \frac {1}{2}}\right )\,|\,\frac {3}{4} \, \sqrt {\frac {11}{3}} - \frac {7}{4}) + 2 \, \sqrt {-2 \, x^{4} - 3 \, x^{2} + 3} {\left (2 \, x^{3} + 7 \, x\right )}}{66 \, {\left (2 \, x^{4} + 3 \, x^{2} - 3\right )}} \] Input:
integrate(1/(-2*x^4-3*x^2+3)^(3/2),x, algorithm="fricas")
Output:
-1/66*((sqrt(11/3)*sqrt(3)*(2*x^4 + 3*x^2 - 3) + sqrt(3)*(2*x^4 + 3*x^2 - 3))*sqrt(1/2*sqrt(11/3) + 1/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(11/3) + 1 /2)), 3/4*sqrt(11/3) - 7/4) - (3*sqrt(11/3)*sqrt(3)*(2*x^4 + 3*x^2 - 3) - sqrt(3)*(2*x^4 + 3*x^2 - 3))*sqrt(1/2*sqrt(11/3) + 1/2)*elliptic_f(arcsin( x*sqrt(1/2*sqrt(11/3) + 1/2)), 3/4*sqrt(11/3) - 7/4) + 2*sqrt(-2*x^4 - 3*x ^2 + 3)*(2*x^3 + 7*x))/(2*x^4 + 3*x^2 - 3)
\[ \int \frac {1}{\left (3-3 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- 2 x^{4} - 3 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(-2*x**4-3*x**2+3)**(3/2),x)
Output:
Integral((-2*x**4 - 3*x**2 + 3)**(-3/2), x)
\[ \int \frac {1}{\left (3-3 x^2-2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-2 \, x^{4} - 3 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-2*x^4-3*x^2+3)^(3/2),x, algorithm="maxima")
Output:
integrate((-2*x^4 - 3*x^2 + 3)^(-3/2), x)
\[ \int \frac {1}{\left (3-3 x^2-2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-2 \, x^{4} - 3 \, x^{2} + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-2*x^4-3*x^2+3)^(3/2),x, algorithm="giac")
Output:
integrate((-2*x^4 - 3*x^2 + 3)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (3-3 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-2\,x^4-3\,x^2+3\right )}^{3/2}} \,d x \] Input:
int(1/(3 - 2*x^4 - 3*x^2)^(3/2),x)
Output:
int(1/(3 - 2*x^4 - 3*x^2)^(3/2), x)
\[ \int \frac {1}{\left (3-3 x^2-2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-2 x^{4}-3 x^{2}+3}}{4 x^{8}+12 x^{6}-3 x^{4}-18 x^{2}+9}d x \] Input:
int(1/(-2*x^4-3*x^2+3)^(3/2),x)
Output:
int(sqrt( - 2*x**4 - 3*x**2 + 3)/(4*x**8 + 12*x**6 - 3*x**4 - 18*x**2 + 9) ,x)