Integrand size = 16, antiderivative size = 127 \[ \int \frac {1}{\left (2-3 x^2-3 x^4\right )^{3/2}} \, dx=\frac {x \left (7+3 x^2\right )}{22 \sqrt {2-3 x^2-3 x^4}}-\frac {1}{22} \sqrt {\frac {1}{2} \left (3+\sqrt {33}\right )} E\left (\arcsin \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )+\frac {1}{2} \sqrt {\frac {1}{66} \left (3+\sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \] Output:
1/22*x*(3*x^2+7)/(-3*x^4-3*x^2+2)^(1/2)-1/44*(6+2*33^(1/2))^(1/2)*Elliptic E(6^(1/2)/(-3+33^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))+1/132*(198+6 6*33^(1/2))^(1/2)*EllipticF(6^(1/2)/(-3+33^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1 /4*I*6^(1/2))
Result contains complex when optimal does not.
Time = 9.52 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (2-3 x^2-3 x^4\right )^{3/2}} \, dx=\frac {12 x \left (7+3 x^2\right )-6 i \sqrt {-3+\sqrt {33}} \sqrt {4-6 x^2-6 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right )|-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )+\frac {6 i \left (-11+\sqrt {33}\right ) \sqrt {4-6 x^2-6 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )}{\sqrt {-3+\sqrt {33}}}}{264 \sqrt {2-3 x^2-3 x^4}} \] 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], -7/4 - Sqrt[33]/4] + ((6*I)*(-1 1 + Sqrt[33])*Sqrt[4 - 6*x^2 - 6*x^4]*EllipticF[I*ArcSinh[Sqrt[6/(3 + Sqrt [33])]*x], -7/4 - Sqrt[33]/4])/Sqrt[-3 + Sqrt[33]])/(264*Sqrt[2 - 3*x^2 - 3*x^4])
Time = 0.55 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12, 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.
| \(\displaystyle \int \frac {1}{\left (-3 x^4-3 x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (3 x^2+7\right )}{22 \sqrt {-3 x^4-3 x^2+2}}-\frac {1}{66} \int -\frac {3 \left (4-3 x^2\right )}{\sqrt {-3 x^4-3 x^2+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \int \frac {4-3 x^2}{\sqrt {-3 x^4-3 x^2+2}}dx+\frac {x \left (3 x^2+7\right )}{22 \sqrt {-3 x^4-3 x^2+2}}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {1}{11} \sqrt {3} \int \frac {4-3 x^2}{\sqrt {-6 x^2+\sqrt {33}-3} \sqrt {6 x^2+\sqrt {33}+3}}dx+\frac {x \left (3 x^2+7\right )}{22 \sqrt {-3 x^4-3 x^2+2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {1}{11} \sqrt {3} \left (\frac {1}{2} \left (11+\sqrt {33}\right ) \int \frac {1}{\sqrt {-6 x^2+\sqrt {33}-3} \sqrt {6 x^2+\sqrt {33}+3}}dx-\frac {1}{2} \int \frac {\sqrt {6 x^2+\sqrt {33}+3}}{\sqrt {-6 x^2+\sqrt {33}-3}}dx\right )+\frac {x \left (3 x^2+7\right )}{22 \sqrt {-3 x^4-3 x^2+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {1}{11} \sqrt {3} \left (\frac {\left (11+\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{2 \sqrt {6 \left (3+\sqrt {33}\right )}}-\frac {1}{2} \int \frac {\sqrt {6 x^2+\sqrt {33}+3}}{\sqrt {-6 x^2+\sqrt {33}-3}}dx\right )+\frac {x \left (3 x^2+7\right )}{22 \sqrt {-3 x^4-3 x^2+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{11} \sqrt {3} \left (\frac {\left (11+\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{2 \sqrt {6 \left (3+\sqrt {33}\right )}}-\frac {1}{2} \sqrt {\frac {1}{6} \left (3+\sqrt {33}\right )} E\left (\arcsin \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )\right )+\frac {x \left (3 x^2+7\right )}{22 \sqrt {-3 x^4-3 x^2+2}}\) |
Input:
Int[(2 - 3*x^2 - 3*x^4)^(-3/2),x]
Output:
(x*(7 + 3*x^2))/(22*Sqrt[2 - 3*x^2 - 3*x^4]) + (Sqrt[3]*(-1/2*(Sqrt[(3 + S qrt[33])/6]*EllipticE[ArcSin[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4 ]) + ((11 + Sqrt[33])*EllipticF[ArcSin[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + S qrt[33])/4])/(2*Sqrt[6*(3 + Sqrt[33])])))/11
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. 217 vs. \(2 (99 ) = 198\).
Time = 2.78 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.72
| method | result | size |
| risch | \(\frac {x \left (3 x^{2}+7\right )}{22 \sqrt {-3 x^{4}-3 x^{2}+2}}+\frac {4 \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {3+\sqrt {33}}\, \sqrt {-3 x^{4}-3 x^{2}+2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {3+\sqrt {33}}\, \sqrt {-3 x^{4}-3 x^{2}+2}\, \left (-3+\sqrt {33}\right )}\) | \(218\) |
| default | \(\frac {\frac {7}{22} x +\frac {3}{22} x^{3}}{\sqrt {-3 x^{4}-3 x^{2}+2}}+\frac {4 \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {3+\sqrt {33}}\, \sqrt {-3 x^{4}-3 x^{2}+2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {3+\sqrt {33}}\, \sqrt {-3 x^{4}-3 x^{2}+2}\, \left (-3+\sqrt {33}\right )}\) | \(219\) |
| elliptic | \(\frac {\frac {7}{22} x +\frac {3}{22} x^{3}}{\sqrt {-3 x^{4}-3 x^{2}+2}}+\frac {4 \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{11 \sqrt {3+\sqrt {33}}\, \sqrt {-3 x^{4}-3 x^{2}+2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3+\sqrt {33}}\, x}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )\right )}{11 \sqrt {3+\sqrt {33}}\, \sqrt {-3 x^{4}-3 x^{2}+2}\, \left (-3+\sqrt {33}\right )}\) | \(219\) |
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))+12/1 1/(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+33^(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)))
Time = 0.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\left (2-3 x^2-3 x^4\right )^{3/2}} \, dx=-\frac {3 \, {\left (\sqrt {33} \sqrt {2} {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )} + 3 \, \sqrt {2} {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} + 3} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} + 3}\right )\,|\,\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) - {\left (7 \, \sqrt {33} \sqrt {2} {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )} - 3 \, \sqrt {2} {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} + 3} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} + 3}\right )\,|\,\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) + 24 \, \sqrt {-3 \, x^{4} - 3 \, x^{2} + 2} {\left (3 \, x^{3} + 7 \, x\right )}}{528 \, {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )}} \] 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) - (7*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_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)
\[ \int \frac {1}{\left (2-3 x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- 3 x^{4} - 3 x^{2} + 2\right )^{\frac {3}{2}}}\, dx \] 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)
\[ \int \frac {1}{\left (2-3 x^2-3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, x^{4} - 3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] 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)
\[ \int \frac {1}{\left (2-3 x^2-3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, x^{4} - 3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \,d x } \] 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)
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/(2 - 3*x^4 - 3*x^2)^(3/2),x)
Output:
int(1/(2 - 3*x^4 - 3*x^2)^(3/2), x)
\[ \int \frac {1}{\left (2-3 x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-3 x^{4}-3 x^{2}+2}}{9 x^{8}+18 x^{6}-3 x^{4}-12 x^{2}+4}d x \] 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)