Integrand size = 16, antiderivative size = 229 \[ \int \frac {1}{\left (-2+3 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {x \left (7+3 x^2\right )}{22 \sqrt {-2+3 x^2+3 x^4}}+\frac {\sqrt {3+\sqrt {33}} \sqrt {4-\left (3-\sqrt {33}\right ) x^2} \sqrt {4-\left (3+\sqrt {33}\right ) x^2} E\left (\arcsin \left (\frac {1}{2} \sqrt {3+\sqrt {33}} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{88 \sqrt {-2+3 x^2+3 x^4}}-\frac {\sqrt {\frac {1}{33} \left (3+\sqrt {33}\right )} \sqrt {4-\left (3-\sqrt {33}\right ) x^2} \sqrt {4-\left (3+\sqrt {33}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {3+\sqrt {33}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right )}{8 \sqrt {-2+3 x^2+3 x^4}} \] Output:
-1/22*x*(3*x^2+7)/(3*x^4+3*x^2-2)^(1/2)+1/88*(3+33^(1/2))^(1/2)*(4-(3-33^( 1/2))*x^2)^(1/2)*(4-(3+33^(1/2))*x^2)^(1/2)*EllipticE(1/2*(3+33^(1/2))^(1/ 2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))/(3*x^4+3*x^2-2)^(1/2)-1/264*(99+33*33^( 1/2))^(1/2)*(4-(3-33^(1/2))*x^2)^(1/2)*(4-(3+33^(1/2))*x^2)^(1/2)*Elliptic F(1/2*(3+33^(1/2))^(1/2)*x,1/4*I*22^(1/2)-1/4*I*6^(1/2))/(3*x^4+3*x^2-2)^( 1/2)
Result contains complex when optimal does not.
Time = 7.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-2+3 x^2+3 x^4\right )^{3/2}} \, dx=\frac {-12 x \left (7+3 x^2\right )+6 i \sqrt {-3+\sqrt {33}} \sqrt {4-6 x^2-6 x^4} E\left (i \text {arcsinh}\left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right )|-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )-\frac {6 i \left (-11+\sqrt {33}\right ) \sqrt {4-6 x^2-6 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )}{\sqrt {-3+\sqrt {33}}}}{264 \sqrt {-2+3 x^2+3 x^4}} \] 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]*Ell ipticE[I*ArcSinh[Sqrt[6/(3 + Sqrt[33])]*x], -7/4 - Sqrt[33]/4] - ((6*I)*(- 11 + Sqrt[33])*Sqrt[4 - 6*x^2 - 6*x^4]*EllipticF[I*ArcSinh[Sqrt[6/(3 + Sqr t[33])]*x], -7/4 - Sqrt[33]/4])/Sqrt[-3 + Sqrt[33]])/(264*Sqrt[-2 + 3*x^2 + 3*x^4])
Time = 0.78 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.61, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1405, 27, 1501, 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.
| \(\displaystyle \int \frac {1}{\left (3 x^4+3 x^2-2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {1}{66} \int -\frac {3 \left (4-3 x^2\right )}{\sqrt {3 x^4+3 x^2-2}}dx-\frac {x \left (3 x^2+7\right )}{22 \sqrt {3 x^4+3 x^2-2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{22} \int \frac {4-3 x^2}{\sqrt {3 x^4+3 x^2-2}}dx-\frac {x \left (3 x^2+7\right )}{22 \sqrt {3 x^4+3 x^2-2}}\) |
| \(\Big \downarrow \) 1501 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{2} \int \frac {6 x^2-\sqrt {33}+3}{\sqrt {3 x^4+3 x^2-2}}dx-\frac {1}{2} \left (11-\sqrt {33}\right ) \int \frac {1}{\sqrt {3 x^4+3 x^2-2}}dx\right )-\frac {x \left (3 x^2+7\right )}{22 \sqrt {3 x^4+3 x^2-2}}\) |
| \(\Big \downarrow \) 1411 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{2} \int \frac {6 x^2-\sqrt {33}+3}{\sqrt {3 x^4+3 x^2-2}}dx-\frac {\left (11-\sqrt {33}\right ) \sqrt {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-4}}\right ),\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{4 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {3 x^4+3 x^2-2}}\right )-\frac {x \left (3 x^2+7\right )}{22 \sqrt {3 x^4+3 x^2-2}}\) |
| \(\Big \downarrow \) 1498 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{2} \left (\frac {x \left (6 x^2+\sqrt {33}+3\right )}{\sqrt {3 x^4+3 x^2-2}}-\frac {\sqrt [4]{33} \sqrt {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-4} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-4}}\right )|\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{\sqrt {2} \sqrt {\frac {1}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {3 x^4+3 x^2-2}}\right )-\frac {\left (11-\sqrt {33}\right ) \sqrt {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-4} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-4}}\right ),\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{4 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {3 x^4+3 x^2-2}}\right )-\frac {x \left (3 x^2+7\right )}{22 \sqrt {3 x^4+3 x^2-2}}\) |
Input:
Int[(-2 + 3*x^2 + 3*x^4)^(-3/2),x]
Output:
-1/22*(x*(7 + 3*x^2))/Sqrt[-2 + 3*x^2 + 3*x^4] + (((x*(3 + Sqrt[33] + 6*x^ 2))/Sqrt[-2 + 3*x^2 + 3*x^4] - (33^(1/4)*Sqrt[(4 - (3 - Sqrt[33])*x^2)/(4 - (3 + Sqrt[33])*x^2)]*Sqrt[-4 + (3 + Sqrt[33])*x^2]*EllipticE[ArcSin[(Sqr t[2]*33^(1/4)*x)/Sqrt[-4 + (3 + Sqrt[33])*x^2]], (11 + Sqrt[33])/22])/(Sqr t[2]*Sqrt[(4 - (3 + Sqrt[33])*x^2)^(-1)]*Sqrt[-2 + 3*x^2 + 3*x^4]))/2 - (( 11 - Sqrt[33])*Sqrt[(4 - (3 - Sqrt[33])*x^2)/(4 - (3 + Sqrt[33])*x^2)]*Sqr t[-4 + (3 + Sqrt[33])*x^2]*EllipticF[ArcSin[(Sqrt[2]*33^(1/4)*x)/Sqrt[-4 + (3 + Sqrt[33])*x^2]], (11 + Sqrt[33])/22])/(4*Sqrt[2]*33^(1/4)*Sqrt[(4 - (3 + Sqrt[33])*x^2)^(-1)]*Sqrt[-2 + 3*x^2 + 3*x^4]))/22
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 = 2.27 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(-\frac {x \left (3 x^{2}+7\right )}{22 \sqrt {3 x^{4}+3 x^{2}-2}}-\frac {4 \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{11 \sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}\, \left (3+\sqrt {33}\right )}\) | \(228\) |
| default | \(-\frac {6 \left (\frac {7}{132} x +\frac {1}{44} x^{3}\right )}{\sqrt {3 x^{4}+3 x^{2}-2}}-\frac {4 \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{11 \sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}\, \left (3+\sqrt {33}\right )}\) | \(229\) |
| elliptic | \(-\frac {6 \left (\frac {7}{132} x +\frac {1}{44} x^{3}\right )}{\sqrt {3 x^{4}+3 x^{2}-2}}-\frac {4 \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{11 \sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}}+\frac {12 \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )\right )}{11 \sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}\, \left (3+\sqrt {33}\right )}\) | \(229\) |
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*6^(1/2)+1/4*I*22^(1/2))+12/11 /(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)/(3+33^(1/2))*(EllipticF(1/2*(3-33^(1/2)) ^(1/2)*x,1/4*I*6^(1/2)+1/4*I*22^(1/2))-EllipticE(1/2*(3-33^(1/2))^(1/2)*x, 1/4*I*6^(1/2)+1/4*I*22^(1/2)))
Time = 0.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (-2+3 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {3 \, {\left (\sqrt {33} \sqrt {-2} {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )} + 3 \, \sqrt {-2} {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} + 3} E(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} + 3}\right )\,|\,\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) - {\left (7 \, \sqrt {33} \sqrt {-2} {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )} - 3 \, \sqrt {-2} {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )}\right )} \sqrt {\sqrt {33} + 3} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} + 3}\right )\,|\,\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) + 24 \, \sqrt {3 \, x^{4} + 3 \, x^{2} - 2} {\left (3 \, x^{3} + 7 \, x\right )}}{528 \, {\left (3 \, x^{4} + 3 \, x^{2} - 2\right )}} \] 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/(3*x^2 + 3*x^4 - 2)^(3/2),x)
Output:
int(1/(3*x^2 + 3*x^4 - 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)