Integrand size = 16, antiderivative size = 123 \[ \int \frac {1}{\left (2-2 x^2-3 x^4\right )^{3/2}} \, dx=\frac {x \left (8+3 x^2\right )}{28 \sqrt {2-2 x^2-3 x^4}}-\frac {1}{28} \sqrt {1+\sqrt {7}} E\left (\arcsin \left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )+\frac {1}{4} \sqrt {\frac {1}{7} \left (1+\sqrt {7}\right )} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right ) \] Output:
1/28*x*(3*x^2+8)/(-3*x^4-2*x^2+2)^(1/2)-1/28*(1+7^(1/2))^(1/2)*EllipticE(3 ^(1/2)/(-1+7^(1/2))^(1/2)*x,1/6*I*42^(1/2)-1/6*I*6^(1/2))+1/28*(7+7*7^(1/2 ))^(1/2)*EllipticF(3^(1/2)/(-1+7^(1/2))^(1/2)*x,1/6*I*42^(1/2)-1/6*I*6^(1/ 2))
Result contains complex when optimal does not.
Time = 9.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (2-2 x^2-3 x^4\right )^{3/2}} \, dx=\frac {1}{28} \left (\frac {x \left (8+3 x^2\right )}{\sqrt {2-2 x^2-3 x^4}}-i \sqrt {-1+\sqrt {7}} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right )|-\frac {4}{3}-\frac {\sqrt {7}}{3}\right )+\frac {i \left (-7+\sqrt {7}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right ),-\frac {4}{3}-\frac {\sqrt {7}}{3}\right )}{\sqrt {-1+\sqrt {7}}}\right ) \] Input:
Integrate[(2 - 2*x^2 - 3*x^4)^(-3/2),x]
Output:
((x*(8 + 3*x^2))/Sqrt[2 - 2*x^2 - 3*x^4] - I*Sqrt[-1 + Sqrt[7]]*EllipticE[ I*ArcSinh[Sqrt[3/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3] + (I*(-7 + Sqrt[7])* EllipticF[I*ArcSinh[Sqrt[3/(1 + Sqrt[7])]*x], -4/3 - Sqrt[7]/3])/Sqrt[-1 + Sqrt[7]])/28
Time = 0.51 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1405, 27, 1494, 27, 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.
| \(\displaystyle \int \frac {1}{\left (-3 x^4-2 x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (3 x^2+8\right )}{28 \sqrt {-3 x^4-2 x^2+2}}-\frac {1}{56} \int -\frac {6 \left (2-x^2\right )}{\sqrt {-3 x^4-2 x^2+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{28} \int \frac {2-x^2}{\sqrt {-3 x^4-2 x^2+2}}dx+\frac {x \left (3 x^2+8\right )}{28 \sqrt {-3 x^4-2 x^2+2}}\) |
| \(\Big \downarrow \) 1494 |
| \(\displaystyle \frac {3}{14} \sqrt {3} \int \frac {2-x^2}{2 \sqrt {-3 x^2+\sqrt {7}-1} \sqrt {3 x^2+\sqrt {7}+1}}dx+\frac {x \left (3 x^2+8\right )}{28 \sqrt {-3 x^4-2 x^2+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{28} \sqrt {3} \int \frac {2-x^2}{\sqrt {-3 x^2+\sqrt {7}-1} \sqrt {3 x^2+\sqrt {7}+1}}dx+\frac {x \left (3 x^2+8\right )}{28 \sqrt {-3 x^4-2 x^2+2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {3}{28} \sqrt {3} \left (\frac {1}{3} \left (7+\sqrt {7}\right ) \int \frac {1}{\sqrt {-3 x^2+\sqrt {7}-1} \sqrt {3 x^2+\sqrt {7}+1}}dx-\frac {1}{3} \int \frac {\sqrt {3 x^2+\sqrt {7}+1}}{\sqrt {-3 x^2+\sqrt {7}-1}}dx\right )+\frac {x \left (3 x^2+8\right )}{28 \sqrt {-3 x^4-2 x^2+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {3}{28} \sqrt {3} \left (\frac {\left (7+\sqrt {7}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{3 \sqrt {3 \left (1+\sqrt {7}\right )}}-\frac {1}{3} \int \frac {\sqrt {3 x^2+\sqrt {7}+1}}{\sqrt {-3 x^2+\sqrt {7}-1}}dx\right )+\frac {x \left (3 x^2+8\right )}{28 \sqrt {-3 x^4-2 x^2+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {3}{28} \sqrt {3} \left (\frac {\left (7+\sqrt {7}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{3 \sqrt {3 \left (1+\sqrt {7}\right )}}-\frac {1}{3} \sqrt {\frac {1}{3} \left (1+\sqrt {7}\right )} E\left (\arcsin \left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )\right )+\frac {x \left (3 x^2+8\right )}{28 \sqrt {-3 x^4-2 x^2+2}}\) |
Input:
Int[(2 - 2*x^2 - 3*x^4)^(-3/2),x]
Output:
(x*(8 + 3*x^2))/(28*Sqrt[2 - 2*x^2 - 3*x^4]) + (3*Sqrt[3]*(-1/3*(Sqrt[(1 + Sqrt[7])/3]*EllipticE[ArcSin[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[7])/3] ) + ((7 + Sqrt[7])*EllipticF[ArcSin[Sqrt[3/(-1 + Sqrt[7])]*x], (-4 + Sqrt[ 7])/3])/(3*Sqrt[3*(1 + Sqrt[7])])))/28
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. 229 vs. \(2 (97 ) = 194\).
Time = 2.29 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.87
| method | result | size |
| risch | \(\frac {x \left (3 x^{2}+8\right )}{28 \sqrt {-3 x^{4}-2 x^{2}+2}}+\frac {3 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {\sqrt {7}}{2}+\frac {1}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}}+\frac {6 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {\sqrt {7}}{2}+\frac {1}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}\, \left (-2+2 \sqrt {7}\right )}\) | \(230\) |
| default | \(\frac {\frac {2}{7} x +\frac {3}{28} x^{3}}{\sqrt {-3 x^{4}-2 x^{2}+2}}+\frac {3 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {\sqrt {7}}{2}+\frac {1}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}}+\frac {6 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {\sqrt {7}}{2}+\frac {1}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}\, \left (-2+2 \sqrt {7}\right )}\) | \(231\) |
| elliptic | \(\frac {\frac {2}{7} x +\frac {3}{28} x^{3}}{\sqrt {-3 x^{4}-2 x^{2}+2}}+\frac {3 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {\sqrt {7}}{2}+\frac {1}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{7 \sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}}+\frac {6 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {\sqrt {7}}{2}+\frac {1}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {2+2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )\right )}{7 \sqrt {2+2 \sqrt {7}}\, \sqrt {-3 x^{4}-2 x^{2}+2}\, \left (-2+2 \sqrt {7}\right )}\) | \(231\) |
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*7^(1/2)+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*7^(1/2)+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 = 177, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (2-2 x^2-3 x^4\right )^{3/2}} \, dx=-\frac {{\left (\sqrt {7} \sqrt {2} {\left (3 \, x^{4} + 2 \, x^{2} - 2\right )} + \sqrt {2} {\left (3 \, x^{4} + 2 \, x^{2} - 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {7} + \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {7} + \frac {1}{2}}\right )\,|\,\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) - {\left (3 \, \sqrt {7} \sqrt {2} {\left (3 \, x^{4} + 2 \, x^{2} - 2\right )} - \sqrt {2} {\left (3 \, x^{4} + 2 \, x^{2} - 2\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {7} + \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {7} + \frac {1}{2}}\right )\,|\,\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) + 2 \, \sqrt {-3 \, x^{4} - 2 \, x^{2} + 2} {\left (3 \, x^{3} + 8 \, x\right )}}{56 \, {\left (3 \, x^{4} + 2 \, x^{2} - 2\right )}} \] 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) - (3*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_f(arcsin(x*sqrt(1/2*sqrt(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/(2 - 3*x^4 - 2*x^2)^(3/2),x)
Output:
int(1/(2 - 3*x^4 - 2*x^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)