\(\int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx\) [18]

Optimal result

Integrand size = 16, antiderivative size = 48 \[ \int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx=\sqrt {\frac {2}{-3+\sqrt {33}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7-\sqrt {33}\right )\right ) \]

[Out]

EllipticF(x*6^(1/2)/(3+33^(1/2))^(1/2),1/4*I*6^(1/2)+1/4*I*22^(1/2))*2^(1/2)/(-3+33^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 430} \[ \int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx=\sqrt {\frac {2}{\sqrt {33}-3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7-\sqrt {33}\right )\right ) \]

[In]

Int[1/Sqrt[2 + 3*x^2 - 3*x^4],x]

[Out]

Sqrt[2/(-3 + Sqrt[33])]*EllipticF[ArcSin[Sqrt[6/(3 + Sqrt[33])]*x], (-7 - Sqrt[33])/4]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 1109

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rubi steps \begin{align*} \text {integral}& = \left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {3+\sqrt {33}-6 x^2} \sqrt {-3+\sqrt {33}+6 x^2}} \, dx \\ & = \sqrt {\frac {2}{-3+\sqrt {33}}} F\left (\sin ^{-1}\left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7-\sqrt {33}\right )\right ) \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx=-i \sqrt {\frac {2}{3+\sqrt {33}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \]

[In]

Integrate[1/Sqrt[2 + 3*x^2 - 3*x^4],x]

[Out]

(-I)*Sqrt[2/(3 + Sqrt[33])]*EllipticF[I*ArcSinh[Sqrt[6/(-3 + Sqrt[33])]*x], (-7 + Sqrt[33])/4]

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (37 ) = 74\).

Time = 0.36 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.67

method	result	size
default	\(\frac {2 \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {-3+\sqrt {33}}}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{\sqrt {-3+\sqrt {33}}\, \sqrt {-3 x^{4}+3 x^{2}+2}}\)	\(80\)
elliptic	\(\frac {2 \sqrt {1-\left (-\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {-3+\sqrt {33}}}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{\sqrt {-3+\sqrt {33}}\, \sqrt {-3 x^{4}+3 x^{2}+2}}\)	\(80\)

[In]

int(1/(-3*x^4+3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/(-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*x*(-3+33^(1/2))^(1/2),1/4*I*6^(1/2)+1/4*I*22^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx=\frac {1}{24} \, {\left (\sqrt {33} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {\sqrt {33} - 3} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} - 3}\right )\,|\,-\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) \]

[In]

integrate(1/(-3*x^4+3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/24*(sqrt(33)*sqrt(2) + 3*sqrt(2))*sqrt(sqrt(33) - 3)*elliptic_f(arcsin(1/2*x*sqrt(sqrt(33) - 3)), -1/4*sqrt(
33) - 7/4)

Sympy [F]

\[ \int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {- 3 x^{4} + 3 x^{2} + 2}}\, dx \]

[In]

integrate(1/(-3*x**4+3*x**2+2)**(1/2),x)

[Out]

Integral(1/sqrt(-3*x**4 + 3*x**2 + 2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} + 3 \, x^{2} + 2}} \,d x } \]

[In]

integrate(1/(-3*x^4+3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-3*x^4 + 3*x^2 + 2), x)

Giac [F]

\[ \int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} + 3 \, x^{2} + 2}} \,d x } \]

[In]

integrate(1/(-3*x^4+3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-3*x^4 + 3*x^2 + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2+3 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {-3\,x^4+3\,x^2+2}} \,d x \]

[In]

int(1/(3*x^2 - 3*x^4 + 2)^(1/2),x)

[Out]

int(1/(3*x^2 - 3*x^4 + 2)^(1/2), x)