\(\int \frac {1}{\sqrt {3+4 x^2-2 x^4}} \, dx\) [30]

Optimal result

Integrand size = 16, antiderivative size = 44 \[ \int \frac {1}{\sqrt {3+4 x^2-2 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{2+\sqrt {10}}} x\right ),\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right )}{\sqrt {-2+\sqrt {10}}} \]

[Out]

EllipticF(x*2^(1/2)/(2+10^(1/2))^(1/2),1/3*I*6^(1/2)+1/3*I*15^(1/2))/(-2+10^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 44, 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} \[ \int \frac {1}{\sqrt {3+4 x^2-2 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{2+\sqrt {10}}} x\right ),\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right )}{\sqrt {\sqrt {10}-2}} \]

[In]

Int[1/Sqrt[3 + 4*x^2 - 2*x^4],x]

[Out]

EllipticF[ArcSin[Sqrt[2/(2 + Sqrt[10])]*x], (-7 - 2*Sqrt[10])/3]/Sqrt[-2 + Sqrt[10]]

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 {2}\right ) \int \frac {1}{\sqrt {4+2 \sqrt {10}-4 x^2} \sqrt {-4+2 \sqrt {10}+4 x^2}} \, dx \\ & = \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{2+\sqrt {10}}} x\right )|\frac {1}{3} \left (-7-2 \sqrt {10}\right )\right )}{\sqrt {-2+\sqrt {10}}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {3+4 x^2-2 x^4}} \, dx=-\frac {i \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{-2+\sqrt {10}}} x\right ),-\frac {7}{3}+\frac {2 \sqrt {10}}{3}\right )}{\sqrt {2+\sqrt {10}}} \]

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[Sqrt[2/(-2 + Sqrt[10])]*x], -7/3 + (2*Sqrt[10])/3])/Sqrt[2 + Sqrt[10]]

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (34 ) = 68\).

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.91

method	result	size
default	\(\frac {3 \sqrt {1-\left (-\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+3 \sqrt {10}}}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {-6+3 \sqrt {10}}\, \sqrt {-2 x^{4}+4 x^{2}+3}}\)	\(84\)
elliptic	\(\frac {3 \sqrt {1-\left (-\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+3 \sqrt {10}}}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {-6+3 \sqrt {10}}\, \sqrt {-2 x^{4}+4 x^{2}+3}}\)	\(84\)

[In]

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

[Out]

3/(-6+3*10^(1/2))^(1/2)*(1-(-2/3+1/3*10^(1/2))*x^2)^(1/2)*(1-(-2/3-1/3*10^(1/2))*x^2)^(1/2)/(-2*x^4+4*x^2+3)^(
1/2)*EllipticF(1/3*x*(-6+3*10^(1/2))^(1/2),1/3*I*6^(1/2)+1/3*I*15^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {3+4 x^2-2 x^4}} \, dx=\frac {1}{6} \, {\left (\sqrt {10} + 2\right )} \sqrt {\sqrt {10} - 2} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {10} - 2}\right )\,|\,-\frac {2}{3} \, \sqrt {10} - \frac {7}{3}) \]

[In]

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

[Out]

1/6*(sqrt(10) + 2)*sqrt(sqrt(10) - 2)*elliptic_f(arcsin(1/3*sqrt(3)*x*sqrt(sqrt(10) - 2)), -2/3*sqrt(10) - 7/3
)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(4*x^2 - 2*x^4 + 3)^(1/2),x)

[Out]

int(1/(4*x^2 - 2*x^4 + 3)^(1/2), x)