\(\int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx\) [132]

Optimal result

Integrand size = 16, antiderivative size = 45 \[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx=\sqrt {\frac {2}{-5+\sqrt {89}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4 x}{\sqrt {5+\sqrt {89}}}\right ),\frac {1}{32} \left (-57-5 \sqrt {89}\right )\right ) \]

[Out]

EllipticF(4*x/(5+89^(1/2))^(1/2),5/8*I+1/8*I*89^(1/2))*2^(1/2)/(-5+89^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, 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} \[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx=\sqrt {\frac {2}{\sqrt {89}-5}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4 x}{\sqrt {5+\sqrt {89}}}\right ),\frac {1}{32} \left (-57-5 \sqrt {89}\right )\right ) \]

[In]

Int[1/Sqrt[2 + 5*x^2 - 8*x^4],x]

[Out]

Sqrt[2/(-5 + Sqrt[89])]*EllipticF[ArcSin[(4*x)/Sqrt[5 + Sqrt[89]]], (-57 - 5*Sqrt[89])/32]

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 (4 \sqrt {2}\right ) \int \frac {1}{\sqrt {5+\sqrt {89}-16 x^2} \sqrt {-5+\sqrt {89}+16 x^2}} \, dx \\ & = \sqrt {\frac {2}{-5+\sqrt {89}}} F\left (\sin ^{-1}\left (\frac {4 x}{\sqrt {5+\sqrt {89}}}\right )|\frac {1}{32} \left (-57-5 \sqrt {89}\right )\right ) \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx=-i \sqrt {\frac {2}{5+\sqrt {89}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {4 x}{\sqrt {-5+\sqrt {89}}}\right ),\frac {1}{32} \left (-57+5 \sqrt {89}\right )\right ) \]

[In]

Integrate[1/Sqrt[2 + 5*x^2 - 8*x^4],x]

[Out]

(-I)*Sqrt[2/(5 + Sqrt[89])]*EllipticF[I*ArcSinh[(4*x)/Sqrt[-5 + Sqrt[89]]], (-57 + 5*Sqrt[89])/32]

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (31 ) = 62\).

Time = 0.62 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.69

method	result	size
default	\(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {89}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {89}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {-5+\sqrt {89}}}{2}, \frac {5 i}{8}+\frac {i \sqrt {89}}{8}\right )}{\sqrt {-5+\sqrt {89}}\, \sqrt {-8 x^{4}+5 x^{2}+2}}\)	\(76\)
elliptic	\(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {89}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {89}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {-5+\sqrt {89}}}{2}, \frac {5 i}{8}+\frac {i \sqrt {89}}{8}\right )}{\sqrt {-5+\sqrt {89}}\, \sqrt {-8 x^{4}+5 x^{2}+2}}\)	\(76\)

[In]

int(1/(-8*x^4+5*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/(-5+89^(1/2))^(1/2)*(1-(-5/4+1/4*89^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*89^(1/2))*x^2)^(1/2)/(-8*x^4+5*x^2+2)^(1/
2)*EllipticF(1/2*x*(-5+89^(1/2))^(1/2),5/8*I+1/8*I*89^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx=\frac {1}{64} \, {\left (\sqrt {89} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {89} - 5} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {89} - 5}\right )\,|\,-\frac {5}{32} \, \sqrt {89} - \frac {57}{32}) \]

[In]

integrate(1/(-8*x^4+5*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/64*(sqrt(89)*sqrt(2) + 5*sqrt(2))*sqrt(sqrt(89) - 5)*elliptic_f(arcsin(1/2*x*sqrt(sqrt(89) - 5)), -5/32*sqrt
(89) - 57/32)

Sympy [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx=\int \frac {1}{\sqrt {- 8 x^{4} + 5 x^{2} + 2}}\, dx \]

[In]

integrate(1/(-8*x**4+5*x**2+2)**(1/2),x)

[Out]

Integral(1/sqrt(-8*x**4 + 5*x**2 + 2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx=\int { \frac {1}{\sqrt {-8 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]

[In]

integrate(1/(-8*x^4+5*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-8*x^4 + 5*x^2 + 2), x)

Giac [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx=\int { \frac {1}{\sqrt {-8 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]

[In]

integrate(1/(-8*x^4+5*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-8*x^4 + 5*x^2 + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2+5 x^2-8 x^4}} \, dx=\int \frac {1}{\sqrt {-8\,x^4+5\,x^2+2}} \,d x \]

[In]

int(1/(5*x^2 - 8*x^4 + 2)^(1/2),x)

[Out]

int(1/(5*x^2 - 8*x^4 + 2)^(1/2), x)