\(\int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx\) [27]

Optimal result

Integrand size = 16, antiderivative size = 45 \[ \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx=\sqrt {\frac {2}{-7+\sqrt {73}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right ) \]

[Out]

EllipticF(2*x/(7+73^(1/2))^(1/2),7/12*I*6^(1/2)+1/12*I*438^(1/2))*2^(1/2)/(-7+73^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.04 (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} \[ \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx=\sqrt {\frac {2}{\sqrt {73}-7}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right ) \]

[In]

Int[1/Sqrt[3 + 7*x^2 - 2*x^4],x]

[Out]

Sqrt[2/(-7 + Sqrt[73])]*EllipticF[ArcSin[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/12]

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 {7+\sqrt {73}-4 x^2} \sqrt {-7+\sqrt {73}+4 x^2}} \, dx \\ & = \sqrt {\frac {2}{-7+\sqrt {73}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right ) \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx=-i \sqrt {\frac {2}{7+\sqrt {73}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {-7+\sqrt {73}}}\right ),\frac {1}{12} \left (-61+7 \sqrt {73}\right )\right ) \]

[In]

Integrate[1/Sqrt[3 + 7*x^2 - 2*x^4],x]

[Out]

(-I)*Sqrt[2/(7 + Sqrt[73])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-7 + Sqrt[73]]], (-61 + 7*Sqrt[73])/12]

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. \(2 (35 ) = 70\).

Time = 0.38 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.87

method	result	size
default	\(\frac {6 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-42+6 \sqrt {73}}}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{\sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}}\)	\(84\)
elliptic	\(\frac {6 \sqrt {1-\left (-\frac {7}{6}+\frac {\sqrt {73}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-42+6 \sqrt {73}}}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{\sqrt {-42+6 \sqrt {73}}\, \sqrt {-2 x^{4}+7 x^{2}+3}}\)	\(84\)

[In]

int(1/(-2*x^4+7*x^2+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

6/(-42+6*73^(1/2))^(1/2)*(1-(-7/6+1/6*73^(1/2))*x^2)^(1/2)*(1-(-7/6-1/6*73^(1/2))*x^2)^(1/2)/(-2*x^4+7*x^2+3)^
(1/2)*EllipticF(1/6*x*(-42+6*73^(1/2))^(1/2),7/12*I*6^(1/2)+1/12*I*438^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx=\frac {1}{72} \, {\left (\sqrt {73} \sqrt {6} \sqrt {3} + 7 \, \sqrt {6} \sqrt {3}\right )} \sqrt {\sqrt {73} - 7} F(\arcsin \left (\frac {1}{6} \, \sqrt {6} x \sqrt {\sqrt {73} - 7}\right )\,|\,-\frac {7}{12} \, \sqrt {73} - \frac {61}{12}) \]

[In]

integrate(1/(-2*x^4+7*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

1/72*(sqrt(73)*sqrt(6)*sqrt(3) + 7*sqrt(6)*sqrt(3))*sqrt(sqrt(73) - 7)*elliptic_f(arcsin(1/6*sqrt(6)*x*sqrt(sq
rt(73) - 7)), -7/12*sqrt(73) - 61/12)

Sympy [F]

\[ \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx=\int \frac {1}{\sqrt {- 2 x^{4} + 7 x^{2} + 3}}\, dx \]

[In]

integrate(1/(-2*x**4+7*x**2+3)**(1/2),x)

[Out]

Integral(1/sqrt(-2*x**4 + 7*x**2 + 3), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx=\int { \frac {1}{\sqrt {-2 \, x^{4} + 7 \, x^{2} + 3}} \,d x } \]

[In]

integrate(1/(-2*x^4+7*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-2*x^4 + 7*x^2 + 3), x)

Giac [F]

\[ \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx=\int { \frac {1}{\sqrt {-2 \, x^{4} + 7 \, x^{2} + 3}} \,d x } \]

[In]

integrate(1/(-2*x^4+7*x^2+3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-2*x^4 + 7*x^2 + 3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {3+7 x^2-2 x^4}} \, dx=\int \frac {1}{\sqrt {-2\,x^4+7\,x^2+3}} \,d x \]

[In]

int(1/(7*x^2 - 2*x^4 + 3)^(1/2),x)

[Out]

int(1/(7*x^2 - 2*x^4 + 3)^(1/2), x)