\(\int \frac {1}{\sqrt {-3+7 x^2-2 x^4}} \, dx\) [95]

Optimal result

Integrand size = 16, antiderivative size = 19 \[ \int \frac {1}{\sqrt {-3+7 x^2-2 x^4}} \, dx=-\frac {\operatorname {EllipticF}\left (\arccos \left (\frac {x}{\sqrt {3}}\right ),\frac {6}{5}\right )}{\sqrt {5}} \]

[Out]

-1/5*(x^2)^(1/2)/x*EllipticF(1/3*(-3*x^2+9)^(1/2),1/5*30^(1/2))*5^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, 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, 431} \[ \int \frac {1}{\sqrt {-3+7 x^2-2 x^4}} \, dx=-\frac {\operatorname {EllipticF}\left (\arccos \left (\frac {x}{\sqrt {3}}\right ),\frac {6}{5}\right )}{\sqrt {5}} \]

[In]

Int[1/Sqrt[-3 + 7*x^2 - 2*x^4],x]

[Out]

-(EllipticF[ArcCos[x/Sqrt[3]], 6/5]/Sqrt[5])

Rule 431

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(-(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a -
b*(c/d)])^(-1))*EllipticF[ArcCos[Rt[-d/c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c
] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]

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 {12-4 x^2} \sqrt {-2+4 x^2}} \, dx \\ & = -\frac {F\left (\cos ^{-1}\left (\frac {x}{\sqrt {3}}\right )|\frac {6}{5}\right )}{\sqrt {5}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(19)=38\).

Time = 10.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.05 \[ \int \frac {1}{\sqrt {-3+7 x^2-2 x^4}} \, dx=\frac {\sqrt {1-2 x^2} \sqrt {1-\frac {x^2}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),\frac {1}{6}\right )}{\sqrt {2} \sqrt {-3+7 x^2-2 x^4}} \]

[In]

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

[Out]

(Sqrt[1 - 2*x^2]*Sqrt[1 - x^2/3]*EllipticF[ArcSin[Sqrt[2]*x], 1/6])/(Sqrt[2]*Sqrt[-3 + 7*x^2 - 2*x^4])

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.53

method	result	size
default	\(\frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {-2 x^{2}+1}\, F\left (\frac {x \sqrt {3}}{3}, \sqrt {6}\right )}{3 \sqrt {-2 x^{4}+7 x^{2}-3}}\)	\(48\)
elliptic	\(\frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {-2 x^{2}+1}\, F\left (\frac {x \sqrt {3}}{3}, \sqrt {6}\right )}{3 \sqrt {-2 x^{4}+7 x^{2}-3}}\)	\(48\)

[In]

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

[Out]

1/3*3^(1/2)*(-3*x^2+9)^(1/2)*(-2*x^2+1)^(1/2)/(-2*x^4+7*x^2-3)^(1/2)*EllipticF(1/3*x*3^(1/2),6^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {-3+7 x^2-2 x^4}} \, dx=-\frac {1}{6} \, \sqrt {2} \sqrt {-3} F(\arcsin \left (\sqrt {2} x\right )\,|\,\frac {1}{6}) \]

[In]

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

[Out]

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

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)