\(\int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+x^2}} \, dx\) [245]

Optimal result

Integrand size = 21, antiderivative size = 32 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right )}{\sqrt {2} \sqrt {-1+x^2}} \]

[Out]

1/2*EllipticF(x,1/2*6^(1/2))*(-x^2+1)^(1/2)*2^(1/2)/(x^2-1)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {432, 430} \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\arcsin (x),\frac {3}{2}\right )}{\sqrt {2} \sqrt {x^2-1}} \]

[In]

Int[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 + x^2]),x]

[Out]

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

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 432

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

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{\sqrt {3} \sqrt {-1+x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {F\left (x , \frac {\sqrt {6}}{2}\right ) \sqrt {-x^{2}+1}\, \sqrt {2}}{2 \sqrt {x^{2}-1}}\)	\(29\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (x^{2}-1\right )}\, \sqrt {-x^{2}+1}\, \sqrt {-6 x^{2}+4}\, F\left (x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {-3 x^{2}+2}\, \sqrt {x^{2}-1}\, \sqrt {-3 x^{4}+5 x^{2}-2}}\)	\(74\)

[In]

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

[Out]

1/2*EllipticF(x,1/2*6^(1/2))*(-x^2+1)^(1/2)*2^(1/2)/(x^2-1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+x^2}} \, dx=-\frac {1}{2} \, \sqrt {-2} F(\arcsin \left (x\right )\,|\,\frac {3}{2}) \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.68 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+x^2}} \, dx=\begin {cases} - \frac {\sqrt {3} i F\left (\operatorname {asin}{\left (\frac {\sqrt {6} x}{2} \right )}\middle | \frac {2}{3}\right )}{3} & \text {for}\: x > - \frac {\sqrt {6}}{3} \wedge x < \frac {\sqrt {6}}{3} \end {cases} \]

[In]

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

[Out]

Piecewise((-sqrt(3)*I*elliptic_f(asin(sqrt(6)*x/2), 2/3)/3, (x > -sqrt(6)/3) & (x < sqrt(6)/3)))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/((x^2 - 1)^(1/2)*(2 - 3*x^2)^(1/2)),x)

[Out]

int(1/((x^2 - 1)^(1/2)*(2 - 3*x^2)^(1/2)), x)