3.3.93 \(\int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx\) [293]

Optimal. Leaf size=6 \[ -F\left (\left .\cos ^{-1}(x)\right |2\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {431} \begin {gather*} -F(\text {ArcCos}(x)|2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-EllipticF[ArcCos[x], 2]

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-x^2} \sqrt {-1+2 x^2}} \, dx &=-F\left (\left .\cos ^{-1}(x)\right |2\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(6)=12\).
time = 0.23, size = 27, normalized size = 4.50 \begin {gather*} \frac {\sqrt {1-2 x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {-1+2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 25, normalized size = 4.17

method	result	size
default	\(\frac {\EllipticF \left (x , \sqrt {2}\right ) \sqrt {-2 x^{2}+1}}{\sqrt {2 x^{2}-1}}\)	\(25\)
elliptic	\(\frac {\sqrt {-\left (2 x^{2}-1\right ) \left (x^{2}-1\right )}\, \sqrt {-2 x^{2}+1}\, \EllipticF \left (x , \sqrt {2}\right )}{\sqrt {2 x^{2}-1}\, \sqrt {-2 x^{4}+3 x^{2}-1}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {2 x^{2} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.17 \begin {gather*} \int \frac {1}{\sqrt {1-x^2}\,\sqrt {2\,x^2-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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