3.221 \(\int \frac{1}{\sqrt{2-2 x^2} \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=8 \[ \frac{\tanh ^{-1}(x)}{\sqrt{2}} \]

[Out]

ArcTanh[x]/Sqrt[2]

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Rubi [A]  time = 0.00771095, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\tanh ^{-1}(x)}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[x]/Sqrt[2]

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Rubi in Sympy [A]  time = 2.77489, size = 8, normalized size = 1. \[ \frac{\sqrt{2} \operatorname{atanh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-2*x**2+2)**(1/2)/(-x**2+1)**(1/2),x)

[Out]

sqrt(2)*atanh(x)/2

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Mathematica [B]  time = 0.00920111, size = 26, normalized size = 3.25 \[ -\frac{\frac{1}{2} \log (1-x)-\frac{1}{2} \log (x+1)}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-((Log[1 - x]/2 - Log[1 + x]/2)/Sqrt[2])

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Maple [A]  time = 0.052, size = 8, normalized size = 1. \[{\frac{{\it Artanh} \left ( x \right ) \sqrt{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^2+2)^(1/2)/(-x^2+1)^(1/2),x)

[Out]

1/2*arctanh(x)*2^(1/2)

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Maxima [A]  time = 1.54241, size = 22, normalized size = 2.75 \[ \frac{1}{4} \, \sqrt{2}{\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*(log(x + 1) - log(x - 1))

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Fricas [A]  time = 0.249576, size = 95, normalized size = 11.88 \[ \frac{1}{8} \, \sqrt{2} \log \left (\frac{4 \,{\left (x^{3} + x\right )} \sqrt{-x^{2} + 1} \sqrt{-2 \, x^{2} + 2} - \sqrt{2}{\left (x^{6} + 5 \, x^{4} - 5 \, x^{2} - 1\right )}}{x^{6} - 3 \, x^{4} + 3 \, x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*log((4*(x^3 + x)*sqrt(-x^2 + 1)*sqrt(-2*x^2 + 2) - sqrt(2)*(x^6 + 5*
x^4 - 5*x^2 - 1))/(x^6 - 3*x^4 + 3*x^2 - 1))

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Sympy [A]  time = 8.93193, size = 22, normalized size = 2.75 \[ - \sqrt{2} \left (\begin{cases} - \frac{\operatorname{acoth}{\left (x \right )}}{2} & \text{for}\: x^{2} > 1 \\- \frac{\operatorname{atanh}{\left (x \right )}}{2} & \text{for}\: x^{2} < 1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-2*x**2+2)**(1/2)/(-x**2+1)**(1/2),x)

[Out]

-sqrt(2)*Piecewise((-acoth(x)/2, x**2 > 1), (-atanh(x)/2, x**2 < 1))

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GIAC/XCAS [A]  time = 0.265517, size = 28, normalized size = 3.5 \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*ln(abs(x + 1)) - 1/4*sqrt(2)*ln(abs(x - 1))