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

Optimal. Leaf size=28 \[ \frac{\sqrt{x^2+1} \tan ^{-1}(x)}{\sqrt{2} \sqrt{-x^2-1}} \]

[Out]

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

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Rubi [A]  time = 0.0113364, antiderivative size = 28, 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 \[ \frac{\sqrt{x^2+1} \tan ^{-1}(x)}{\sqrt{2} \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.12783, size = 24, normalized size = 0.86 \[ \frac{\sqrt{2 x^{2} + 2} \operatorname{atan}{\left (x \right )}}{2 \sqrt{- x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2*x**2 + 2)*atan(x)/(2*sqrt(-x**2 - 1))

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Mathematica [A]  time = 0.0168138, size = 26, normalized size = 0.93 \[ \frac{\left (x^2+1\right ) \tan ^{-1}(x)}{\sqrt{2} \sqrt{-\left (x^2+1\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 24, normalized size = 0.9 \[ -{\frac{\arctan \left ( x \right ) \sqrt{2}}{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-x^2-1)^(1/2)*2^(1/2)/(x^2+1)^(1/2)*arctan(x)

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Maxima [A]  time = 1.52956, size = 9, normalized size = 0.32 \[ -\frac{1}{2} i \, \sqrt{2} \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*I*sqrt(2)*arctan(x)

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Fricas [A]  time = 0.233575, size = 136, normalized size = 4.86 \[ \frac{1}{8} \, \sqrt{2}{\left (\log \left (\frac{2 \, \sqrt{2}{\left (x^{4} + \sqrt{2} \sqrt{2 \, x^{2} + 2} \sqrt{-x^{2} - 1} x - 1\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \log \left (-\frac{2 \, \sqrt{2}{\left (x^{4} - \sqrt{2} \sqrt{2 \, x^{2} + 2} \sqrt{-x^{2} - 1} x - 1\right )}}{x^{4} + 2 \, x^{2} + 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{2} \int \frac{1}{\sqrt{- x^{2} - 1} \sqrt{x^{2} + 1}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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