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

Optimal. Leaf size=25 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0270475, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.09703, size = 22, normalized size = 0.88 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- x^{2} + 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0397045, size = 25, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 28, normalized size = 1.1 \[ -{\frac{\sqrt{2}}{2}\arctan \left ({\frac{x\sqrt{2}}{{x}^{2}-1}\sqrt{-{x}^{2}+1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+1)/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 1\right )} \sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.212849, size = 65, normalized size = 2.6 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x^{2} - 1\right )} + \sqrt{2} \sqrt{-x^{2} + 1}}{2 \,{\left (\sqrt{-x^{2} + 1} x - x\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.236704, size = 69, normalized size = 2.76 \[ \frac{1}{4} \, \sqrt{2}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{2} x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*(pi*sign(x) + 2*arctan(-1/4*sqrt(2)*x*((sqrt(-x^2 + 1) - 1)^2/x^2 -
1)/(sqrt(-x^2 + 1) - 1)))