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

Optimal. Leaf size=16 \[ -\log \left (\sqrt{1-x^2}+1\right ) \]

[Out]

-Log[1 + Sqrt[1 - x^2]]

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Rubi [A]  time = 0.0791616, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\log \left (\sqrt{1-x^2}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Log[1 + Sqrt[1 - x^2]]

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Rubi in Sympy [A]  time = 3.25225, size = 12, normalized size = 0.75 \[ - \log{\left (\sqrt{- x^{2} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(sqrt(-x**2 + 1) + 1)

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Mathematica [A]  time = 0.012734, size = 16, normalized size = 1. \[ -\log \left (\sqrt{1-x^2}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Log[1 + Sqrt[1 - x^2]]

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Maple [B]  time = 0.023, size = 59, normalized size = 3.7 \[ -\ln \left ( x \right ) -{\frac{1}{2}\sqrt{- \left ( -1+x \right ) ^{2}+2-2\,x}}-{\frac{1}{2}\sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x}}+\sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1-x^2+(-x^2+1)^(1/2)),x)

[Out]

-ln(x)-1/2*(-(-1+x)^2+2-2*x)^(1/2)-1/2*(-(1+x)^2+2+2*x)^(1/2)+(-x^2+1)^(1/2)-arc
tanh(1/(-x^2+1)^(1/2))

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Maxima [A]  time = 1.34201, size = 19, normalized size = 1.19 \[ -\log \left (\sqrt{-x^{2} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(sqrt(-x^2 + 1) + 1)

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Fricas [A]  time = 0.234352, size = 28, normalized size = 1.75 \[ -\log \left (x\right ) + \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x) + log((sqrt(-x^2 + 1) - 1)/x)

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Sympy [A]  time = 1.91802, size = 17, normalized size = 1.06 \[ - \begin{cases} \log{\left (\sqrt{- x^{2} + 1} + 1 \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Piecewise((log(sqrt(-x**2 + 1) + 1), (x > -1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.227232, size = 19, normalized size = 1.19 \[ -{\rm ln}\left (\sqrt{-x^{2} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(sqrt(-x^2 + 1) + 1)