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

Optimal. Leaf size=55 \[ \sqrt{1-x^2}-\sqrt{1-x^2} \log \left (\sqrt{1-x^2}+1\right )-\log \left (\sqrt{1-x^2}+1\right ) \]

[Out]

Sqrt[1 - x^2] - Log[1 + Sqrt[1 - x^2]] - Sqrt[1 - x^2]*Log[1 + Sqrt[1 - x^2]]

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Rubi [A]  time = 0.0868472, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \sqrt{1-x^2}-\sqrt{1-x^2} \log \left (\sqrt{1-x^2}+1\right )-\log \left (\sqrt{1-x^2}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - x^2] - Log[1 + Sqrt[1 - x^2]] - Sqrt[1 - x^2]*Log[1 + Sqrt[1 - x^2]]

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Rubi in Sympy [A]  time = 7.69162, size = 39, normalized size = 0.71 \[ - \sqrt{- x^{2} + 1} \log{\left (\sqrt{- x^{2} + 1} + 1 \right )} + \sqrt{- x^{2} + 1} - \log{\left (\sqrt{- x^{2} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 1)*log(sqrt(-x**2 + 1) + 1) + sqrt(-x**2 + 1) - log(sqrt(-x**2 + 1
) + 1)

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Mathematica [A]  time = 0.0158318, size = 41, normalized size = 0.75 \[ \sqrt{1-x^2}-\left (\sqrt{1-x^2}+1\right ) \log \left (\sqrt{1-x^2}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 37, normalized size = 0.7 \[ -\ln \left ( 1+\sqrt{-{x}^{2}+1} \right ) \left ( 1+\sqrt{-{x}^{2}+1} \right ) +1+\sqrt{-{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.42518, size = 49, normalized size = 0.89 \[ -{\left (\sqrt{-x^{2} + 1} + 1\right )} \log \left (\sqrt{-x^{2} + 1} + 1\right ) + \sqrt{-x^{2} + 1} + 1 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(sqrt(-x^2 + 1) + 1)*log(sqrt(-x^2 + 1) + 1) + sqrt(-x^2 + 1) + 1

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Fricas [A]  time = 0.213015, size = 47, normalized size = 0.85 \[ -{\left (\sqrt{-x^{2} + 1} + 1\right )} \log \left (\sqrt{-x^{2} + 1} + 1\right ) + \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(sqrt(-x^2 + 1) + 1)*log(sqrt(-x^2 + 1) + 1) + sqrt(-x^2 + 1)

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Sympy [A]  time = 12.9463, size = 31, normalized size = 0.56 \[ \sqrt{- x^{2} + 1} - \left (\sqrt{- x^{2} + 1} + 1\right ) \log{\left (\sqrt{- x^{2} + 1} + 1 \right )} + 1 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x**2 + 1) - (sqrt(-x**2 + 1) + 1)*log(sqrt(-x**2 + 1) + 1) + 1

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GIAC/XCAS [A]  time = 0.195137, size = 49, normalized size = 0.89 \[ -{\left (\sqrt{-x^{2} + 1} + 1\right )}{\rm ln}\left (\sqrt{-x^{2} + 1} + 1\right ) + \sqrt{-x^{2} + 1} + 1 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(sqrt(-x^2 + 1) + 1)*ln(sqrt(-x^2 + 1) + 1) + sqrt(-x^2 + 1) + 1