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

Optimal. Leaf size=26 \[ \sqrt{x^2+1}+x \log \left (x-\sqrt{x^2+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0115053, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \sqrt{x^2+1}+x \log \left (x-\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Log[x - Sqrt[1 + x^2]],x]

[Out]

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

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Rubi in Sympy [A]  time = 0.867183, size = 20, normalized size = 0.77 \[ x \log{\left (x - \sqrt{x^{2} + 1} \right )} + \sqrt{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.021139, size = 26, normalized size = 1. \[ \sqrt{x^2+1}+x \log \left (x-\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[x - Sqrt[1 + x^2]],x]

[Out]

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

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Maple [A]  time = 0.003, size = 23, normalized size = 0.9 \[ x\ln \left ( x-\sqrt{{x}^{2}+1} \right ) +\sqrt{{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x-(x^2+1)^(1/2)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ x \log \left (x - \sqrt{x^{2} + 1}\right ) - x + \arctan \left (x\right ) + \int -\frac{x}{x^{3} -{\left (x^{2} + 1\right )}^{\frac{3}{2}} + x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*log(x - sqrt(x^2 + 1)) - x + arctan(x) + integrate(-x/(x^3 - (x^2 + 1)^(3/2) +
x), x)

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Fricas [A]  time = 0.217292, size = 30, normalized size = 1.15 \[ x \log \left (x - \sqrt{x^{2} + 1}\right ) + \sqrt{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.7891, size = 20, normalized size = 0.77 \[ x \log{\left (x - \sqrt{x^{2} + 1} \right )} + \sqrt{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222063, size = 30, normalized size = 1.15 \[ x{\rm ln}\left (x - \sqrt{x^{2} + 1}\right ) + \sqrt{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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