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

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

[Out]

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

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Rubi [A]  time = 0.019533, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{x^2}{2}-\frac{1}{2} \sqrt{x^2+1} x-\frac{1}{2} \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x - Sqrt[1 + x^2])^(-1),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0250543, size = 23, normalized size = 0.82 \[ \frac{1}{2} \left (-x \left (\sqrt{x^2+1}+x\right )-\sinh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 21, normalized size = 0.8 \[ -{\frac{{x}^{2}}{2}}-{\frac{{\it Arcsinh} \left ( x \right ) }{2}}-{\frac{x}{2}\sqrt{{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-(x^2+1)^(1/2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x - sqrt(x^2 + 1)), x)

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Fricas [A]  time = 0.266012, size = 88, normalized size = 3.14 \[ \frac{x^{2} +{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) - \sqrt{x^{2} + 1} x}{2 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.46705, size = 58, normalized size = 2.07 \[ - \frac{x \operatorname{asinh}{\left (x \right )}}{2 x - 2 \sqrt{x^{2} + 1}} + \frac{x}{2 x - 2 \sqrt{x^{2} + 1}} + \frac{\sqrt{x^{2} + 1} \operatorname{asinh}{\left (x \right )}}{2 x - 2 \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.266995, size = 41, normalized size = 1.46 \[ -\frac{1}{2} \, x^{2} - \frac{1}{2} \, \sqrt{x^{2} + 1} x + \frac{1}{2} \,{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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