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

Optimal. Leaf size=40 \[ -\frac{1}{2} \log \left (x^2+1\right )+\tanh ^{-1}\left (\frac{x}{\sqrt{2 x^2+1}}\right )-\sqrt{2} \sinh ^{-1}\left (\sqrt{2} x\right ) \]

[Out]

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

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Rubi [A]  time = 0.090177, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{1}{2} \log \left (x^2+1\right )+\tanh ^{-1}\left (\frac{x}{\sqrt{2 x^2+1}}\right )-\sqrt{2} \sinh ^{-1}\left (\sqrt{2} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x - sqrt(2*x**2 + 1)), x)

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Mathematica [A]  time = 0.0466097, size = 74, normalized size = 1.85 \[ \frac{1}{4} \left (-2 \log \left (x^2+1\right )-\log \left (3 x^2-2 \sqrt{2 x^2+1} x+1\right )+\log \left (3 x^2+2 \sqrt{2 x^2+1} x+1\right )-4 \sqrt{2} \sinh ^{-1}\left (\sqrt{2} x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-4*Sqrt[2]*ArcSinh[Sqrt[2]*x] - 2*Log[1 + x^2] - Log[1 + 3*x^2 - 2*x*Sqrt[1 + 2
*x^2]] + Log[1 + 3*x^2 + 2*x*Sqrt[1 + 2*x^2]])/4

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Maple [A]  time = 0.012, size = 33, normalized size = 0.8 \[{\it Artanh} \left ({x{\frac{1}{\sqrt{2\,{x}^{2}+1}}}} \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-{\it Arcsinh} \left ( \sqrt{2}x \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-(2*x^2+1)^(1/2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x - sqrt(2*x^2 + 1)), x)

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Fricas [A]  time = 0.276397, size = 159, normalized size = 3.98 \[ \sqrt{2} \log \left (-\frac{2 \, x^{2} - \sqrt{2 \, x^{2} + 1}{\left (\sqrt{2} x + 1\right )} + \sqrt{2} x + 1}{\sqrt{2 \, x^{2} + 1} - 1}\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) - \frac{1}{2} \, \log \left (\frac{2 \, x^{2} - \sqrt{2 \, x^{2} + 1}{\left (x + 1\right )} + x + 1}{x^{2}}\right ) + \frac{1}{2} \, \log \left (\frac{2 \, x^{2} + \sqrt{2 \, x^{2} + 1}{\left (x - 1\right )} - x + 1}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.550336, size = 27, normalized size = 0.68 \[ - \log{\left (x - \sqrt{2 x^{2} + 1} \right )} - \sqrt{2} \operatorname{asinh}{\left (\sqrt{2} x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268594, size = 119, normalized size = 2.98 \[ \sqrt{2}{\rm ln}\left (-\sqrt{2} x + \sqrt{2 \, x^{2} + 1}\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) - \frac{1}{2} \,{\rm ln}\left (\frac{{\left (\sqrt{2} x - \sqrt{2 \, x^{2} + 1}\right )}^{2} - 2 \, \sqrt{2} + 3}{{\left (\sqrt{2} x - \sqrt{2 \, x^{2} + 1}\right )}^{2} + 2 \, \sqrt{2} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*ln(-sqrt(2)*x + sqrt(2*x^2 + 1)) - 1/2*ln(x^2 + 1) - 1/2*ln(((sqrt(2)*x
- sqrt(2*x^2 + 1))^2 - 2*sqrt(2) + 3)/((sqrt(2)*x - sqrt(2*x^2 + 1))^2 + 2*sqrt(
2) + 3))