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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]