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

Optimal. Leaf size=42 \[ \frac{\sqrt{2 x^2+1}}{2 x}+x-\frac{1}{2 x}-\frac{\sinh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.222706, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\sqrt{2 x^2+1}}{2 x}+x-\frac{1}{2 x}-\frac{\sinh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.87343, size = 49, normalized size = 1.17 \[ x - \frac{\sqrt{2} \log{\left (\sqrt{2} x + \sqrt{2 x^{2} + 1} \right )}}{2} - \frac{\sqrt{2}}{\sqrt{2} x + \sqrt{2 x^{2} + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0227639, size = 42, normalized size = 1. \[ \frac{\sqrt{2 x^2+1}}{2 x}+x-\frac{1}{2 x}-\frac{\sinh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 45, normalized size = 1.1 \[ x-{\frac{1}{2\,x}}+{\frac{1}{2\,x} \left ( 2\,{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-x\sqrt{2\,{x}^{2}+1}-{\frac{{\it Arcsinh} \left ( \sqrt{2}x \right ) \sqrt{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x-1/2/x+1/2/x*(2*x^2+1)^(3/2)-x*(2*x^2+1)^(1/2)-1/2*arcsinh(2^(1/2)*x)*2^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.270421, size = 142, normalized size = 3.38 \[ \frac{{\left (\sqrt{2} \sqrt{2 \, x^{2} + 1} x - \sqrt{2} x\right )} \log \left (-\frac{2 \, x^{2} - \sqrt{2 \, x^{2} + 1}{\left (\sqrt{2} x + 1\right )} + \sqrt{2} x + 1}{\sqrt{2 \, x^{2} + 1} - 1}\right ) + 2 \, \sqrt{2 \, x^{2} + 1}{\left (x^{2} - 1\right )} + 2}{2 \,{\left (\sqrt{2 \, x^{2} + 1} x - x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273292, size = 77, normalized size = 1.83 \[ \frac{1}{2} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} x + \sqrt{2 \, x^{2} + 1}\right ) + x - \frac{\sqrt{2}}{{\left (\sqrt{2} x - \sqrt{2 \, x^{2} + 1}\right )}^{2} - 1} - \frac{1}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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