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

Optimal. Leaf size=27 \[ \sinh ^{-1}(x)-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt{x^2+1}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0350852, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \sinh ^{-1}(x)-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt{x^2+1}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x^2]/(2 + x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.0803, size = 27, normalized size = 1. \[ \operatorname{asinh}{\left (x \right )} - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{2 \sqrt{x^{2} + 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0193743, size = 27, normalized size = 1. \[ \sinh ^{-1}(x)-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt{x^2+1}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 23, normalized size = 0.9 \[{\it Arcsinh} \left ( x \right ) -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{x\sqrt{2}}{2}{\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)^(1/2)/(x^2+2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{x^{2} + 1} x}{x^{2} + 2} + \int \frac{\sqrt{x^{2} + 1} x^{4}}{x^{6} + 5 \, x^{4} + 8 \, x^{2} + 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 + 1)*x/(x^2 + 2) + integrate(sqrt(x^2 + 1)*x^4/(x^6 + 5*x^4 + 8*x^2 + 4
), x)

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Fricas [A]  time = 0.25256, size = 143, normalized size = 5.3 \[ -\frac{1}{4} \, \sqrt{2}{\left (2 \, \sqrt{2} \log \left (-x + \sqrt{x^{2} + 1}\right ) - \log \left (-\frac{4 \, x^{2} - \sqrt{2}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} + 2 \, \sqrt{x^{2} + 1}{\left (\sqrt{2}{\left (x^{3} + 2 \, x\right )} - 2 \, x\right )} + 8}{2 \, x^{4} + 5 \, x^{2} - 2 \,{\left (x^{3} + 2 \, x\right )} \sqrt{x^{2} + 1} + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*(2*sqrt(2)*log(-x + sqrt(x^2 + 1)) - log(-(4*x^2 - sqrt(2)*(2*x^4 +
5*x^2 + 6) + 2*sqrt(x^2 + 1)*(sqrt(2)*(x^3 + 2*x) - 2*x) + 8)/(2*x^4 + 5*x^2 -
2*(x^3 + 2*x)*sqrt(x^2 + 1) + 2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217259, size = 86, normalized size = 3.19 \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\frac{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 2 \, \sqrt{2} + 3}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} + 2 \, \sqrt{2} + 3}\right ) -{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*ln(((x - sqrt(x^2 + 1))^2 - 2*sqrt(2) + 3)/((x - sqrt(x^2 + 1))^2 +
2*sqrt(2) + 3)) - ln(-x + sqrt(x^2 + 1))