3.145 \(\int \frac{1}{\sqrt{8+4 x+x^2}} \, dx\)

Optimal. Leaf size=8 \[ \sinh ^{-1}\left (\frac{x+2}{2}\right ) \]

[Out]

ArcSinh[(2 + x)/2]

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Rubi [A]  time = 0.0129142, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \sinh ^{-1}\left (\frac{x+2}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[8 + 4*x + x^2],x]

[Out]

ArcSinh[(2 + x)/2]

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Rubi in Sympy [A]  time = 0.665103, size = 19, normalized size = 2.38 \[ \operatorname{atanh}{\left (\frac{2 x + 4}{2 \sqrt{x^{2} + 4 x + 8}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+4*x+8)**(1/2),x)

[Out]

atanh((2*x + 4)/(2*sqrt(x**2 + 4*x + 8)))

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Mathematica [A]  time = 0.00797654, size = 8, normalized size = 1. \[ \sinh ^{-1}\left (\frac{x+2}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[8 + 4*x + x^2],x]

[Out]

ArcSinh[(2 + x)/2]

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Maple [A]  time = 0.006, size = 7, normalized size = 0.9 \[{\it Arcsinh} \left ( 1+{\frac{x}{2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+4*x+8)^(1/2),x)

[Out]

arcsinh(1+1/2*x)

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Maxima [A]  time = 1.63778, size = 8, normalized size = 1. \[ \operatorname{arsinh}\left (\frac{1}{2} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + 4*x + 8),x, algorithm="maxima")

[Out]

arcsinh(1/2*x + 1)

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Fricas [A]  time = 0.201239, size = 24, normalized size = 3. \[ -\log \left (-x + \sqrt{x^{2} + 4 \, x + 8} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + 4*x + 8),x, algorithm="fricas")

[Out]

-log(-x + sqrt(x^2 + 4*x + 8) - 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 4 x + 8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+4*x+8)**(1/2),x)

[Out]

Integral(1/sqrt(x**2 + 4*x + 8), x)

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GIAC/XCAS [A]  time = 0.212927, size = 24, normalized size = 3. \[ -{\rm ln}\left (-x + \sqrt{x^{2} + 4 \, x + 8} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + 4*x + 8),x, algorithm="giac")

[Out]

-ln(-x + sqrt(x^2 + 4*x + 8) - 2)