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

Optimal. Leaf size=6 \[ \sinh ^{-1}\left (\frac {x}{2}\right ) \]

[Out]

arcsinh(1/2*x)

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Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {215} \[ \sinh ^{-1}\left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4 + x^2],x]

[Out]

ArcSinh[x/2]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {4+x^2}} \, dx &=\sinh ^{-1}\left (\frac {x}{2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 1.00 \[ \sinh ^{-1}\left (\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4 + x^2],x]

[Out]

ArcSinh[x/2]

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fricas [B]  time = 0.42, size = 14, normalized size = 2.33 \[ -\log \left (-x + \sqrt {x^{2} + 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+4)^(1/2),x, algorithm="fricas")

[Out]

-log(-x + sqrt(x^2 + 4))

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giac [B]  time = 1.07, size = 14, normalized size = 2.33 \[ -\log \left (-x + \sqrt {x^{2} + 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+4)^(1/2),x, algorithm="giac")

[Out]

-log(-x + sqrt(x^2 + 4))

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maple [A]  time = 0.00, size = 5, normalized size = 0.83 \[ \arcsinh \left (\frac {x}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+4)^(1/2),x)

[Out]

arcsinh(1/2*x)

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maxima [A]  time = 0.97, size = 4, normalized size = 0.67 \[ \operatorname {arsinh}\left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+4)^(1/2),x, algorithm="maxima")

[Out]

arcsinh(1/2*x)

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mupad [B]  time = 0.03, size = 4, normalized size = 0.67 \[ \mathrm {asinh}\left (\frac {x}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 + 4)^(1/2),x)

[Out]

asinh(x/2)

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sympy [A]  time = 0.14, size = 3, normalized size = 0.50 \[ \operatorname {asinh}{\left (\frac {x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+4)**(1/2),x)

[Out]

asinh(x/2)

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