3.3.69 \(\int \frac {1}{\sqrt {4+x^2}} \, dx\) [269]

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 = {221} \begin {gather*} \sinh ^{-1}\left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4 + x^2],x]

[Out]

ArcSinh[x/2]

Rule 221

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 = 0.01, size = 12, normalized size = 2.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {4+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4 + x^2],x]

[Out]

ArcTanh[x/Sqrt[4 + x^2]]

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Maple [A]
time = 0.05, size = 5, normalized size = 0.83

method	result	size
default	\(\arcsinh \left (\frac {x}{2}\right )\)	\(5\)
meijerg	\(\arcsinh \left (\frac {x}{2}\right )\)	\(5\)
trager	\(\ln \left (x +\sqrt {x^{2}+4}\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsinh(1/2*x)

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Maxima [A]
time = 3.49, size = 4, normalized size = 0.67 \begin {gather*} \operatorname {arsinh}\left (\frac {1}{2} \, x\right ) \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (4) = 8\).
time = 1.03, size = 14, normalized size = 2.33 \begin {gather*} -\log \left (-x + \sqrt {x^{2} + 4}\right ) \end {gather*}

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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Sympy [A]
time = 0.05, size = 3, normalized size = 0.50 \begin {gather*} \operatorname {asinh}{\left (\frac {x}{2} \right )} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (4) = 8\).
time = 0.44, size = 25, normalized size = 4.17 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 4} x - 2 \, \log \left (-x + \sqrt {x^{2} + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 4, normalized size = 0.67 \begin {gather*} \mathrm {asinh}\left (\frac {x}{2}\right ) \end {gather*}

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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