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

Optimal result

Integrand size = 9, antiderivative size = 6 \[ \int \frac {1}{\sqrt {4+x^2}} \, dx=\text {arcsinh}\left (\frac {x}{2}\right ) \]

[Out]

arcsinh(1/2*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {221} \[ \int \frac {1}{\sqrt {4+x^2}} \, dx=\text {arcsinh}\left (\frac {x}{2}\right ) \]

[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*} \text {integral}& = \text {arcsinh}\left (\frac {x}{2}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(16\) vs. \(2(6)=12\).

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\sqrt {4+x^2}} \, dx=-\log \left (-x+\sqrt {4+x^2}\right ) \]

[In]

Integrate[1/Sqrt[4 + x^2],x]

[Out]

-Log[-x + Sqrt[4 + x^2]]

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83

method	result	size
default	\(\operatorname {arcsinh}\left (\frac {x}{2}\right )\)	\(5\)
meijerg	\(\operatorname {arcsinh}\left (\frac {x}{2}\right )\)	\(5\)
pseudoelliptic	\(\operatorname {arctanh}\left (\frac {\sqrt {x^{2}+4}}{x}\right )\)	\(13\)
trager	\(-\ln \left (x -\sqrt {x^{2}+4}\right )\)	\(15\)

[In]

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

[Out]

arcsinh(1/2*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (4) = 8\).

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 2.33 \[ \int \frac {1}{\sqrt {4+x^2}} \, dx=-\log \left (-x + \sqrt {x^{2} + 4}\right ) \]

[In]

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

[Out]

-log(-x + sqrt(x^2 + 4))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\sqrt {4+x^2}} \, dx=\operatorname {asinh}{\left (\frac {x}{2} \right )} \]

[In]

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

[Out]

asinh(x/2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {4+x^2}} \, dx=\operatorname {arsinh}\left (\frac {1}{2} \, x\right ) \]

[In]

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

[Out]

arcsinh(1/2*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (4) = 8\).

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 4.17 \[ \int \frac {1}{\sqrt {4+x^2}} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 4} x - 2 \, \log \left (-x + \sqrt {x^{2} + 4}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {4+x^2}} \, dx=\mathrm {asinh}\left (\frac {x}{2}\right ) \]

[In]

int(1/(x^2 + 4)^(1/2),x)

[Out]

asinh(x/2)