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

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

[Out]

ArcSinh[x]

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Rubi [A]  time = 0.0007175, antiderivative size = 2, normalized size of antiderivative = 1., 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} \[ \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + x^2],x]

[Out]

ArcSinh[x]

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{1+x^2}} \, dx &=\sinh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0033069, size = 2, normalized size = 1. \[ \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + x^2],x]

[Out]

ArcSinh[x]

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Maple [A]  time = 0.002, size = 3, normalized size = 1.5 \begin{align*}{\it Arcsinh} \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(x)

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Maxima [A]  time = 1.49112, size = 3, normalized size = 1.5 \begin{align*} \operatorname{arsinh}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(x)

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Fricas [B]  time = 1.43456, size = 35, normalized size = 17.5 \begin{align*} -\log \left (-x + \sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(x^2 + 1))

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Sympy [A]  time = 0.123759, size = 2, normalized size = 1. \begin{align*} \operatorname{asinh}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(x)

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Giac [B]  time = 1.12184, size = 19, normalized size = 9.5 \begin{align*} -\log \left (-x + \sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(x^2 + 1))