3.3.60 \(\int \frac {1}{\sqrt {1+x^2}} \, dx\) [260]

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

[Out]

arcsinh(x)

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Rubi [A]
time = 0.00, antiderivative size = 2, 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}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + x^2],x]

[Out]

ArcSinh[x]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(12\) vs. \(2(2)=4\).
time = 0.00, size = 12, normalized size = 6.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + x^2],x]

[Out]

ArcTanh[x/Sqrt[1 + x^2]]

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Maple [A]
time = 0.03, size = 3, normalized size = 1.50

method	result	size
default	\(\arcsinh \left (x \right )\)	\(3\)
meijerg	\(\arcsinh \left (x \right )\)	\(3\)
trager	\(\ln \left (x +\sqrt {x^{2}+1}\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(x)

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Maxima [A]
time = 4.33, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {arsinh}\left (x\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (2) = 4\).
time = 0.66, size = 14, normalized size = 7.00 \begin {gather*} -\log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}

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 = 0.04, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {asinh}{\left (x \right )} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (2) = 4\).
time = 0.45, size = 25, normalized size = 12.50 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.00, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {asinh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 + 1)^(1/2),x)

[Out]

asinh(x)

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