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

Optimal. Leaf size=14 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{a^2+x^2}}\right ) \]

[Out]

ArcTanh[x/Sqrt[a^2 + x^2]]

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Rubi [A]  time = 0.0023598, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac{x}{\sqrt{a^2+x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a^2 + x^2],x]

[Out]

ArcTanh[x/Sqrt[a^2 + x^2]]

Rule 217

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [B]  time = 0.0025837, size = 42, normalized size = 3. \[ \frac{1}{2} \log \left (\frac{x}{\sqrt{a^2+x^2}}+1\right )-\frac{1}{2} \log \left (1-\frac{x}{\sqrt{a^2+x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a^2 + x^2],x]

[Out]

-Log[1 - x/Sqrt[a^2 + x^2]]/2 + Log[1 + x/Sqrt[a^2 + x^2]]/2

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \begin{align*} \ln \left ( x+\sqrt{{a}^{2}+{x}^{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x+(a^2+x^2)^(1/2))

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Maxima [A]  time = 0.918816, size = 11, normalized size = 0.79 \begin{align*} \operatorname{arsinh}\left (\frac{x}{\sqrt{a^{2}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(x/sqrt(a^2))

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Fricas [A]  time = 2.02888, size = 38, normalized size = 2.71 \begin{align*} -\log \left (-x + \sqrt{a^{2} + x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(a^2 + x^2))

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Sympy [A]  time = 0.938693, size = 3, normalized size = 0.21 \begin{align*} \operatorname{asinh}{\left (\frac{x}{a} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(x/a)

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Giac [A]  time = 1.06161, size = 22, normalized size = 1.57 \begin{align*} -\log \left (-x + \sqrt{a^{2} + x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(a^2 + x^2))