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

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [B]  time = 0.00, size = 42, normalized size = 3.00 \[ \frac {1}{2} \log \left (\frac {x}{\sqrt {a^2+x^2}}+1\right )-\frac {1}{2} \log \left (1-\frac {x}{\sqrt {a^2+x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a^2 + x^2],x]

[Out]

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

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fricas [A]  time = 0.40, size = 16, normalized size = 1.14 \[ -\log \left (-x + \sqrt {a^{2} + x^{2}}\right ) \]

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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giac [A]  time = 0.85, size = 16, normalized size = 1.14 \[ -\log \left (-x + \sqrt {a^{2} + x^{2}}\right ) \]

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))

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \[ \ln \left (x +\sqrt {a^{2}+x^{2}}\right ) \]

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 = 0.44, size = 6, normalized size = 0.43 \[ \operatorname {arsinh}\left (\frac {x}{a}\right ) \]

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/a)

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mupad [B]  time = 0.19, size = 12, normalized size = 0.86 \[ \ln \left (x+\sqrt {a^2+x^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2 + x^2)^(1/2),x)

[Out]

log(x + (a^2 + x^2)^(1/2))

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sympy [A]  time = 1.03, size = 3, normalized size = 0.21 \[ \operatorname {asinh}{\left (\frac {x}{a} \right )} \]

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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