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

Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-a^2}}\right ) \]

[Out]

ArcTanh[x/Sqrt[-a^2 + x^2]]

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Rubi [A]  time = 0.00715674, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-a^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-a^2 + x^2],x]

[Out]

ArcTanh[x/Sqrt[-a^2 + x^2]]

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Rubi in Sympy [A]  time = 0.927429, size = 12, normalized size = 0.75 \[ \operatorname{atanh}{\left (\frac{x}{\sqrt{- a^{2} + x^{2}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-a**2+x**2)**(1/2),x)

[Out]

atanh(x/sqrt(-a**2 + x**2))

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Mathematica [B]  time = 0.00492518, size = 46, normalized size = 2.88 \[ \frac{1}{2} \log \left (\frac{x}{\sqrt{x^2-a^2}}+1\right )-\frac{1}{2} \log \left (1-\frac{x}{\sqrt{x^2-a^2}}\right ) \]

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 = 0.004, size = 15, normalized size = 0.9 \[ \ln \left ( x+\sqrt{-{a}^{2}+{x}^{2}} \right ) \]

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 = 1.34544, size = 24, normalized size = 1.5 \[ \log \left (2 \, x + 2 \, \sqrt{-a^{2} + x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-a^2 + x^2),x, algorithm="maxima")

[Out]

log(2*x + 2*sqrt(-a^2 + x^2))

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Fricas [A]  time = 0.207408, size = 24, normalized size = 1.5 \[ -\log \left (-x + \sqrt{-a^{2} + x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-a^2 + x^2),x, algorithm="fricas")

[Out]

-log(-x + sqrt(-a^2 + x^2))

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Sympy [A]  time = 1.76401, size = 19, normalized size = 1.19 \[ \begin{cases} \operatorname{acosh}{\left (\frac{x}{a} \right )} & \text{for}\: \left |{\frac{x^{2}}{a^{2}}}\right | > 1 \\- i \operatorname{asin}{\left (\frac{x}{a} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-a**2+x**2)**(1/2),x)

[Out]

Piecewise((acosh(x/a), Abs(x**2/a**2) > 1), (-I*asin(x/a), True))

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GIAC/XCAS [A]  time = 0.231935, size = 26, normalized size = 1.62 \[ -{\rm ln}\left ({\left | -x + \sqrt{-a^{2} + x^{2}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-a^2 + x^2),x, algorithm="giac")

[Out]

-ln(abs(-x + sqrt(-a^2 + x^2)))