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

Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2-x^2}}{a}\right )}{a} \]

[Out]

-(ArcTanh[Sqrt[a^2 - x^2]/a]/a)

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Rubi [A]  time = 0.0391589, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2-x^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a^2 - x^2]),x]

[Out]

-(ArcTanh[Sqrt[a^2 - x^2]/a]/a)

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Rubi in Sympy [A]  time = 3.12027, size = 15, normalized size = 0.65 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a^{2} - x^{2}}}{a} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(a**2 - x**2)/a)/a

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Mathematica [A]  time = 0.0141429, size = 32, normalized size = 1.39 \[ \frac{\log (x)}{a}-\frac{\log \left (a \sqrt{a^2-x^2}+a^2\right )}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[a^2 - x^2]),x]

[Out]

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

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Maple [A]  time = 0.006, size = 37, normalized size = 1.6 \[ -{1\ln \left ({\frac{1}{x} \left ( 2\,{a}^{2}+2\,\sqrt{{a}^{2}}\sqrt{{a}^{2}-{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a^2-x^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227966, size = 34, normalized size = 1.48 \[ \frac{\log \left (-\frac{a - \sqrt{a^{2} - x^{2}}}{x}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(-(a - sqrt(a^2 - x^2))/x)/a

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Sympy [A]  time = 1.77494, size = 22, normalized size = 0.96 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{a}{x} \right )}}{a} & \text{for}\: \left |{\frac{a^{2}}{x^{2}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{a}{x} \right )}}{a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209626, size = 58, normalized size = 2.52 \[ -\frac{{\rm ln}\left ({\left | a + \sqrt{a^{2} - x^{2}} \right |}\right )}{2 \, a} + \frac{{\rm ln}\left ({\left | -a + \sqrt{a^{2} - x^{2}} \right |}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*ln(abs(a + sqrt(a^2 - x^2)))/a + 1/2*ln(abs(-a + sqrt(a^2 - x^2)))/a