Optimal. Leaf size=23 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {266, 63, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a^2-x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2-x} x} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,\sqrt {a^2-x^2}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 23, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 25, normalized size = 1.09 \[ \frac {\log \left (-\frac {a - \sqrt {a^{2} - x^{2}}}{x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.85, size = 43, normalized size = 1.87 \[ -\frac {\log \left ({\left | a + \sqrt {a^{2} - x^{2}} \right |}\right )}{2 \, a} + \frac {\log \left ({\left | -a + \sqrt {a^{2} - x^{2}} \right |}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 37, normalized size = 1.61
method | result | size |
default | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-x^{2}}}{x}\right )}{\sqrt {a^{2}}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 34, normalized size = 1.48 \[ -\frac {\log \left (\frac {2 \, a^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {a^{2} - x^{2}} a}{{\left | x \right |}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 21, normalized size = 0.91 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.09, 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.
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