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 = {272, 65, 212}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a^2-x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a^2-x} x} \, dx,x,x^2\right )\\ &=-\text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(23)=46\).
time = 0.04, size = 53, normalized size = 2.30 \begin {gather*} -\frac {\log \left (a+\sqrt {a^2-x^2}\right )}{2 a}+\frac {\log \left (-a^2+a \sqrt {a^2-x^2}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, 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 = 2.03, size = 34, normalized size = 1.48 \begin {gather*} -\frac {\log \left (\frac {2 \, a^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {a^{2} - x^{2}} a}{{\left | x \right |}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 25, normalized size = 1.09 \begin {gather*} \frac {\log \left (-\frac {a - \sqrt {a^{2} - x^{2}}}{x}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.47, size = 22, normalized size = 0.96 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (21) = 42\).
time = 0.71, size = 43, normalized size = 1.87 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {\sqrt {a^2-x^2}}{a}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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