Optimal. Leaf size=22 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {-a^2+x^2}}{a}\right )}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, 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, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
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}{x \sqrt {-a^2+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 {\tan ^{-1}\left (\frac {\sqrt {-a^2+x^2}}{a}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {-a^2+x^2}}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 41, normalized size = 1.86
method | result | size |
default | \(-\frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}}{x}\right )}{\sqrt {-a^{2}}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.41, size = 12, normalized size = 0.55 \begin {gather*} -\frac {\arcsin \left (\frac {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 = 1.09, size = 26, normalized size = 1.18 \begin {gather*} \frac {2 \, \arctan \left (-\frac {x - \sqrt {-a^{2} + x^{2}}}{a}\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 = 0.45, size = 22, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {a}{x} \right )}}{a} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {\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 [A]
time = 0.79, size = 20, normalized size = 0.91 \begin {gather*} \frac {\arctan \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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Mupad [B]
time = 0.26, size = 24, normalized size = 1.09 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {x^2-a^2}}{\sqrt {a^2}}\right )}{\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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