Optimal. Leaf size=16 \[ \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {217, 203} \[ \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a^2-x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {a^2-x^2}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 23, normalized size = 1.44 \[ -2 \, \arctan \left (-\frac {a - \sqrt {a^{2} - x^{2}}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.04, size = 9, normalized size = 0.56 \[ \arcsin \left (\frac {x}{a}\right ) \mathrm {sgn}\relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.94 \[ \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 6, normalized size = 0.38 \[ \arcsin \left (\frac {x}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 14, normalized size = 0.88 \[ \mathrm {atan}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 19, normalized size = 1.19 \[ \begin {cases} - i \operatorname {acosh}{\left (\frac {x}{a} \right )} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\\operatorname {asin}{\left (\frac {x}{a} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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