Optimal. Leaf size=13 \[ \tan (x) \sqrt {a \cos ^2(x)} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3176, 3207, 2637} \[ \tan (x) \sqrt {a \cos ^2(x)} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3176
Rule 3207
Rubi steps
\begin {align*} \int \sqrt {a-a \sin ^2(x)} \, dx &=\int \sqrt {a \cos ^2(x)} \, dx\\ &=\left (\sqrt {a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx\\ &=\sqrt {a \cos ^2(x)} \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ \tan (x) \sqrt {a \cos ^2(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 15, normalized size = 1.15 \[ \frac {\sqrt {a \cos \relax (x)^{2}} \sin \relax (x)}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 27, normalized size = 2.08 \[ -\frac {2 \, \sqrt {a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )}{\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 15, normalized size = 1.15 \[ \frac {a \cos \relax (x ) \sin \relax (x )}{\sqrt {a \left (\cos ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 6, normalized size = 0.46 \[ \sqrt {a} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 46, normalized size = 3.54 \[ \frac {\sqrt {2}\,\sqrt {a}\,\sqrt {\cos \left (2\,x\right )+1}\,\left (\cos \left (2\,x\right )-1+\sin \left (2\,x\right )\,1{}\mathrm {i}\right )}{2\,\left (\cos \left (2\,x\right )\,1{}\mathrm {i}-\sin \left (2\,x\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- a \sin ^{2}{\relax (x )} + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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