Optimal. Leaf size=13 \[ \sqrt {a \cos ^2(x)} \tan (x) \]
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Rubi [A]
time = 0.02, 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 = {3255, 3286,
2717} \begin {gather*} \tan (x) \sqrt {a \cos ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3255
Rule 3286
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 \begin {gather*} \sqrt {a \cos ^2(x)} \tan (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.24, size = 15, normalized size = 1.15
method | result | size |
default | \(\frac {a \cos \left (x \right ) \sin \left (x \right )}{\sqrt {a \left (\cos ^{2}\left (x \right )\right )}}\) | \(15\) |
risch | \(-\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, {\mathrm e}^{2 i x}}{2 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{2 \,{\mathrm e}^{2 i x}+2}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 6, normalized size = 0.46 \begin {gather*} \sqrt {a} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 15, normalized size = 1.15 \begin {gather*} \frac {\sqrt {a \cos \left (x\right )^{2}} \sin \left (x\right )}{\cos \left (x\right )} \end {gather*}
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 \begin {gather*} \int \sqrt {- a \sin ^{2}{\left (x \right )} + a}\, dx \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. 27 vs.
\(2 (11) = 22\).
time = 0.44, size = 27, normalized size = 2.08 \begin {gather*} -\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 )} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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