Integrand size = 13, antiderivative size = 13 \[ \int \sqrt {a-a \sin ^2(x)} \, dx=\sqrt {a \cos ^2(x)} \tan (x) \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3255, 3286, 2717} \[ \int \sqrt {a-a \sin ^2(x)} \, dx=\tan (x) \sqrt {a \cos ^2(x)} \]
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Rule 2717
Rule 3255
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \sqrt {a-a \sin ^2(x)} \, dx=\sqrt {a \cos ^2(x)} \tan (x) \]
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Time = 0.62 (sec) , antiderivative size = 15, normalized size of antiderivative = 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 | \(-\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, {\mathrm e}^{2 i x}}{2 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{2 \,{\mathrm e}^{2 i x}+2}\) | \(67\) |
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none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \sqrt {a-a \sin ^2(x)} \, dx=\frac {\sqrt {a \cos \left (x\right )^{2}} \sin \left (x\right )}{\cos \left (x\right )} \]
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\[ \int \sqrt {a-a \sin ^2(x)} \, dx=\int \sqrt {- a \sin ^{2}{\left (x \right )} + a}\, dx \]
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none
Time = 0.41 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int \sqrt {a-a \sin ^2(x)} \, dx=\sqrt {a} \sin \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (11) = 22\).
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08 \[ \int \sqrt {a-a \sin ^2(x)} \, dx=-\frac {2 \, \sqrt {a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )}{\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )} \]
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Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 3.54 \[ \int \sqrt {a-a \sin ^2(x)} \, dx=\frac {\sqrt {2}\,\sqrt {a}\,\sqrt {\cos \left (2\,x\right )+1}\,\left (\cos \left (2\,x\right )-1+\sin \left (2\,x\right )\,1{}\mathrm {i}\right )}{2\,\left (\cos \left (2\,x\right )\,1{}\mathrm {i}-\sin \left (2\,x\right )+1{}\mathrm {i}\right )} \]
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