Integrand size = 10, antiderivative size = 14 \[ \int \sqrt {a \sin ^2(x)} \, dx=-\cot (x) \sqrt {a \sin ^2(x)} \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3286, 2718} \[ \int \sqrt {a \sin ^2(x)} \, dx=-\cot (x) \sqrt {a \sin ^2(x)} \]
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Rule 2718
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \left (\csc (x) \sqrt {a \sin ^2(x)}\right ) \int \sin (x) \, dx \\ & = -\cot (x) \sqrt {a \sin ^2(x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \sin ^2(x)} \, dx=-\cot (x) \sqrt {a \sin ^2(x)} \]
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Time = 0.49 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {a \cos \left (x \right ) \sin \left (x \right )}{\sqrt {a \left (\sin ^{2}\left (x \right )\right )}}\) | \(16\) |
risch | \(-\frac {i \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, {\mathrm e}^{2 i x}}{2 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{2 \left ({\mathrm e}^{2 i x}-1\right )}\) | \(69\) |
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none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \sqrt {a \sin ^2(x)} \, dx=-\frac {\sqrt {-a \cos \left (x\right )^{2} + a} \cos \left (x\right )}{\sin \left (x\right )} \]
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Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21 \[ \int \sqrt {a \sin ^2(x)} \, dx=- \frac {\sqrt {a \sin ^{2}{\left (x \right )}} \cos {\left (x \right )}}{\sin {\left (x \right )}} \]
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none
Time = 0.40 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \sqrt {a \sin ^2(x)} \, dx=-\frac {\sqrt {a}}{\sqrt {\tan \left (x\right )^{2} + 1}} \]
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none
Time = 0.34 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21 \[ \int \sqrt {a \sin ^2(x)} \, dx=-{\left (\cos \left (x\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \mathrm {sgn}\left (\sin \left (x\right )\right )\right )} \sqrt {a} \]
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Time = 13.75 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.86 \[ \int \sqrt {a \sin ^2(x)} \, dx=-\frac {\sqrt {2}\,\sqrt {a}\,\sqrt {2\,{\sin \left (x\right )}^2}\,\left (-{\sin \left (x\right )}^2+\frac {\sin \left (2\,x\right )\,1{}\mathrm {i}}{2}+1\right )}{{\sin \left (x\right )}^2\,2{}\mathrm {i}+\sin \left (2\,x\right )} \]
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