Integrand size = 10, antiderivative size = 16 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 3852, 8} \[ \int \sqrt {a \csc ^4(x)} \, dx=\sin (x) (-\cos (x)) \sqrt {a \csc ^4(x)} \]
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Rule 8
Rule 3852
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^2(x) \, dx \\ & = -\left (\left (\sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \text {Subst}(\int 1 \, dx,x,\cot (x))\right ) \\ & = -\cos (x) \sqrt {a \csc ^4(x)} \sin (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]
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Time = 0.50 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
method | result | size |
default | \(-\frac {\cos \left (x \right ) \sin \left (x \right ) \sqrt {a \csc \left (x \right )^{4}}\, \sqrt {16}}{4}\) | \(18\) |
risch | \(2 i \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left (1-{\mathrm e}^{-2 i x}\right )\) | \(31\) |
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none
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}} \cos \left (x\right ) \sin \left (x\right ) \]
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\[ \int \sqrt {a \csc ^4(x)} \, dx=\int \sqrt {a \csc ^{4}{\left (x \right )}}\, dx \]
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none
Time = 0.32 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\frac {\sqrt {a}}{\tan \left (x\right )} \]
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none
Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\frac {\sqrt {a}}{\tan \left (x\right )} \]
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Time = 20.74 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.44 \[ \int \sqrt {a \csc ^4(x)} \, dx=-\sqrt {a}\,\mathrm {cot}\left (x\right ) \]
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