Integrand size = 10, antiderivative size = 14 \[ \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]
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Time = 0.02 (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 = {4207, 197} \[ \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]
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Rule 197
Rule 4207
Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]
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Time = 0.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57
method | result | size |
default | \(\frac {\sin \left (x \right ) \sqrt {4}}{2 \sqrt {a \csc \left (x \right )^{2}}\, \left (\cos \left (x \right )-1\right )}\) | \(22\) |
risch | \(-\frac {i {\mathrm e}^{2 i x}}{2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i}{2 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\) | \(69\) |
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none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx=-\frac {\sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \cos \left (x\right ) \sin \left (x\right )}{a} \]
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Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx=- \frac {\cot {\left (x \right )}}{\sqrt {a \csc ^{2}{\left (x \right )}}} \]
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx=-\frac {1}{\sqrt {\tan \left (x\right )^{2} + 1} \sqrt {a}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.43 \[ \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx=\frac {2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{\sqrt {a}} + \frac {2}{\sqrt {a} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )} \]
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Time = 16.95 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx=-\frac {\sin \left (2\,x\right )}{2\,\sqrt {a}\,\sqrt {{\sin \left (x\right )}^2}} \]
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