Integrand size = 10, antiderivative size = 26 \[ \int \sqrt {a \csc ^2(x)} \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 26, 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 = {4207, 223, 212} \[ \int \sqrt {a \csc ^2(x)} \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right ) \]
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Rule 212
Rule 223
Rule 4207
Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\cot (x)\right )\right ) \\ & = -\left (a \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\right )\right ) \\ & = -\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \sqrt {a \csc ^2(x)} \, dx=\sqrt {a \csc ^2(x)} \left (-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin (x) \]
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Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {\sqrt {a \csc \left (x \right )^{2}}\, \sin \left (x \right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) \sqrt {4}}{2}\) | \(24\) |
risch | \(-2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )+2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )\) | \(64\) |
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Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \sqrt {a \csc ^2(x)} \, dx=\left [\frac {1}{2} \, \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \sin \left (x\right ), \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \cos \left (x\right ) \sin \left (x\right )}{a}\right )\right ] \]
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\[ \int \sqrt {a \csc ^2(x)} \, dx=\int \sqrt {a \csc ^{2}{\left (x \right )}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \sqrt {a \csc ^2(x)} \, dx=-\sqrt {-a} {\left (\arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )} \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.50 \[ \int \sqrt {a \csc ^2(x)} \, dx=\sqrt {a} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
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Timed out. \[ \int \sqrt {a \csc ^2(x)} \, dx=\int \sqrt {\frac {a}{{\sin \left (x\right )}^2}} \,d x \]
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