Integrand size = 24, antiderivative size = 50 \[ \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
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Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3255, 3284, 52, 65, 212} \[ \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f} \]
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Rule 52
Rule 65
Rule 212
Rule 3255
Rule 3284
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {a \cos ^2(e+f x)} \cot (e+f x) \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {a x}}{1-x} \, dx,x,\cos ^2(e+f x)\right )}{2 f} \\ & = \frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {a \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f} \\ & = \frac {\sqrt {a \cos ^2(e+f x)}}{f}-\frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cos ^2(e+f x)}\right )}{f} \\ & = -\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a \cos ^2(e+f x)}}{f} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a \cos ^2(e+f x)}}{f} \]
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Time = 0.83 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {a \cos \left (f x +e \right ) \left (2 \cos \left (f x +e \right )+\ln \left (\cos \left (f x +e \right )-1\right )-\ln \left (1+\cos \left (f x +e \right )\right )\right )}{2 \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(54\) |
risch | \(\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{2 i \left (f x +e \right )}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {\ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(232\) |
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Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.14 \[ \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (2 \, \cos \left (f x + e\right ) - \log \left (-\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right )\right )}}{2 \, f \cos \left (f x + e\right )} \]
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\[ \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \cot {\left (e + f x \right )}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.40 \[ \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {\sqrt {a} \log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right ) - \sqrt {-a \sin \left (f x + e\right )^{2} + a}}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (42) = 84\).
Time = 0.56 (sec) , antiderivative size = 524, normalized size of antiderivative = 10.48 \[ \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \cot (e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\int \mathrm {cot}\left (e+f\,x\right )\,\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2} \,d x \]
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