Integrand size = 24, antiderivative size = 31 \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \]
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Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3255, 3284, 65, 212} \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \]
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Rule 65
Rule 212
Rule 3255
Rule 3284
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot (e+f x)}{\sqrt {a \cos ^2(e+f x)}} \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cos ^2(e+f x)}\right )}{a f} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f} \]
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Time = 0.60 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\cos \left (f x +e \right ) \operatorname {arctanh}\left (\cos \left (f x +e \right )\right )}{\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(31\) |
risch | \(-\frac {2 \ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}+\frac {2 \ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}\) | \(104\) |
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none
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.71 \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\left [-\frac {\sqrt {a \cos \left (f x + e\right )^{2}} \log \left (-\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right )}{2 \, a f \cos \left (f x + e\right )}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a \cos \left (f x + e\right )^{2}} \sqrt {-a}}{a}\right )}{a f}\right ] \]
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\[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\int \frac {\cot {\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right )}{\sqrt {a} f} \]
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Exception generated. \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\cot (e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\int \frac {\mathrm {cot}\left (e+f\,x\right )}{\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2}} \,d x \]
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