Integrand size = 26, antiderivative size = 25 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}} \]
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Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3255, 3286, 2686, 8} \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}} \]
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Rule 8
Rule 2686
Rule 3255
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^2(e+f x)}{\sqrt {a \cos ^2(e+f x)}} \, dx \\ & = \frac {\cos (e+f x) \int \cot (e+f x) \csc (e+f x) \, dx}{\sqrt {a \cos ^2(e+f x)}} \\ & = -\frac {\cos (e+f x) \text {Subst}(\int 1 \, dx,x,\csc (e+f x))}{f \sqrt {a \cos ^2(e+f x)}} \\ & = -\frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}} \]
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Time = 0.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28
method | result | size |
default | \(-\frac {\cos \left (f x +e \right )}{\sin \left (f x +e \right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(32\) |
risch | \(-\frac {2 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\) | \(57\) |
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none
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{a f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \]
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\[ \int \frac {\cot ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\int \frac {\cot ^{2}{\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (23) = 46\).
Time = 0.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.60 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {2 \, {\left (\cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) - \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + \sin \left (f x + e\right )\right )} \sqrt {a}}{{\left (a \cos \left (2 \, f x + 2 \, e\right )^{2} + a \sin \left (2 \, f x + 2 \, e\right )^{2} - 2 \, a \cos \left (2 \, f x + 2 \, e\right ) + a\right )} f} \]
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Exception generated. \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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Time = 13.61 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx=-\frac {\sqrt {2\,a\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )}}{a\,f\,\sin \left (2\,e+2\,f\,x\right )} \]
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