Integrand size = 21, antiderivative size = 90 \[ \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx=-\frac {(b \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),\cos ^2(e+f x)\right ) (a \sec (e+f x))^m \sin ^2(e+f x)^{\frac {1+n}{2}}}{b f (1-m+n)} \]
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Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2698, 2682, 2656} \[ \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx=-\frac {\sin ^2(e+f x)^{\frac {n+1}{2}} (a \sec (e+f x))^m (b \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {1}{2} (-m+n+1),\frac {1}{2} (-m+n+3),\cos ^2(e+f x)\right )}{b f (-m+n+1)} \]
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Rule 2656
Rule 2682
Rule 2698
Rubi steps \begin{align*} \text {integral}& = \left (\left (\frac {\cos (e+f x)}{a}\right )^m (a \sec (e+f x))^m\right ) \int \left (\frac {\cos (e+f x)}{a}\right )^{-m} (b \cot (e+f x))^n \, dx \\ & = -\frac {\left (\left (\frac {\cos (e+f x)}{a}\right )^{-1+m-n} (b \cot (e+f x))^{1+n} (a \sec (e+f x))^m (-\sin (e+f x))^{1+n}\right ) \int \left (\frac {\cos (e+f x)}{a}\right )^{-m+n} (-\sin (e+f x))^{-n} \, dx}{a b} \\ & = -\frac {(b \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),\cos ^2(e+f x)\right ) (a \sec (e+f x))^m \sin ^2(e+f x)^{\frac {1+n}{2}}}{b f (1-m+n)} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx=-\frac {b (b \cot (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}\left (1-\frac {m}{2},\frac {1-n}{2},\frac {3-n}{2},-\tan ^2(e+f x)\right ) (a \sec (e+f x))^m \sec ^2(e+f x)^{-m/2}}{f (-1+n)} \]
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\[\int \left (b \cot \left (f x +e \right )\right )^{n} \left (a \sec \left (f x +e \right )\right )^{m}d x\]
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\[ \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx=\int { \left (b \cot \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx=\int \left (a \sec {\left (e + f x \right )}\right )^{m} \left (b \cot {\left (e + f x \right )}\right )^{n}\, dx \]
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\[ \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx=\int { \left (b \cot \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx=\int { \left (b \cot \left (f x + e\right )\right )^{n} \left (a \sec \left (f x + e\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (b \cot (e+f x))^n (a \sec (e+f x))^m \, dx=\int {\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (\frac {a}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]
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