Integrand size = 17, antiderivative size = 40 \[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \]
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Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2686, 371} \[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=-\frac {(b \sec (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {m+2}{2},\sec ^2(e+f x)\right )}{f m} \]
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Rule 371
Rule 2686
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {(b x)^{-1+m}}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \]
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\[\int \cot \left (f x +e \right ) \left (b \sec \left (f x +e \right )\right )^{m}d x\]
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\[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right ) \,d x } \]
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\[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=\int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot {\left (e + f x \right )}\, dx \]
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\[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right ) \,d x } \]
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\[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right ) \,d x } \]
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Timed out. \[ \int \cot (e+f x) (b \sec (e+f x))^m \, dx=\int \mathrm {cot}\left (e+f\,x\right )\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]
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