Integrand size = 19, antiderivative size = 63 \[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=-\frac {\cos ^2(e+f x)^{\frac {1}{2} (-3+m)} \cot ^3(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2} (-3+m),-\frac {1}{2},\sin ^2(e+f x)\right ) (b \sec (e+f x))^m}{3 f} \]
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Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697} \[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=-\frac {\cot ^3(e+f x) \cos ^2(e+f x)^{\frac {m-3}{2}} (b \sec (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m-3}{2},-\frac {1}{2},\sin ^2(e+f x)\right )}{3 f} \]
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Rule 2697
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^2(e+f x)^{\frac {1}{2} (-3+m)} \cot ^3(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2} (-3+m),-\frac {1}{2},\sin ^2(e+f x)\right ) (b \sec (e+f x))^m}{3 f} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m \sqrt {-\tan ^2(e+f x)}}{f m} \]
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\[\int \left (\cot ^{4}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{m}d x\]
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\[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \]
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\[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot ^{4}{\left (e + f x \right )}\, dx \]
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\[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \]
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\[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]
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