Integrand size = 19, antiderivative size = 63 \[ \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx=\frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2} (-3+m),-\frac {1}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (-3+m)} \tan ^3(e+f x)}{3 f} \]
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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} \[ \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx=\frac {\tan ^3(e+f x) \sin ^2(e+f x)^{\frac {m-3}{2}} (b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m-3}{2},-\frac {1}{2},\cos ^2(e+f x)\right )}{3 f} \]
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Rule 2697
Rubi steps \begin{align*} \text {integral}& = \frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{2} (-3+m),-\frac {1}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (-3+m)} \tan ^3(e+f x)}{3 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(240\) vs. \(2(63)=126\).
Time = 8.73 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.81 \[ \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx=-\frac {\cos (e+f x) (b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3}{2},\cos ^2(e+f x)\right ) \sin (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-1+m)}}{f}+\frac {4 (b \csc (e+f x))^m \left (\frac {m \operatorname {Hypergeometric2F1}\left (1-m,\frac {1}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2},-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )^{-m} \tan \left (\frac {1}{2} (e+f x)\right )}{-1+m}-\frac {1}{2} \tan (e+f x)\right )}{f}+\frac {(b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (-1-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{-m/2} \tan (e+f x)}{f (1-m)} \]
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\[\int \left (b \csc \left (f x +e \right )\right )^{m} \left (\tan ^{4}\left (f x +e \right )\right )d x\]
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\[ \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4} \,d x } \]
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\[ \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx=\int \left (b \csc {\left (e + f x \right )}\right )^{m} \tan ^{4}{\left (e + f x \right )}\, dx \]
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\[ \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4} \,d x } \]
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\[ \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int (b \csc (e+f x))^m \tan ^4(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]
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