Integrand size = 17, antiderivative size = 62 \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\frac {\cot ^m(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \]
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Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2684, 3557, 371} \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\frac {\cot ^m(e+f x) \tan ^{n+1}(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m+n+1),\frac {1}{2} (-m+n+3),-\tan ^2(e+f x)\right )}{f (-m+n+1)} \]
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Rule 371
Rule 2684
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \left (\cot ^m(e+f x) \tan ^m(e+f x)\right ) \int \tan ^{-m+n}(e+f x) \, dx \\ & = \frac {\left (\cot ^m(e+f x) \tan ^m(e+f x)\right ) \text {Subst}\left (\int \frac {x^{-m+n}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cot ^m(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\frac {\cot ^m(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \]
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\[\int \left (\cot ^{m}\left (f x +e \right )\right ) \left (\tan ^{n}\left (f x +e \right )\right )d x\]
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\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \]
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\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int \tan ^{n}{\left (e + f x \right )} \cot ^{m}{\left (e + f x \right )}\, dx \]
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\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \]
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\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \]
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Timed out. \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^m\,{\mathrm {tan}\left (e+f\,x\right )}^n \,d x \]
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