Integrand size = 25, antiderivative size = 101 \[ \int (d \cos (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {(d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (1-m+n p)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+n p),\frac {1}{2} (1-m+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)} \]
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Time = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3740, 2683, 2697} \[ \int (d \cos (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {\tan (e+f x) (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (-m+n p+1)} \left (b (c \tan (e+f x))^n\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (n p+1),\frac {1}{2} (-m+n p+1),\frac {1}{2} (n p+3),\sin ^2(e+f x)\right )}{f (n p+1)} \]
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Rule 2683
Rule 2697
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (d \cos (e+f x))^m (c \tan (e+f x))^{n p} \, dx \\ & = \left ((d \cos (e+f x))^m \left (\frac {\sec (e+f x)}{d}\right )^m (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \left (\frac {\sec (e+f x)}{d}\right )^{-m} (c \tan (e+f x))^{n p} \, dx \\ & = \frac {(d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (1-m+n p)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+n p),\frac {1}{2} (1-m+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.90 \[ \int (d \cos (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {(d \cos (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{m/2} \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)} \]
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\[\int \left (d \cos \left (f x +e \right )\right )^{m} \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}d x\]
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\[ \int (d \cos (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (d \cos (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \left (d \cos {\left (e + f x \right )}\right )^{m}\, dx \]
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\[ \int (d \cos (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (d \cos (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (d \cos (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int {\left (d\,\cos \left (e+f\,x\right )\right )}^m\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]
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