Integrand size = 27, antiderivative size = 27 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=(d \csc (e+f x))^m \left (\frac {\sin (e+f x)}{d}\right )^m \text {Int}\left (\left (\frac {\sin (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \left ((d \csc (e+f x))^m \left (\frac {\sin (e+f x)}{d}\right )^m\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-m} \left (a+b (c \tan (e+f x))^n\right )^p \, dx \\ \end{align*}
Not integrable
Time = 5.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
\[\int \left (d \csc \left (f x +e \right )\right )^{m} \left (a +b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}d x\]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]
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Not integrable
Time = 166.56 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \left (d \csc {\left (e + f x \right )}\right )^{m} \left (a + b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p}\, dx \]
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Not integrable
Time = 7.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]
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Not integrable
Time = 1.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]
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Not integrable
Time = 12.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int (d \csc (e+f x))^m \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int {\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]
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