Integrand size = 25, antiderivative size = 25 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\text {Int}\left (\cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p,x\right ) \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 25, 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 = {} \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx \\ \end{align*}
Not integrable
Time = 2.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx \]
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Not integrable
Time = 0.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
\[\int \cot \left (f x +e \right )^{2} \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2} \,d x } \]
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Not integrable
Time = 73.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p} \cot ^{2}{\left (e + f x \right )}\, dx \]
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Not integrable
Time = 6.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2} \,d x } \]
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Not integrable
Time = 0.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2} \,d x } \]
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Not integrable
Time = 21.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]
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