Integrand size = 21, antiderivative size = 21 \[ \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx=\text {Int}\left (\cos ^2(e+f x) (a+b \sec (e+f x))^m,x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 21, 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 = {} \[ \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx=\int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx \\ \end{align*}
Not integrable
Time = 14.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx=\int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx \]
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Not integrable
Time = 1.68 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
\[\int \cos \left (f x +e \right )^{2} \left (a +b \sec \left (f x +e \right )\right )^{m}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]
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Not integrable
Time = 37.77 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx=\int \left (a + b \sec {\left (e + f x \right )}\right )^{m} \cos ^{2}{\left (e + f x \right )}\, dx \]
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Not integrable
Time = 2.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]
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Not integrable
Time = 0.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2} \,d x } \]
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Not integrable
Time = 14.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]
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