Integrand size = 25, antiderivative size = 25 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx=\text {Int}\left (\frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}},x\right ) \]
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Not integrable
Time = 0.08 (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 = {} \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx=\int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 22.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx=\int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx \]
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Not integrable
Time = 0.56 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
\[\int \frac {\left (d \sec \left (f x +e \right )\right )^{n}}{\left (a +b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{n}}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 1.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx=\int \frac {\left (d \sec {\left (e + f x \right )}\right )^{n}}{\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Not integrable
Time = 1.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{n}}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 1.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (d \sec \left (f x + e\right )\right )^{n}}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 17.78 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n}{{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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