Integrand size = 32, antiderivative size = 148 \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx=\frac {4 i a^2 (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n}}{f \left (2+3 n+n^2\right )}+\frac {i a (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{2-n}}{f (2+n)}+\frac {8 i a^3 (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n}}{f n \left (2+3 n+n^2\right )} \]
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Time = 0.26 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3575, 3574} \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx=\frac {8 i a^3 (a+i a \tan (e+f x))^{-n} (d \sec (e+f x))^{2 n}}{f n \left (n^2+3 n+2\right )}+\frac {4 i a^2 (a+i a \tan (e+f x))^{1-n} (d \sec (e+f x))^{2 n}}{f \left (n^2+3 n+2\right )}+\frac {i a (a+i a \tan (e+f x))^{2-n} (d \sec (e+f x))^{2 n}}{f (n+2)} \]
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Rule 3574
Rule 3575
Rubi steps \begin{align*} \text {integral}& = \frac {i a (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{2-n}}{f (2+n)}+\frac {(4 a) \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{2-n} \, dx}{2+n} \\ & = \frac {4 i a^2 (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n}}{f \left (2+3 n+n^2\right )}+\frac {i a (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{2-n}}{f (2+n)}+\frac {\left (8 a^2\right ) \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n} \, dx}{2+3 n+n^2} \\ & = \frac {4 i a^2 (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{1-n}}{f \left (2+3 n+n^2\right )}+\frac {i a (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{2-n}}{f (2+n)}+\frac {8 i a^3 (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{-n}}{f n \left (2+3 n+n^2\right )} \\ \end{align*}
Time = 3.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87 \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx=\frac {i a^3 \sec ^2(e+f x) (d \sec (e+f x))^{2 n} (\cos (3 f x)+i \sin (3 f x)) \left (2 (2+n)+\left (4+3 n+n^2\right ) \cos (2 (e+f x))+i n (3+n) \sin (2 (e+f x))\right ) (a+i a \tan (e+f x))^{-n}}{f n (1+n) (2+n) (\cos (f x)+i \sin (f x))^3} \]
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\[\int \left (d \sec \left (f x +e \right )\right )^{2 n} \left (a +i a \tan \left (f x +e \right )\right )^{3-n}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.16 \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx=\frac {{\left ({\left (i \, n^{2} + 3 i \, n + 2 i\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (i \, n^{2} + 5 i \, n + 6 i\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, {\left (-i \, n - 3 i\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i\right )} \left (\frac {2 \, d e^{\left (i \, f x + i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{2 \, n} e^{\left (-i \, e n + {\left (-i \, f n + 3 i \, f\right )} x - 6 i \, f x - {\left (n - 3\right )} \log \left (\frac {2 \, d e^{\left (i \, f x + i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right ) - {\left (n - 3\right )} \log \left (\frac {a}{d}\right ) - 3 i \, e\right )}}{2 \, {\left (f n^{3} + 3 \, f n^{2} + 2 \, f n\right )}} \]
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\[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{2 n} \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{3 - n}\, dx \]
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Exception generated. \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{2 \, n} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{-n + 3} \,d x } \]
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Time = 13.76 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.17 \[ \int (d \sec (e+f x))^{2 n} (a+i a \tan (e+f x))^{3-n} \, dx=-\left (\cos \left (6\,e+6\,f\,x\right )-\sin \left (6\,e+6\,f\,x\right )\,1{}\mathrm {i}\right )\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{2\,n}\,\left (\frac {{\left (a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}\right )}^{3-n}}{f\,n\,\left (n^2\,1{}\mathrm {i}+n\,3{}\mathrm {i}+2{}\mathrm {i}\right )}+\frac {\left (\cos \left (4\,e+4\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}\right )\,{\left (a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}\right )}^{3-n}\,\left (n^2+5\,n+6\right )}{2\,f\,n\,\left (n^2\,1{}\mathrm {i}+n\,3{}\mathrm {i}+2{}\mathrm {i}\right )}+\frac {\left (\cos \left (6\,e+6\,f\,x\right )+\sin \left (6\,e+6\,f\,x\right )\,1{}\mathrm {i}\right )\,{\left (a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}\right )}^{3-n}\,\left (n^2+3\,n+2\right )}{2\,f\,n\,\left (n^2\,1{}\mathrm {i}+n\,3{}\mathrm {i}+2{}\mathrm {i}\right )}+\frac {\left (2\,n+6\right )\,\left (\cos \left (2\,e+2\,f\,x\right )+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )\,{\left (a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}\right )}^{3-n}}{2\,f\,n\,\left (n^2\,1{}\mathrm {i}+n\,3{}\mathrm {i}+2{}\mathrm {i}\right )}\right ) \]
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