Integrand size = 38, antiderivative size = 167 \[ \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-n),\frac {1}{2} (2-m-n),\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (m+n) \sqrt {\sin ^2(c+d x)}}+\frac {C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-m-n),\frac {1}{2} (1-m-n),\cos ^2(c+d x)\right ) \sec ^{1+m}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (1+m+n) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {20, 4132, 3857, 2722, 12} \[ \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B \sin (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-n),\frac {1}{2} (-m-n+2),\cos ^2(c+d x)\right )}{d (m+n) \sqrt {\sin ^2(c+d x)}}+\frac {C \sin (c+d x) \sec ^{m+1}(c+d x) (b \sec (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-n-1),\frac {1}{2} (-m-n+1),\cos ^2(c+d x)\right )}{d (m+n+1) \sqrt {\sin ^2(c+d x)}} \]
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Rule 12
Rule 20
Rule 2722
Rule 3857
Rule 4132
Rubi steps \begin{align*} \text {integral}& = \left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{m+n}(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \\ & = \left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int C \sec ^{2+m+n}(c+d x) \, dx+\left (B \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{1+m+n}(c+d x) \, dx \\ & = \left (B \cos ^{m+n}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n\right ) \int \cos ^{-1-m-n}(c+d x) \, dx+\left (C \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{2+m+n}(c+d x) \, dx \\ & = \frac {B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-n),\frac {1}{2} (2-m-n),\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (m+n) \sqrt {\sin ^2(c+d x)}}+\left (C \cos ^{m+n}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n\right ) \int \cos ^{-2-m-n}(c+d x) \, dx \\ & = \frac {B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-n),\frac {1}{2} (2-m-n),\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (m+n) \sqrt {\sin ^2(c+d x)}}+\frac {C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-m-n),\frac {1}{2} (1-m-n),\cos ^2(c+d x)\right ) \sec ^{1+m}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (1+m+n) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.77 \[ \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\csc (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \left (B (2+m+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\sec ^2(c+d x)\right )+C (1+m+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+m+n),\frac {1}{2} (4+m+n),\sec ^2(c+d x)\right ) \sec (c+d x)\right ) \sqrt {-\tan ^2(c+d x)}}{d (1+m+n) (2+m+n)} \]
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\[\int \sec \left (d x +c \right )^{m} \left (b \sec \left (d x +c \right )\right )^{n} \left (B \sec \left (d x +c \right )+C \sec \left (d x +c \right )^{2}\right )d x\]
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\[ \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m} \,d x } \]
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\[ \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (b \sec {\left (c + d x \right )}\right )^{n} \left (B + C \sec {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )} \sec ^{m}{\left (c + d x \right )}\, dx \]
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\[ \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m} \,d x } \]
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\[ \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \sec ^m(c+d x) (b \sec (c+d x))^n \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m \,d x \]
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