Integrand size = 33, antiderivative size = 140 \[ \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)}}-\frac {2 (A+C (3-2 n)-2 A n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (5-2 n),\frac {1}{4} (9-2 n),\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (5-2 n) \sec ^{\frac {5}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {20, 4131, 3857, 2722} \[ \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (5-2 n),\frac {1}{4} (9-2 n),\cos ^2(c+d x)\right )}{d (1-2 n) (5-2 n) \sqrt {\sin ^2(c+d x)} \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt {\sec (c+d x)}} \]
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Rule 20
Rule 2722
Rule 3857
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac {3}{2}+n}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx \\ & = -\frac {2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)}}+\frac {\left (\left (C \left (-\frac {3}{2}+n\right )+A \left (-\frac {1}{2}+n\right )\right ) \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{-\frac {3}{2}+n}(c+d x) \, dx}{-\frac {1}{2}+n} \\ & = -\frac {2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)}}+\frac {\left (\left (C \left (-\frac {3}{2}+n\right )+A \left (-\frac {1}{2}+n\right )\right ) \cos ^{\frac {1}{2}+n}(c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{\frac {3}{2}-n}(c+d x) \, dx}{-\frac {1}{2}+n} \\ & = -\frac {2 C (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) \sqrt {\sec (c+d x)}}-\frac {2 (A (1-2 n)+C (3-2 n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (5-2 n),\frac {1}{4} (9-2 n),\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (5-2 n) \sec ^{\frac {5}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \csc (c+d x) (b \sec (c+d x))^n \left (A (1+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-3+2 n),\frac {1}{4} (1+2 n),\sec ^2(c+d x)\right )+C (-3+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (1+2 n),\frac {1}{4} (5+2 n),\sec ^2(c+d x)\right ) \sec ^2(c+d x)\right ) \sqrt {-\tan ^2(c+d x)}}{d (-3+2 n) (1+2 n) \sec ^{\frac {5}{2}}(c+d x)} \]
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\[\int \frac {\left (b \sec \left (d x +c \right )\right )^{n} \left (A +C \sec \left (d x +c \right )^{2}\right )}{\sec \left (d x +c \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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