Integrand size = 23, antiderivative size = 76 \[ \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=-\frac {c \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(a+b x)\right ) (b \sec (a+b x))^{-1+n} \sqrt {c \sin (a+b x)}}{(1-n) \sqrt [4]{\sin ^2(a+b x)}} \]
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Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2667, 2656} \[ \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=-\frac {c \sqrt {c \sin (a+b x)} (b \sec (a+b x))^{n-1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(a+b x)\right )}{(1-n) \sqrt [4]{\sin ^2(a+b x)}} \]
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Rule 2656
Rule 2667
Rubi steps \begin{align*} \text {integral}& = \left (b^2 (b \cos (a+b x))^{-1+n} (b \sec (a+b x))^{-1+n}\right ) \int (b \cos (a+b x))^{-n} (c \sin (a+b x))^{3/2} \, dx \\ & = -\frac {c \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(a+b x)\right ) (b \sec (a+b x))^{-1+n} \sqrt {c \sin (a+b x)}}{(1-n) \sqrt [4]{\sin ^2(a+b x)}} \\ \end{align*}
Time = 31.91 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.37 \[ \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=\frac {2 \cos ^2(a+b x)^{\frac {1}{2} (-1+n)} (b \sec (a+b x))^{-1+n} (c \sin (a+b x))^{5/2} \left (9 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1}{2} (-1+n),\frac {9}{4},\sin ^2(a+b x)\right )+5 \operatorname {Hypergeometric2F1}\left (\frac {9}{4},\frac {1+n}{2},\frac {13}{4},\sin ^2(a+b x)\right ) \sin ^2(a+b x)\right )}{45 c} \]
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\[\int \left (b \sec \left (b x +a \right )\right )^{n} \left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}}d x\]
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\[ \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \left (b \sec \left (b x + a\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \left (b \sec \left (b x + a\right )\right )^{n} \,d x } \]
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\[ \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \left (b \sec \left (b x + a\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx=\int {\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}\,{\left (\frac {b}{\cos \left (a+b\,x\right )}\right )}^n \,d x \]
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