Integrand size = 23, antiderivative size = 77 \[ \int \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \, dx=\frac {\cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (d \sec (a+b x))^{3/2} (c \sin (a+b x))^{1+m}}{b c d (1+m)} \]
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Time = 0.06 (sec) , antiderivative size = 77, 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 = {2666, 2657} \[ \int \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \, dx=\frac {\cos ^2(a+b x)^{3/4} (d \sec (a+b x))^{3/2} (c \sin (a+b x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(a+b x)\right )}{b c d (m+1)} \]
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Rule 2657
Rule 2666
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d \cos (a+b x))^{3/2} (d \sec (a+b x))^{3/2}\right ) \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx}{d^2} \\ & = \frac {\cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (d \sec (a+b x))^{3/2} (c \sin (a+b x))^{1+m}}{b c d (1+m)} \\ \end{align*}
Time = 5.46 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.38 \[ \int \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \, dx=-\frac {\csc ^2(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (1-2 m),\frac {1-m}{2},\frac {1}{4} (5-2 m),\sec ^2(a+b x)\right ) \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \sin (2 (a+b x)) \left (-\tan ^2(a+b x)\right )^{\frac {1-m}{2}}}{b (-1+2 m)} \]
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\[\int \sqrt {d \sec \left (b x +a \right )}\, \left (c \sin \left (b x +a \right )\right )^{m}d x\]
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\[ \int \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \, dx=\int { \sqrt {d \sec \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{m} \,d x } \]
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\[ \int \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{m} \sqrt {d \sec {\left (a + b x \right )}}\, dx \]
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\[ \int \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \, dx=\int { \sqrt {d \sec \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{m} \,d x } \]
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\[ \int \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \, dx=\int { \sqrt {d \sec \left (b x + a\right )} \left (c \sin \left (b x + a\right )\right )^{m} \,d x } \]
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Timed out. \[ \int \sqrt {d \sec (a+b x)} (c \sin (a+b x))^m \, dx=\int {\left (c\,\sin \left (a+b\,x\right )\right )}^m\,\sqrt {\frac {d}{\cos \left (a+b\,x\right )}} \,d x \]
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