Integrand size = 23, antiderivative size = 75 \[ \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx=\frac {d \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m) (d \cos (a+b x))^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2657} \[ \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx=\frac {d \cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(a+b x)\right )}{b c (m+1) (d \cos (a+b x))^{3/2}} \]
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Rule 2657
Rubi steps \begin{align*} \text {integral}& = \frac {d \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 ) (c \sin (a+b x))^{1+m}}{b c (1+m) (d \cos (a+b x))^{3/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx=\frac {\cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^m \tan (a+b x)}{b (1+m) \sqrt {d \cos (a+b x)}} \]
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\[\int \frac {\left (c \sin \left (b x +a \right )\right )^{m}}{\sqrt {d \cos \left (b x +a \right )}}d x\]
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\[ \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{m}}{\sqrt {d \cos \left (b x + a\right )}} \,d x } \]
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\[ \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {\left (c \sin {\left (a + b x \right )}\right )^{m}}{\sqrt {d \cos {\left (a + b x \right )}}}\, dx \]
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\[ \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{m}}{\sqrt {d \cos \left (b x + a\right )}} \,d x } \]
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\[ \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{m}}{\sqrt {d \cos \left (b x + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {(c \sin (a+b x))^m}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \]
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