Integrand size = 25, antiderivative size = 97 \[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2^{1+\frac {p}{2}} a (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (-\frac {p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-p/2}}{d e (1+p) \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2768, 72, 71} \[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=-\frac {a 2^{\frac {p}{2}+1} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \operatorname {Hypergeometric2F1}\left (-\frac {p}{2},\frac {p+1}{2},\frac {p+3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d e (p+1) \sqrt {a \sin (c+d x)+a}} \]
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Rule 71
Rule 72
Rule 2768
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 (e \cos (c+d x))^{1+p} (a-a \sin (c+d x))^{\frac {1}{2} (-1-p)} (a+a \sin (c+d x))^{\frac {1}{2} (-1-p)}\right ) \text {Subst}\left (\int (a-a x)^{\frac {1}{2} (-1+p)} (a+a x)^{\frac {1}{2}+\frac {1}{2} (-1+p)} \, dx,x,\sin (c+d x)\right )}{d e} \\ & = \frac {\left (2^{p/2} a^2 (e \cos (c+d x))^{1+p} (a-a \sin (c+d x))^{\frac {1}{2} (-1-p)} (a+a \sin (c+d x))^{\frac {1}{2} (-1-p)+\frac {p}{2}} \left (\frac {a+a \sin (c+d x)}{a}\right )^{-p/2}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{2}+\frac {1}{2} (-1+p)} (a-a x)^{\frac {1}{2} (-1+p)} \, dx,x,\sin (c+d x)\right )}{d e} \\ & = -\frac {2^{1+\frac {p}{2}} a (e \cos (c+d x))^{1+p} \operatorname {Hypergeometric2F1}\left (-\frac {p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-p/2}}{d e (1+p) \sqrt {a+a \sin (c+d x)}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01 \[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2^{1+\frac {p}{2}} a \cos (c+d x) (e \cos (c+d x))^p \operatorname {Hypergeometric2F1}\left (-\frac {p}{2},\frac {1+p}{2},\frac {3+p}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-p/2}}{d (1+p) \sqrt {a (1+\sin (c+d x))}} \]
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\[\int \left (e \cos \left (d x +c \right )\right )^{p} \sqrt {a +a \sin \left (d x +c \right )}d x\]
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\[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
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\[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{p}\, dx \]
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\[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
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\[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{p} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^p \sqrt {a+a \sin (c+d x)} \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^p\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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