Integrand size = 27, antiderivative size = 78 \[ \int \frac {\sqrt [3]{e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {3 a (e \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{6},\frac {5}{3},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/6}}{2\ 2^{5/6} d e (a+a \sin (c+d x))^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2768, 72, 71} \[ \int \frac {\sqrt [3]{e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {3 a (\sin (c+d x)+1)^{5/6} (e \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{6},\frac {5}{3},\frac {1}{2} (1-\sin (c+d x))\right )}{2\ 2^{5/6} d e (a \sin (c+d x)+a)^{3/2}} \]
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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))^{4/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a-a x} (a+a x)^{5/6}} \, dx,x,\sin (c+d x)\right )}{d e (a-a \sin (c+d x))^{2/3} (a+a \sin (c+d x))^{2/3}} \\ & = \frac {\left (a^2 (e \cos (c+d x))^{4/3} \left (\frac {a+a \sin (c+d x)}{a}\right )^{5/6}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {x}{2}\right )^{5/6} \sqrt [3]{a-a x}} \, dx,x,\sin (c+d x)\right )}{2^{5/6} d e (a-a \sin (c+d x))^{2/3} (a+a \sin (c+d x))^{3/2}} \\ & = -\frac {3 a (e \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{6},\frac {5}{3},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/6}}{2\ 2^{5/6} d e (a+a \sin (c+d x))^{3/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [3]{e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {3 (e \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{6},\frac {5}{3},\frac {1}{2} (1-\sin (c+d x))\right )}{2\ 2^{5/6} d e \sqrt [6]{1+\sin (c+d x)} \sqrt {a (1+\sin (c+d x))}} \]
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\[\int \frac {\left (e \cos \left (d x +c \right )\right )^{\frac {1}{3}}}{\sqrt {a +a \sin \left (d x +c \right )}}d x\]
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\[ \int \frac {\sqrt [3]{e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {1}{3}}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\sqrt [3]{e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\sqrt [3]{e \cos {\left (c + d x \right )}}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {\sqrt [3]{e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {1}{3}}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\sqrt [3]{e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {1}{3}}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{1/3}}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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