Integrand size = 25, antiderivative size = 104 \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx=-\frac {216 \sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {5}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{f \sqrt {1+\sin (e+f x)}} \]
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Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2863, 144, 143} \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx=-\frac {8 \sqrt {2} a^3 \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {5}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt {\sin (e+f x)+1}} \]
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Rule 143
Rule 144
Rule 2863
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^3 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(1+x)^{5/2} (c+d x)^n}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = \frac {\left (a^3 \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac {c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \text {Subst}\left (\int \frac {(1+x)^{5/2} \left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^n}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = -\frac {8 \sqrt {2} a^3 \operatorname {AppellF1}\left (\frac {1}{2},-\frac {5}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{f \sqrt {1+\sin (e+f x)}} \\ \end{align*}
\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx=\int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{3} \left (c +d \sin \left (f x +e \right )\right )^{n}d x\]
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\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx=\text {Timed out} \]
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\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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\[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Timed out. \[ \int (3+3 \sin (e+f x))^3 (c+d \sin (e+f x))^n \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]
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