Integrand size = 27, antiderivative size = 101 \[ \int \frac {(c+d \sin (e+f x))^n}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},-n,2,\frac {3}{2},\frac {d (1-\sin (e+f x))}{c+d},\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{6 f \sqrt {3+3 \sin (e+f x)}} \]
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Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.29, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2867, 142, 141} \[ \int \frac {(c+d \sin (e+f x))^n}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {d \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} \operatorname {AppellF1}\left (n+1,\frac {1}{2},2,n+2,\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d)^2 (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}} \]
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Rule 141
Rule 142
Rule 2867
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {a-a x} (a+a x)^2} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = \frac {\left (a^2 \cos (e+f x) \sqrt {\frac {d (a-a \sin (e+f x))}{a c+a d}}\right ) \text {Subst}\left (\int \frac {(c+d x)^n}{(a+a x)^2 \sqrt {\frac {a d}{a c+a d}-\frac {a d x}{a c+a d}}} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = \frac {d \operatorname {AppellF1}\left (1+n,\frac {1}{2},2,2+n,\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right ) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{1+n}}{(c-d)^2 f (1+n) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(314\) vs. \(2(101)=202\).
Time = 5.39 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.11 \[ \int \frac {(c+d \sin (e+f x))^n}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\sec (e+f x) (c+d \sin (e+f x))^n \left (\operatorname {AppellF1}\left (1,\frac {1}{2},-n,2,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \sqrt {2-2 \sin (e+f x)} (1+\sin (e+f x))^2 \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}-\frac {4 (1+\sin (e+f x)) \sqrt {1-\frac {2}{1+\sin (e+f x)}} \left (1+\frac {-1+\frac {c}{d}}{1+\sin (e+f x)}\right )^{-n} \left (2 (1+2 n) \operatorname {AppellF1}\left (\frac {1}{2}-n,-\frac {1}{2},-n,\frac {3}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {-c+d}{d+d \sin (e+f x)}\right )+(-1+2 n) \operatorname {AppellF1}\left (-\frac {1}{2}-n,-\frac {1}{2},-n,\frac {1}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {-c+d}{d+d \sin (e+f x)}\right ) (1+\sin (e+f x))\right )}{-1+4 n^2}\right )}{24 \sqrt {3} f \sqrt {1+\sin (e+f x)}} \]
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\[\int \frac {\left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(c+d \sin (e+f x))^n}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(c+d \sin (e+f x))^n}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (c + d \sin {\left (e + f x \right )}\right )^{n}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(c+d \sin (e+f x))^n}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^n}{(3+3 \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^n}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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