Integrand size = 19, antiderivative size = 109 \[ \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx=-\frac {\cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n)}-\frac {2^{\frac {1}{2}+n} n \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-n} (a+a \sin (c+d x))^n}{d (1+n)} \]
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Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2830, 2731, 2730} \[ \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx=-\frac {2^{n+\frac {1}{2}} n \cos (c+d x) (\sin (c+d x)+1)^{-n-\frac {1}{2}} (a \sin (c+d x)+a)^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{d (n+1)}-\frac {\cos (c+d x) (a \sin (c+d x)+a)^n}{d (n+1)} \]
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Rule 2730
Rule 2731
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n)}+\frac {n \int (a+a \sin (c+d x))^n \, dx}{1+n} \\ & = -\frac {\cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n)}+\frac {\left (n (1+\sin (c+d x))^{-n} (a+a \sin (c+d x))^n\right ) \int (1+\sin (c+d x))^n \, dx}{1+n} \\ & = -\frac {\cos (c+d x) (a+a \sin (c+d x))^n}{d (1+n)}-\frac {2^{\frac {1}{2}+n} n \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-n} (a+a \sin (c+d x))^n}{d (1+n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87 \[ \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx=-\frac {2^n \left (B_{\frac {1}{2} (1+\sin (c+d x))}\left (\frac {1}{2}+n,\frac {1}{2}\right )-2 B_{\frac {1}{2} (1+\sin (c+d x))}\left (\frac {3}{2}+n,\frac {1}{2}\right )\right ) \sqrt {\cos ^2(c+d x)} \sec (c+d x) (1+\sin (c+d x))^{-n} (a (1+\sin (c+d x)))^n}{d} \]
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\[\int \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{n}d x\]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ) \,d x } \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{n} \sin {\left (c + d x \right )}\, dx \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ) \,d x } \]
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\[ \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \sin (c+d x) (a+a \sin (c+d x))^n \, dx=\int \sin \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^n \,d x \]
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