Optimal. Leaf size=74 \[ \frac{2^{n+\frac{1}{2}} \cos (c+d x) (1-\sin (c+d x))^{-n-\frac{1}{2}} (a-a \sin (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{d} \]
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Rubi [A] time = 0.033747, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2652, 2651} \[ \frac{2^{n+\frac{1}{2}} \cos (c+d x) (1-\sin (c+d x))^{-n-\frac{1}{2}} (a-a \sin (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int (a-a \sin (c+d x))^n \, dx &=\left ((1-\sin (c+d x))^{-n} (a-a \sin (c+d x))^n\right ) \int (1-\sin (c+d x))^n \, dx\\ &=\frac{2^{\frac{1}{2}+n} \cos (c+d x) \, _2F_1\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}\\ \end{align*}
Mathematica [A] time = 0.142221, size = 91, normalized size = 1.23 \[ \frac{\cos (c+d x) \cos ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )^{-n-\frac{1}{2}} (a-a \sin (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left (\frac{1}{4} (2 c+2 d x-\pi )\right )\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.326, size = 0, normalized size = 0. \begin{align*} \int \left ( a-a\sin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a \sin \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a \sin \left (d x + c\right ) + a\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- a \sin{\left (c + d x \right )} + a\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a \sin \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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