Optimal. Leaf size=75 \[ -\frac{2^{n+\frac{1}{2}} \sin (c+d x) (1-\cos (c+d x))^{-n-\frac{1}{2}} (a-a \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\cos (c+d x)+1)\right )}{d} \]
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Rubi [A] time = 0.0334317, antiderivative size = 75, 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} \[ -\frac{2^{n+\frac{1}{2}} \sin (c+d x) (1-\cos (c+d x))^{-n-\frac{1}{2}} (a-a \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\cos (c+d x)+1)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int (a-a \cos (c+d x))^n \, dx &=\left ((1-\cos (c+d x))^{-n} (a-a \cos (c+d x))^n\right ) \int (1-\cos (c+d x))^n \, dx\\ &=-\frac{2^{\frac{1}{2}+n} (1-\cos (c+d x))^{-\frac{1}{2}-n} (a-a \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1+\cos (c+d x))\right ) \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0779838, size = 75, normalized size = 1. \[ \frac{\sqrt{2} \sqrt{\cos (c+d x)+1} \tan \left (\frac{1}{2} (c+d x)\right ) (a-a \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\sin ^2\left (\frac{1}{2} (c+d x)\right )\right )}{2 d n+d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.483, size = 0, normalized size = 0. \begin{align*} \int \left ( a-\cos \left ( dx+c \right ) a \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 \cos \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 \cos \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 \cos{\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 \cos \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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