Optimal. Leaf size=58 \[ -\frac{2^{n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d \sqrt{\sin (c+d x)+1}} \]
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Rubi [A] time = 0.0139806, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2651} \[ -\frac{2^{n+\frac{1}{2}} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d \sqrt{\sin (c+d x)+1}} \]
Antiderivative was successfully verified.
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Rule 2651
Rubi steps
\begin{align*} \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 )}{d \sqrt{1+\sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.12066, size = 88, normalized size = 1.52 \[ \frac{\sqrt{2} \cos (c+d x) (\sin (c+d x)+1)^n \, _2F_1\left (\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\frac{1}{4} \cos ^2(c+d x) \csc ^2\left (\frac{1}{4} (2 c+2 d x-\pi )\right )\right )}{(2 d n+d) \sqrt{1-\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.26, size = 0, normalized size = 0. \begin{align*} \int \left ( 1+\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 (\sin \left (d x + c\right ) + 1\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 (\sin \left (d x + c\right ) + 1\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 (\sin{\left (c + d x \right )} + 1\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 (\sin \left (d x + c\right ) + 1\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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