Optimal. Leaf size=60 \[ -\frac {2^{2 n+\frac {1}{2}} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt {\sin (c+d x)+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2651} \[ -\frac {2^{2 n+\frac {1}{2}} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt {\sin (c+d x)+1}} \]
Antiderivative was successfully verified.
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Rule 2651
Rubi steps
\begin {align*} \int (2+2 \sin (c+d x))^n \, dx &=-\frac {2^{\frac {1}{2}+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*}
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Mathematica [A] time = 0.11, size = 90, normalized size = 1.50 \[ \frac {2^{n+\frac {1}{2}} \cos (c+d x) (\sin (c+d x)+1)^n \, _2F_1\left (\frac {1}{2},n+\frac {1}{2};n+\frac {3}{2};\frac {1}{4} \cos ^2(c+d x) \csc ^2\left (\frac {1}{4} (2 c+2 d x-\pi )\right )\right )}{(2 d n+d) \sqrt {1-\sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, \sin \left (d x + c\right ) + 2\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (2 \, \sin \left (d x + c\right ) + 2\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \left (2+2 \sin \left (d x +c \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (2 \, \sin \left (d x + c\right ) + 2\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (2\,\sin \left (c+d\,x\right )+2\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2^{n} \int \left (\sin {\left (c + d x \right )} + 1\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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