Optimal. Leaf size=59 \[ \frac {2^{\frac {1}{2}+2 n} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d \sqrt {1+\cos (c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 59, 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 = {2730}
\begin {gather*} \frac {2^{2 n+\frac {1}{2}} \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\cos (c+d x))\right )}{d \sqrt {\cos (c+d x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
Rubi steps
\begin {align*} \int (2+2 \cos (c+d x))^n \, dx &=\frac {2^{\frac {1}{2}+2 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 \sqrt {1+\cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 77, normalized size = 1.31 \begin {gather*} -\frac {2^{1+n} (1+\cos (c+d x))^n \cot \left (\frac {1}{2} (c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1}{2}+n;\frac {3}{2}+n;\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{d+2 d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (2+2 \cos \left (d x +c \right )\right )^{n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} 2^{n} \int \left (\cos {\left (c + d x \right )} + 1\right )^{n}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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