Optimal. Leaf size=58 \[ -\frac {2^{\frac {1}{2}+n} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt {1+\sin (c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2730}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
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*}
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Mathematica [A]
time = 0.09, size = 88, normalized size = 1.52 \begin {gather*} \frac {\sqrt {2} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}+n;\frac {3}{2}+n;\frac {1}{4} \cos ^2(c+d x) \csc ^2\left (\frac {1}{4} (2 c-\pi +2 d x)\right )\right ) (1+\sin (c+d x))^n}{(d+2 d n) \sqrt {1-\sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (1+\sin \left (d x +c \right )\right )^{n}\, dx\]
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 \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*} \int \left (\sin {\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 (\sin \left (c+d\,x\right )+1\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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