Optimal. Leaf size=65 \[ -\frac{2^{m+\frac{1}{2}} 3^{-m-1} \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{f \sqrt{\sin (e+f x)+1}} \]
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Rubi [A] time = 0.0213046, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {12, 2651} \[ -\frac{2^{m+\frac{1}{2}} 3^{-m-1} \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{f \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2651
Rubi steps
\begin{align*} \int 3^{-1-m} (1+\sin (e+f x))^m \, dx &=3^{-1-m} \int (1+\sin (e+f x))^m \, dx\\ &=-\frac{2^{\frac{1}{2}+m} 3^{-1-m} \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{f \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.145701, size = 95, normalized size = 1.46 \[ \frac{\sqrt{2} 3^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^m \, _2F_1\left (\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};\frac{1}{4} \cos ^2(e+f x) \csc ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{(2 f m+f) \sqrt{1-\sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.297, size = 0, normalized size = 0. \begin{align*} \int{3}^{-1-m} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{m}\, 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*} 3^{-m - 1} \int{\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{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 (3^{-m - 1}{\left (\sin \left (f x + e\right ) + 1\right )}^{m}, 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*} 3^{- m - 1} \int \left (\sin{\left (e + f x \right )} + 1\right )^{m}\, 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 3^{-m - 1}{\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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