Optimal. Leaf size=81 \[ -\frac {2^{m+\frac {1}{2}} 3^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2652, 2651} \[ -\frac {2^{m+\frac {1}{2}} 3^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2651
Rule 2652
Rubi steps
\begin {align*} \int 3^{-1-m} (a+a \sin (e+f x))^m \, dx &=3^{-1-m} \int (a+a \sin (e+f x))^m \, dx\\ &=\left (3^{-1-m} (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \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 ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 97, normalized size = 1.20 \[ \frac {\sqrt {2} 3^{-m-1} \cos (e+f x) (a (\sin (e+f x)+1))^m \, _2F_1\left (\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};\frac {1}{4} \cos ^2(e+f x) \csc ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )}{(2 f m+f) \sqrt {1-\sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (3^{-m - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int 3^{-m - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int 3^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx \]
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 \[ 3^{-m - 1} \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{3^{m+1}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
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 \[ 3^{- m - 1} \int \left (a \sin {\left (e + f x \right )} + a\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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