Optimal. Leaf size=81 \[ -\frac {2^{\frac {1}{2}+m} 3^{-1-m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f} \]
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Rubi [A]
time = 0.03, 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, 2731, 2730}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2730
Rule 2731
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 = 0.18, size = 97, normalized size = 1.20 \begin {gather*} \frac {\sqrt {2} 3^{-1-m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}+m;\frac {3}{2}+m;\frac {1}{4} \cos ^2(e+f x) \csc ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right ) (a (1+\sin (e+f x)))^m}{(f+2 f m) \sqrt {1-\sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, 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 \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*} 3^{- m - 1} \int \left (a \sin {\left (e + f x \right )} + a\right )^{m}\, 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 = -0.01 \begin {gather*} \int \frac {1}{3^{m+1}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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