Optimal. Leaf size=45 \[ \frac {\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2742} \[ \frac {\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (a \sin (e+f x)+a)^m}{f (2 m+1)} \]
Antiderivative was successfully verified.
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Rule 2742
Rubi steps
\begin {align*} \int (3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx &=\frac {\cos (e+f x) (3-3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m}{f (1+2 m)}\\ \end {align*}
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Mathematica [B] time = 0.60, size = 99, normalized size = 2.20 \[ \frac {\sin \left (\frac {1}{4} (2 e+2 f x+\pi )\right ) (6-6 \sin (e+f x))^{-m} \cos ^{-2 m-1}\left (\frac {1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{2 m}}{3 (2 f m+f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 43, normalized size = 0.96 \[ \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \left (3-3 \sin \left (f x +e \right )\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 45, normalized size = 1.00 \[ \frac {\cos \left (e+f\,x\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m}{f\,\left (2\,m+1\right )\,{\left (3-3\,\sin \left (e+f\,x\right )\right )}^{m+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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