Optimal. Leaf size=46 \[ \frac {2 c \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2738} \[ \frac {2 c \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2738
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx &=\frac {2 c \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+2 m) \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 85, normalized size = 1.85 \[ \frac {2 \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (a (\sin (e+f x)+1))^m}{f (2 m+1) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 76, normalized size = 1.65 \[ \frac {2 \, \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{2 \, f m + {\left (2 \, f m + f\right )} \cos \left (f x + e\right ) - {\left (2 \, f m + f\right )} \sin \left (f x + e\right ) + f} \]
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 \[ \int \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {c -c \sin \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 116, normalized size = 2.52 \[ -\frac {2 \, {\left (a^{m} \sqrt {c} + \frac {a^{m} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{f {\left (2 \, m + 1\right )} \sqrt {\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 53, normalized size = 1.15 \[ -\frac {2\,\cos \left (e+f\,x\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{f\,\left (2\,m+1\right )\,\left (\sin \left (e+f\,x\right )-1\right )} \]
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 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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