Optimal. Leaf size=50 \[ \frac {2 \cos (e+f x) (c \sin (e+f x)+c)^{m+1}}{c f (2 m+3) \sqrt {a-a \sin (e+f x)}} \]
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Rubi [A] time = 0.25, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2841, 2738} \[ \frac {2 \cos (e+f x) (c \sin (e+f x)+c)^{m+1}}{c f (2 m+3) \sqrt {a-a \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2738
Rule 2841
Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x) (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx &=\frac {\int \sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{1+m} \, dx}{a c}\\ &=\frac {2 \cos (e+f x) (c+c \sin (e+f x))^{1+m}}{c f (3+2 m) \sqrt {a-a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 85, normalized size = 1.70 \[ \frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3 (c (\sin (e+f x)+1))^m}{f (2 m+3) \sqrt {a-a \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 108, normalized size = 2.16 \[ -\frac {2 \, {\left (\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2\right )} \sqrt {-a \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{2 \, a f m + 3 \, a f + {\left (2 \, a f m + 3 \, a f\right )} \cos \left (f x + e\right ) - {\left (2 \, a f m + 3 \, a f\right )} \sin \left (f x + e\right )} \]
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 \[ \int \frac {{\left (c \sin \left (f x + e\right ) + c\right )}^{m} \cos \left (f x + e\right )^{2}}{\sqrt {-a \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{2}\left (f x +e \right )\right ) \left (c +c \sin \left (f x +e \right )\right )^{m}}{\sqrt {a -a \sin \left (f x +e \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [B] time = 9.11, size = 68, normalized size = 1.36 \[ -\frac {\sqrt {-a\,\left (\sin \left (e+f\,x\right )-1\right )}\,{\left (c\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (2\,\cos \left (e+f\,x\right )+\sin \left (2\,e+2\,f\,x\right )\right )}{a\,f\,\left (2\,m+3\right )\,\left (\sin \left (e+f\,x\right )-1\right )} \]
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 \[ \int \frac {\left (c \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cos ^{2}{\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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