Optimal. Leaf size=45 \[ -\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.31, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2841, 2738} \[ -\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt {a \sin (e+f x)+a}} \]
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))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx &=\frac {\int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx}{a c}\\ &=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 0.51, size = 120, normalized size = 2.67 \[ -\frac {c (\sin (e+f x)-1) \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (15 \sin (e+f x)-\sin (3 (e+f x))+6 \cos (2 (e+f x)))}{12 f \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 78, normalized size = 1.73 \[ \frac {{\left (3 \, c \cos \left (f x + e\right )^{2} - {\left (c \cos \left (f x + e\right )^{2} - 4 \, c\right )} \sin \left (f x + e\right ) - 3 \, c\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.37, size = 147, normalized size = 3.27 \[ \frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\left (\cos ^{3}\left (f x +e \right )\right )-3 \sin \left (f x +e \right ) \cos \left (f x +e \right )-2 \left (\cos ^{2}\left (f x +e \right )\right )-\sin \left (f x +e \right )+4 \cos \left (f x +e \right )-1\right )}{3 f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (\sin \left (f x +e \right ) \cos \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right )-2 \sin \left (f x +e \right )-\cos \left (f x +e \right )+2\right )} \]
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 \[ \int \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{2}}{\sqrt {a \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.96, size = 83, normalized size = 1.84 \[ -\frac {c\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (6\,\cos \left (e+f\,x\right )+6\,\cos \left (3\,e+3\,f\,x\right )+14\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )\right )}{24\,f\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (\sin \left (e+f\,x\right )-1\right )} \]
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 \[ \int \frac {\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}} \cos ^{2}{\left (e + f x \right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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