Optimal. Leaf size=92 \[ -\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{4 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 c f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.390051, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {2841, 2740, 2738} \[ -\frac{\cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}{4 c f}-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 c f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int \cos ^2(e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx &=\frac{\int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx}{a c}\\ &=-\frac{\cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{4 c f}+\frac{\int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx}{2 c}\\ &=-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{6 c f \sqrt{a+a \sin (e+f x)}}-\frac{\cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{4 c f}\\ \end{align*}
Mathematica [A] time = 0.403659, size = 83, normalized size = 0.9 \[ \frac{c \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (8 (9 \sin (e+f x)+\sin (3 (e+f x)))+12 \cos (2 (e+f x))+3 \cos (4 (e+f x)))}{96 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.243, size = 90, normalized size = 1. \begin{align*}{\frac{\sin \left ( fx+e \right ) \left ( 3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+5\,\sin \left ( fx+e \right ) +5 \right ) }{12\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (f x + e\right ) + a}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68568, size = 193, normalized size = 2.1 \begin{align*} \frac{{\left (3 \, c \cos \left (f x + e\right )^{4} + 4 \,{\left (c \cos \left (f x + e\right )^{2} + 2 \, 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}}{12 \, f \cos \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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