Optimal. Leaf size=43 \[ -\frac{a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.0829605, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {2738} \[ -\frac{a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2738
Rubi steps
\begin{align*} \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx &=-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.395553, size = 83, normalized size = 1.93 \[ -\frac{c^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (8 (\sin (3 (e+f x))-7 \sin (e+f x))-28 \cos (2 (e+f x))+\cos (4 (e+f x)))}{32 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.211, size = 103, normalized size = 2.4 \begin{align*}{\frac{\sin \left ( fx+e \right ) \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{6}+\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) - \left ( \cos \left ( fx+e \right ) \right ) ^{2}+4\,\sin \left ( fx+e \right ) +4 \right ) }{4\,f \left ( \cos \left ( fx+e \right ) \right ) ^{7}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.13083, size = 232, normalized size = 5.4 \begin{align*} -\frac{{\left (c^{3} \cos \left (f x + e\right )^{4} - 8 \, c^{3} \cos \left (f x + e\right )^{2} + 7 \, c^{3} + 4 \,{\left (c^{3} \cos \left (f x + e\right )^{2} - 2 \, c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{4 \, 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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