Optimal. Leaf size=89 \[ \frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{5 f} \]
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Rubi [A] time = 0.169982, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2740, 2738} \[ \frac{c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt{c-c \sin (e+f x)}}{5 f} \]
Antiderivative was successfully verified.
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Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac{c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{5 f}+\frac{1}{5} (2 c) \int (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{10 f \sqrt{c-c \sin (e+f x)}}+\frac{c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)}}{5 f}\\ \end{align*}
Mathematica [A] time = 0.963196, size = 93, normalized size = 1.04 \[ -\frac{a^3 c \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (-70 \sin (e+f x)-5 \sin (3 (e+f x))+\sin (5 (e+f x))+20 \cos (2 (e+f x))+5 \cos (4 (e+f x)))}{80 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.161, size = 106, normalized size = 1.2 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( -2\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-2\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +6\,\sin \left ( fx+e \right ) -6 \right ) }{10\,f \left ( \cos \left ( fx+e \right ) \right ) ^{7}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}}} \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{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11897, size = 250, normalized size = 2.81 \begin{align*} -\frac{{\left (5 \, a^{3} c \cos \left (f x + e\right )^{4} - 5 \, a^{3} c + 2 \,{\left (a^{3} c \cos \left (f x + e\right )^{4} - 2 \, a^{3} c \cos \left (f x + e\right )^{2} - 4 \, a^{3} c\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}}{10 \, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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