Optimal. Leaf size=103 \[ \frac{64 a c^4 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f} \]
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Rubi [A] time = 0.222452, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2674, 2673} \[ \frac{64 a c^4 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx &=(a c) \int \cos ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac{2 a c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}+\frac{1}{7} \left (8 a c^2\right ) \int \cos ^2(e+f x) \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{16 a c^3 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}+\frac{1}{35} \left (32 a c^3\right ) \int \frac{\cos ^2(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{64 a c^4 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}\\ \end{align*}
Mathematica [A] time = 0.532364, size = 94, normalized size = 0.91 \[ -\frac{a c^2 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3 (108 \sin (e+f x)+15 \cos (2 (e+f x))-157)}{105 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.52, size = 69, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ){c}^{3} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}a \left ( 15\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}-54\,\sin \left ( fx+e \right ) +71 \right ) }{105\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{c-c\sin \left ( fx+e \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{\left (a \sin \left (f x + e\right ) + a\right )}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09818, size = 385, normalized size = 3.74 \begin{align*} \frac{2 \,{\left (15 \, a c^{2} \cos \left (f x + e\right )^{4} + 39 \, a c^{2} \cos \left (f x + e\right )^{3} - 8 \, a c^{2} \cos \left (f x + e\right )^{2} + 32 \, a c^{2} \cos \left (f x + e\right ) + 64 \, a c^{2} -{\left (15 \, a c^{2} \cos \left (f x + e\right )^{3} - 24 \, a c^{2} \cos \left (f x + e\right )^{2} - 32 \, a c^{2} \cos \left (f x + e\right ) - 64 \, a c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{105 \,{\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\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(-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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