Optimal. Leaf size=137 \[ \frac{256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a c^4 \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}+\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \]
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Rubi [A] time = 0.294873, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2674, 2673} \[ \frac{256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a c^4 \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}+\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \]
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))^{7/2} \, dx &=(a c) \int \cos ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx\\ &=\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac{1}{3} \left (4 a c^2\right ) \int \cos ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac{8 a c^3 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}+\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac{1}{21} \left (32 a c^3\right ) \int \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 \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}+\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac{1}{105} \left (128 a c^4\right ) \int \frac{\cos ^2(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a c^4 \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}+\frac{2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}\\ \end{align*}
Mathematica [A] time = 0.840671, size = 104, normalized size = 0.76 \[ \frac{a c^3 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3 (-1389 \sin (e+f x)+35 \sin (3 (e+f x))-330 \cos (2 (e+f x))+1606)}{630 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.573, size = 79, normalized size = 0.6 \begin{align*}{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ){c}^{4} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}a \left ( 35\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}-165\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}+321\,\sin \left ( fx+e \right ) -319 \right ) }{315\,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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1593, size = 466, normalized size = 3.4 \begin{align*} -\frac{2 \,{\left (35 \, a c^{3} \cos \left (f x + e\right )^{5} - 95 \, a c^{3} \cos \left (f x + e\right )^{4} - 226 \, a c^{3} \cos \left (f x + e\right )^{3} + 32 \, a c^{3} \cos \left (f x + e\right )^{2} - 128 \, a c^{3} \cos \left (f x + e\right ) - 256 \, a c^{3} +{\left (35 \, a c^{3} \cos \left (f x + e\right )^{4} + 130 \, a c^{3} \cos \left (f x + e\right )^{3} - 96 \, a c^{3} \cos \left (f x + e\right )^{2} - 128 \, a c^{3} \cos \left (f x + e\right ) - 256 \, a c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{315 \,{\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] 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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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