Optimal. Leaf size=157 \[ \frac{64 a c^4 (3 A-B) \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 (3 A-B) \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 (3 A-B) \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}-\frac{2 a B c \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \]
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Rubi [A] time = 0.412332, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2967, 2856, 2674, 2673} \[ \frac{64 a c^4 (3 A-B) \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a c^3 (3 A-B) \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a c^2 (3 A-B) \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}-\frac{2 a B c \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2856
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx &=(a c) \int \cos ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac{2 a B c \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac{1}{3} (a (3 A-B) c) \int \cos ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac{2 a (3 A-B) c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}-\frac{2 a B c \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac{1}{21} \left (8 a (3 A-B) c^2\right ) \int \cos ^2(e+f x) \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{16 a (3 A-B) c^3 \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a (3 A-B) c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}-\frac{2 a B c \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac{1}{105} \left (32 a (3 A-B) c^3\right ) \int \frac{\cos ^2(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{64 a (3 A-B) c^4 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac{16 a (3 A-B) c^3 \cos ^3(e+f x)}{105 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a (3 A-B) c^2 \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{21 f}-\frac{2 a B c \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}\\ \end{align*}
Mathematica [A] time = 1.44892, size = 123, normalized size = 0.78 \[ -\frac{a c^2 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3 ((648 A-741 B) \sin (e+f x)+30 (3 A-8 B) \cos (2 (e+f x))-942 A+35 B \sin (3 (e+f x))+664 B)}{630 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.018, size = 103, 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 ( -35\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) + \left ( -162\,A+194\,B \right ) \sin \left ( fx+e \right ) + \left ( -45\,A+120\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+258\,A-226\,B \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 (B \sin \left (f x + e\right ) + A\right )}{\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 = 1.75947, size = 591, normalized size = 3.76 \begin{align*} \frac{2 \,{\left (35 \, B a c^{2} \cos \left (f x + e\right )^{5} + 5 \,{\left (9 \, A - 10 \, B\right )} a c^{2} \cos \left (f x + e\right )^{4} +{\left (117 \, A - 109 \, B\right )} a c^{2} \cos \left (f x + e\right )^{3} - 8 \,{\left (3 \, A - B\right )} a c^{2} \cos \left (f x + e\right )^{2} + 32 \,{\left (3 \, A - B\right )} a c^{2} \cos \left (f x + e\right ) + 64 \,{\left (3 \, A - B\right )} a c^{2} +{\left (35 \, B a c^{2} \cos \left (f x + e\right )^{4} - 5 \,{\left (9 \, A - 17 \, B\right )} a c^{2} \cos \left (f x + e\right )^{3} + 24 \,{\left (3 \, A - B\right )} a c^{2} \cos \left (f x + e\right )^{2} + 32 \,{\left (3 \, A - B\right )} a c^{2} \cos \left (f x + e\right ) + 64 \,{\left (3 \, A - B\right )} a c^{2}\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 (B \sin \left (f x + e\right ) + A\right )}{\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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