Optimal. Leaf size=116 \[ \frac{2 a c^2 (7 A-B) \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 (7 A-B) \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}-\frac{2 a B c \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f} \]
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Rubi [A] time = 0.320017, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2967, 2856, 2674, 2673} \[ \frac{2 a c^2 (7 A-B) \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 (7 A-B) \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}-\frac{2 a B c \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f} \]
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))^{3/2} \, dx &=(a c) \int \cos ^2(e+f x) (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx\\ &=-\frac{2 a B c \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}+\frac{1}{7} (a (7 A-B) c) \int \cos ^2(e+f x) \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{2 a (7 A-B) c^2 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a B c \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}+\frac{1}{35} \left (4 a (7 A-B) c^2\right ) \int \frac{\cos ^2(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{8 a (7 A-B) c^3 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a (7 A-B) c^2 \cos ^3(e+f x)}{35 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a B c \cos ^3(e+f x) \sqrt{c-c \sin (e+f x)}}{7 f}\\ \end{align*}
Mathematica [A] time = 0.980497, size = 104, normalized size = 0.9 \[ \frac{a c \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3 ((66 B-42 A) \sin (e+f x)+98 A+15 B \cos (2 (e+f x))-59 B)}{105 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.917, size = 81, normalized size = 0.7 \begin{align*}{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ){c}^{2} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}a \left ( \sin \left ( fx+e \right ) \left ( 21\,A-33\,B \right ) -15\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}-49\,A+37\,B \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 (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{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71847, size = 454, normalized size = 3.91 \begin{align*} -\frac{2 \,{\left (15 \, B a c \cos \left (f x + e\right )^{4} - 3 \,{\left (7 \, A - 6 \, B\right )} a c \cos \left (f x + e\right )^{3} +{\left (7 \, A - B\right )} a c \cos \left (f x + e\right )^{2} - 4 \,{\left (7 \, A - B\right )} a c \cos \left (f x + e\right ) - 8 \,{\left (7 \, A - B\right )} a c -{\left (15 \, B a c \cos \left (f x + e\right )^{3} + 3 \,{\left (7 \, A - B\right )} a c \cos \left (f x + e\right )^{2} + 4 \,{\left (7 \, A - B\right )} a c \cos \left (f x + e\right ) + 8 \,{\left (7 \, A - B\right )} a c\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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