Optimal. Leaf size=73 \[ \frac {2 a c^2 (5 A+B) \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}-\frac {2 a B c \cos ^3(e+f x)}{5 f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.24, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2967, 2856, 2673} \[ \frac {2 a c^2 (5 A+B) \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}-\frac {2 a B c \cos ^3(e+f x)}{5 f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2856
Rule 2967
Rubi steps
\begin {align*} \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx &=(a c) \int \frac {\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)}{5 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{5} (a (5 A+B) c) \int \frac {\cos ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {2 a (5 A+B) c^2 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}-\frac {2 a B c \cos ^3(e+f x)}{5 f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 87, normalized size = 1.19 \[ \frac {2 a \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3 (5 A+3 B \sin (e+f x)-2 B)}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 130, normalized size = 1.78 \[ -\frac {2 \, {\left (3 \, B a \cos \left (f x + e\right )^{3} + {\left (5 \, A + 4 \, B\right )} a \cos \left (f x + e\right )^{2} - {\left (5 \, A + B\right )} a \cos \left (f x + e\right ) - 2 \, {\left (5 \, A + B\right )} a + {\left (3 \, B a \cos \left (f x + e\right )^{2} - {\left (5 \, A + B\right )} a \cos \left (f x + e\right ) - 2 \, {\left (5 \, A + B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.31, size = 63, normalized size = 0.86 \[ -\frac {2 \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right )^{2} a \left (3 B \sin \left (f x +e \right )+5 A -2 B \right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )} \sqrt {-c \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (a+a\,\sin \left (e+f\,x\right )\right )\,\sqrt {c-c\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int A \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int A \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int B \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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