Optimal. Leaf size=92 \[ \frac {a B \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt {a \sin (e+f x)+a}}-\frac {a (A+B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.31, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2971, 2738} \[ \frac {a B \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt {a \sin (e+f x)+a}}-\frac {a (A+B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2738
Rule 2971
Rubi steps
\begin {align*} \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx &=(A+B) \int \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx-\frac {B \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx}{c}\\ &=-\frac {a (A+B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {a B \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 63, normalized size = 0.68 \[ \frac {\sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (4 A \sin (e+f x)-B \cos (2 (e+f x)))}{4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 61, normalized size = 0.66 \[ -\frac {{\left (B \cos \left (f x + e\right )^{2} - 2 \, A \sin \left (f x + e\right ) - B\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2 \, f \cos \left (f x + e\right )} \]
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 = 0.80, size = 57, normalized size = 0.62 \[ \frac {\left (B \sin \left (f x +e \right )+2 A \right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sin \left (f x +e \right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{2 f \cos \left (f x +e \right )} \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 75, normalized size = 0.82 \[ -\frac {\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (B\,\cos \left (e+f\,x\right )+B\,\cos \left (3\,e+3\,f\,x\right )-4\,A\,\sin \left (2\,e+2\,f\,x\right )\right )}{4\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
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 \[ \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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