Optimal. Leaf size=74 \[ \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{f^2}+\frac {x \tan (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{f} \]
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Rubi [A] time = 0.11, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {4604, 3296, 2638} \[ \frac {\sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{f^2}+\frac {x \tan (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}{f} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 4604
Rubi steps
\begin {align*} \int x \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \, dx &=\left (\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}\right ) \int x \cos (e+f x) \, dx\\ &=\frac {x \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \tan (e+f x)}{f}-\frac {\left (\sec (e+f x) \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}\right ) \int \sin (e+f x) \, dx}{f}\\ &=\frac {\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}{f^2}+\frac {x \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)} \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 44, normalized size = 0.59 \[ \frac {(f x \tan (e+f x)+1) \sqrt {a-a \sin (e+f x)} \sqrt {c (\sin (e+f x)+1)}}{f^2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int x \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}\, dx \]
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 \[ \int \sqrt {-a \sin \left (f x + e\right ) + a} \sqrt {c \sin \left (f x + e\right ) + c} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.69, size = 61, normalized size = 0.82 \[ \frac {\sqrt {-a\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (2\,{\cos \left (e+f\,x\right )}^2+f\,x\,\sin \left (2\,e+2\,f\,x\right )\right )\,\sqrt {c\,\left (\sin \left (e+f\,x\right )+1\right )}}{2\,f^2\,{\cos \left (e+f\,x\right )}^2} \]
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 \[ \int x \sqrt {c \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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