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 = 0.110354, antiderivative size = 74, normalized size of antiderivative = 1., 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 4604
Rule 3296
Rule 2638
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*}
Mathematica [A] time = 0.206221, 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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Maple [F] time = 0.081, size = 0, normalized size = 0. \begin{align*} \int x\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{c+c\sin \left ( fx+e \right ) }\, dx \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 \sqrt{-a \sin \left (f x + e\right ) + a} \sqrt{c \sin \left (f x + e\right ) + c} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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