Optimal. Leaf size=145 \[ -\frac{2 a^2 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a (A+B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.381569, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2973, 2740, 2737, 2667, 31} \[ -\frac{2 a^2 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a (A+B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2973
Rule 2740
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}+(A+B) \int \frac{(a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{a (A+B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}+(2 a (A+B)) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{a (A+B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}+\frac{\left (2 a^2 (A+B) c \cos (e+f x)\right ) \int \frac{\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{a (A+B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}-\frac{\left (2 a^2 (A+B) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{2 a^2 (A+B) \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{a (A+B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{f \sqrt{c-c \sin (e+f x)}}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.709697, size = 136, normalized size = 0.94 \[ -\frac{(a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (4 (A+2 B) \sin (e+f x)+16 (A+B) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-B \cos (2 (e+f x))\right )}{4 f \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.342, size = 495, normalized size = 3.4 \begin{align*} \text{result too large to display} \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 \frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B a \cos \left (f x + e\right )^{2} -{\left (A + B\right )} a \sin \left (f x + e\right ) -{\left (A + B\right )} a\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{c \sin \left (f x + e\right ) - c}, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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