Optimal. Leaf size=118 \[ -\frac{2 (5 A d+5 B c-2 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 a (15 A c+5 A d+5 B c+7 B d) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 B d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f} \]
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Rubi [A] time = 0.249048, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {2968, 3023, 2751, 2646} \[ -\frac{2 (5 A d+5 B c-2 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{15 f}-\frac{2 a (15 A c+5 A d+5 B c+7 B d) \cos (e+f x)}{15 f \sqrt{a \sin (e+f x)+a}}-\frac{2 B d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3023
Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx &=\int \sqrt{a+a \sin (e+f x)} \left (A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)\right ) \, dx\\ &=-\frac{2 B d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}+\frac{2 \int \sqrt{a+a \sin (e+f x)} \left (\frac{1}{2} a (5 A c+3 B d)+\frac{1}{2} a (5 B c+5 A d-2 B d) \sin (e+f x)\right ) \, dx}{5 a}\\ &=-\frac{2 (5 B c+5 A d-2 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{15 f}-\frac{2 B d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}+\frac{1}{15} (15 A c+5 B c+5 A d+7 B d) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{2 a (15 A c+5 B c+5 A d+7 B d) \cos (e+f x)}{15 f \sqrt{a+a \sin (e+f x)}}-\frac{2 (5 B c+5 A d-2 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{15 f}-\frac{2 B d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}\\ \end{align*}
Mathematica [A] time = 0.36511, size = 117, normalized size = 0.99 \[ -\frac{\sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) (2 (5 A d+5 B c+4 B d) \sin (e+f x)+30 A c+20 A d+20 B c-3 B d \cos (2 (e+f x))+19 B d)}{15 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.954, size = 102, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ) a \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 3\,B \left ( \sin \left ( fx+e \right ) \right ) ^{2}d+5\,A\sin \left ( fx+e \right ) d+5\,B\sin \left ( fx+e \right ) c+4\,B\sin \left ( fx+e \right ) d+15\,Ac+10\,Ad+10\,Bc+8\,Bd \right ) }{15\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \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{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99923, size = 436, normalized size = 3.69 \begin{align*} \frac{2 \,{\left (3 \, B d \cos \left (f x + e\right )^{3} -{\left (5 \, B c +{\left (5 \, A + B\right )} d\right )} \cos \left (f x + e\right )^{2} - 5 \,{\left (3 \, A + B\right )} c -{\left (5 \, A + 7 \, B\right )} d -{\left (5 \,{\left (3 \, A + 2 \, B\right )} c +{\left (10 \, A + 11 \, B\right )} d\right )} \cos \left (f x + e\right ) -{\left (3 \, B d \cos \left (f x + e\right )^{2} - 5 \,{\left (3 \, A + B\right )} c -{\left (5 \, A + 7 \, B\right )} d +{\left (5 \, B c +{\left (5 \, A + 4 \, B\right )} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{15 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \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 \sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \left (A + B \sin{\left (e + f x \right )}\right ) \left (c + d \sin{\left (e + f x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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