Optimal. Leaf size=192 \[ \frac{2 a \left (15 c^2+10 c d+7 d^2\right ) (-7 A d+B c-6 B d) \cos (e+f x)}{105 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 d (-7 A d+B c-6 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 a f}+\frac{4 (5 c-d) (-7 A d+B c-6 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.339343, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {2981, 2761, 2751, 2646} \[ \frac{2 a \left (15 c^2+10 c d+7 d^2\right ) (-7 A d+B c-6 B d) \cos (e+f x)}{105 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 d (-7 A d+B c-6 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 a f}+\frac{4 (5 c-d) (-7 A d+B c-6 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2981
Rule 2761
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))^2 \, dx &=-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt{a+a \sin (e+f x)}}+\frac{(7 a A d-B (a c-6 a d)) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{7 a d}\\ &=\frac{2 d (B c-7 A d-6 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 a f}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt{a+a \sin (e+f x)}}+\frac{(2 (7 a A d-B (a c-6 a d))) \int \sqrt{a+a \sin (e+f x)} \left (\frac{1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{35 a^2 d}\\ &=\frac{4 (5 c-d) (B c-7 A d-6 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{105 f}+\frac{2 d (B c-7 A d-6 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 a f}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt{a+a \sin (e+f x)}}+\frac{\left (\left (15 c^2+10 c d+7 d^2\right ) (7 a A d-B (a c-6 a d))\right ) \int \sqrt{a+a \sin (e+f x)} \, dx}{105 a d}\\ &=\frac{2 a (B c-7 A d-6 B d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{105 d f \sqrt{a+a \sin (e+f x)}}+\frac{4 (5 c-d) (B c-7 A d-6 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{105 f}+\frac{2 d (B c-7 A d-6 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 a f}-\frac{2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.746042, size = 176, normalized size = 0.92 \[ -\frac{\sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\left (56 A d (5 c+2 d)+B \left (140 c^2+224 c d+141 d^2\right )\right ) \sin (e+f x)-6 d (7 A d+14 B c+6 B d) \cos (2 (e+f x))+420 A c^2+560 A c d+266 A d^2+280 B c^2+532 B c d-15 B d^2 \sin (3 (e+f x))+228 B d^2\right )}{210 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.049, size = 161, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ) a \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( -15\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ){d}^{2}+ \left ( 70\,Acd+28\,A{d}^{2}+35\,B{c}^{2}+56\,Bcd+39\,B{d}^{2} \right ) \sin \left ( fx+e \right ) + \left ( -21\,A{d}^{2}-42\,Bcd-18\,B{d}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+105\,A{c}^{2}+140\,Acd+77\,A{d}^{2}+70\,B{c}^{2}+154\,Bcd+66\,B{d}^{2} \right ) }{105\,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 )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00698, size = 761, normalized size = 3.96 \begin{align*} \frac{2 \,{\left (15 \, B d^{2} \cos \left (f x + e\right )^{4} + 3 \,{\left (14 \, B c d +{\left (7 \, A + 6 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{3} - 35 \,{\left (3 \, A + B\right )} c^{2} - 14 \,{\left (5 \, A + 7 \, B\right )} c d -{\left (49 \, A + 27 \, B\right )} d^{2} -{\left (35 \, B c^{2} + 14 \,{\left (5 \, A + B\right )} c d +{\left (7 \, A + 36 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} -{\left (35 \,{\left (3 \, A + 2 \, B\right )} c^{2} + 14 \,{\left (10 \, A + 11 \, B\right )} c d + 11 \,{\left (7 \, A + 6 \, B\right )} d^{2}\right )} \cos \left (f x + e\right ) +{\left (15 \, B d^{2} \cos \left (f x + e\right )^{3} + 35 \,{\left (3 \, A + B\right )} c^{2} + 14 \,{\left (5 \, A + 7 \, B\right )} c d +{\left (49 \, A + 27 \, B\right )} d^{2} - 3 \,{\left (14 \, B c d +{\left (7 \, A + B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} -{\left (35 \, B c^{2} + 14 \,{\left (5 \, A + 4 \, B\right )} c d +{\left (28 \, A + 39 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{105 \,{\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 )^{2}\, 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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