Integrand size = 35, antiderivative size = 118 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {2 a (15 A c+5 B c+5 A d+7 B d) \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {2 (5 B c+5 A d-2 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f} \]
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Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3047, 3102, 2830, 2725} \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {2 (5 A d+5 B c-2 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 f}-\frac {2 a (15 A c+5 A d+5 B c+7 B d) \cos (e+f x)}{15 f \sqrt {a \sin (e+f x)+a}}-\frac {2 B d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f} \]
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Rule 2725
Rule 2830
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.74 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.99 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (30 A c+20 B c+20 A d+19 B d-3 B d \cos (2 (e+f x))+2 (5 B c+5 A d+4 B d) \sin (e+f x))}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 1.54 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (3 B \left (\sin ^{2}\left (f x +e \right )\right ) d +5 A \sin \left (f x +e \right ) d +5 B \sin \left (f x +e \right ) c +4 B \sin \left (f x +e \right ) d +15 A c +10 d A +10 B c +8 d B \right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(102\) |
parts | \(\frac {2 A c \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) a}{\cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 \left (d A +B c \right ) \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+2\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d B \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )+4 \sin \left (f x +e \right )+8\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(167\) |
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Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.48 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {2 \, {\left (3 \, B d \cos \left (f x + e\right )^{3} - {\left (5 \, B c + {\left (5 \, A + B\right )} d\right )} \cos \left (f x + e\right )^{2} - 5 \, {\left (3 \, A + B\right )} c - {\left (5 \, A + 7 \, B\right )} d - {\left (5 \, {\left (3 \, A + 2 \, B\right )} c + {\left (10 \, A + 11 \, B\right )} d\right )} \cos \left (f x + e\right ) - {\left (3 \, B d \cos \left (f x + e\right )^{2} - 5 \, {\left (3 \, A + B\right )} c - {\left (5 \, A + 7 \, B\right )} d + {\left (5 \, B c + {\left (5 \, A + 4 \, B\right )} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{15 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )\, dx \]
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\[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.58 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {\sqrt {2} {\left (3 \, B d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 30 \, {\left (2 \, A c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + A d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, {\left (2 \, B c \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, A d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right )\right )} \sqrt {a}}{30 \, f} \]
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Timed out. \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \]
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