Integrand size = 37, antiderivative size = 192 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {2 a (B c-7 A d-6 B d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{105 d f \sqrt {a+a \sin (e+f x)}}+\frac {4 (5 c-d) (B c-7 A d-6 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}+\frac {2 d (B c-7 A d-6 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 a f}-\frac {2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a+a \sin (e+f x)}} \]
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Time = 0.25 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {3060, 2840, 2830, 2725} \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {2 a \left (15 c^2+10 c d+7 d^2\right ) (-7 A d+B c-6 B d) \cos (e+f x)}{105 d f \sqrt {a \sin (e+f x)+a}}+\frac {2 d (-7 A d+B c-6 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 a f}+\frac {4 (5 c-d) (-7 A d+B c-6 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{105 f}-\frac {2 a B \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}} \]
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Rule 2725
Rule 2830
Rule 2840
Rule 3060
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.84 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (420 A c^2+280 B c^2+560 A c d+532 B c d+266 A d^2+228 B d^2-6 d (14 B c+7 A d+6 B d) \cos (2 (e+f x))+\left (56 A d (5 c+2 d)+B \left (140 c^2+224 c d+141 d^2\right )\right ) \sin (e+f x)-15 B d^2 \sin (3 (e+f x))\right )}{210 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 1.76 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (-15 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) d^{2}+\left (-21 A \,d^{2}-42 c d B -18 d^{2} B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (70 A c d +28 A \,d^{2}+35 B \,c^{2}+56 c d B +39 d^{2} B \right ) \sin \left (f x +e \right )+105 A \,c^{2}+140 A c d +77 A \,d^{2}+70 B \,c^{2}+154 c d B +66 d^{2} B \right )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(161\) |
parts | \(\frac {2 A \,c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) a}{\cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 c \left (2 d A +B c \right ) \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+2\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d \left (d A +2 B c \right ) \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )+4 \sin \left (f x +e \right )+8\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d^{2} B \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (5 \left (\sin ^{3}\left (f x +e \right )\right )+6 \left (\sin ^{2}\left (f x +e \right )\right )+8 \sin \left (f x +e \right )+16\right )}{35 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(254\) |
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Time = 0.28 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.59 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {2 \, {\left (15 \, B d^{2} \cos \left (f x + e\right )^{4} + 3 \, {\left (14 \, B c d + {\left (7 \, A + 6 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{3} - 35 \, {\left (3 \, A + B\right )} c^{2} - 14 \, {\left (5 \, A + 7 \, B\right )} c d - {\left (49 \, A + 27 \, B\right )} d^{2} - {\left (35 \, B c^{2} + 14 \, {\left (5 \, A + B\right )} c d + {\left (7 \, A + 36 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (35 \, {\left (3 \, A + 2 \, B\right )} c^{2} + 14 \, {\left (10 \, A + 11 \, B\right )} c d + 11 \, {\left (7 \, A + 6 \, B\right )} d^{2}\right )} \cos \left (f x + e\right ) + {\left (15 \, B d^{2} \cos \left (f x + e\right )^{3} + 35 \, {\left (3 \, A + B\right )} c^{2} + 14 \, {\left (5 \, A + 7 \, B\right )} c d + {\left (49 \, A + 27 \, B\right )} d^{2} - 3 \, {\left (14 \, B c d + {\left (7 \, A + B\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (35 \, B c^{2} + 14 \, {\left (5 \, A + 4 \, B\right )} c d + {\left (28 \, A + 39 \, B\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{105 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{2}\, dx \]
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\[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{2} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.81 \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {\sqrt {2} {\left (15 \, B d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 105 \, {\left (8 \, A c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 4 \, B c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 8 \, A c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 8 \, B c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 4 \, A d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B d^{2} \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 ) + 35 \, {\left (4 \, B c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 8 \, A c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 4 \, B c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, A d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B d^{2} \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 ) + 21 \, {\left (4 \, B c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, A d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right )\right )} \sqrt {a}}{420 \, f} \]
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Timed out. \[ \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]
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