Integrand size = 36, antiderivative size = 116 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 a (7 A-B) c^3 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a (7 A-B) c^2 \cos ^3(e+f x)}{35 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a B c \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f} \]
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Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2935, 2753, 2752} \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 a c^3 (7 A-B) \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a c^2 (7 A-B) \cos ^3(e+f x)}{35 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a B c \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f} \]
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Rule 2752
Rule 2753
Rule 2935
Rule 3046
Rubi steps \begin{align*} \text {integral}& = (a c) \int \cos ^2(e+f x) (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx \\ & = -\frac {2 a B c \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f}+\frac {1}{7} (a (7 A-B) c) \int \cos ^2(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {2 a (7 A-B) c^2 \cos ^3(e+f x)}{35 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a B c \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f}+\frac {1}{35} \left (4 a (7 A-B) c^2\right ) \int \frac {\cos ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {8 a (7 A-B) c^3 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a (7 A-B) c^2 \cos ^3(e+f x)}{35 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a B c \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f} \\ \end{align*}
Time = 2.45 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {a c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (98 A-59 B+15 B \cos (2 (e+f x))+(-42 A+66 B) \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{105 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 1.60 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right )^{2} a \left (-15 B \left (\cos ^{2}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \left (21 A -33 B \right )-49 A +37 B \right )}{105 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(81\) |
parts | \(\frac {2 a A \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-5\right )}{3 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 B a \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (15 \left (\sin ^{3}\left (f x +e \right )\right )-39 \left (\sin ^{2}\left (f x +e \right )\right )+52 \sin \left (f x +e \right )-104\right )}{105 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a \left (A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )-3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(201\) |
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Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.59 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=-\frac {2 \, {\left (15 \, B a c \cos \left (f x + e\right )^{4} - 3 \, {\left (7 \, A - 6 \, B\right )} a c \cos \left (f x + e\right )^{3} + {\left (7 \, A - B\right )} a c \cos \left (f x + e\right )^{2} - 4 \, {\left (7 \, A - B\right )} a c \cos \left (f x + e\right ) - 8 \, {\left (7 \, A - B\right )} a c - {\left (15 \, B a c \cos \left (f x + e\right )^{3} + 3 \, {\left (7 \, A - B\right )} a c \cos \left (f x + e\right )^{2} + 4 \, {\left (7 \, A - B\right )} a c \cos \left (f x + e\right ) + 8 \, {\left (7 \, A - B\right )} a c\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=a \left (\int A c \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int \left (- A c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int B c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \left (- B c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\right )\, dx\right ) \]
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\[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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Time = 0.37 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.67 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {2} {\left (15 \, B a c \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 105 \, {\left (4 \, A a c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 35 \, {\left (2 \, A a c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 21 \, {\left (2 \, A a c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right )\right )} \sqrt {c}}{420 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
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