Integrand size = 38, antiderivative size = 118 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx=-\frac {4 (3 A-5 B) c^2 \cos (e+f x)}{3 a f \sqrt {c-c \sin (e+f x)}}-\frac {(3 A-5 B) c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{a c f} \]
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Time = 0.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2934, 2726, 2725} \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx=-\frac {4 c^2 (3 A-5 B) \cos (e+f x)}{3 a f \sqrt {c-c \sin (e+f x)}}-\frac {c (3 A-5 B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{a c f} \]
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Rule 2725
Rule 2726
Rule 2934
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx}{a c} \\ & = -\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{a c f}-\frac {(3 A-5 B) \int (c-c \sin (e+f x))^{3/2} \, dx}{2 a} \\ & = -\frac {(3 A-5 B) c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{a c f}-\frac {(2 (3 A-5 B) c) \int \sqrt {c-c \sin (e+f x)} \, dx}{3 a} \\ & = -\frac {4 (3 A-5 B) c^2 \cos (e+f x)}{3 a f \sqrt {c-c \sin (e+f x)}}-\frac {(3 A-5 B) c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{a c f} \\ \end{align*}
Time = 9.75 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx=\frac {c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-18 A+27 B+B \cos (2 (e+f x))+(-6 A+14 B) \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{3 a f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))} \]
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Time = 0.83 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {2 c^{2} \left (\sin \left (f x +e \right )-1\right ) \left (-B \left (\cos ^{2}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \left (3 A -7 B \right )+9 A -13 B \right )}{3 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(73\) |
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Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.57 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx=\frac {2 \, {\left (B c \cos \left (f x + e\right )^{2} - {\left (3 \, A - 7 \, B\right )} c \sin \left (f x + e\right ) - {\left (9 \, A - 13 \, B\right )} c\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a f \cos \left (f x + e\right )} \]
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\[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx=\frac {\int \frac {A c \sqrt {- c \sin {\left (e + f x \right )} + c}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {A c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {B c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {B c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\right )\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (108) = 216\).
Time = 0.33 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.49 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (3 \, c^{\frac {3}{2}} + \frac {2 \, c^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {6 \, c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {2 \, c^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, c^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} A}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (7 \, c^{\frac {3}{2}} + \frac {7 \, c^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {12 \, c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {7 \, c^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {7 \, c^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )} B}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}}\right )}}{3 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (108) = 216\).
Time = 0.42 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.99 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx=\frac {4 \, \sqrt {2} \sqrt {c} {\left (\frac {3 \, {\left (A c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )}}{a {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}} - \frac {3 \, A c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 7 \, B c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \frac {6 \, A c {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {18 \, B c {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {3 \, A c {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} - \frac {3 \, B c {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}}{a {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )}^{3}}\right )}}{3 \, f} \]
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Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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