Integrand size = 36, antiderivative size = 122 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} a (A+B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a (3 A+5 B) \cos (e+f x)}{3 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a B \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 c f} \]
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Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3046, 2937, 2830, 2728, 212} \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \sqrt {2} a (A+B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a (3 A+5 B) \cos (e+f x)}{3 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a B \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 c f} \]
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Rule 212
Rule 2728
Rule 2830
Rule 2937
Rule 3046
Rubi steps \begin{align*} \text {integral}& = (a c) \int \frac {\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {2 a B \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 c f}-\frac {(2 a) \int \frac {-\frac {3 A c}{2}-\frac {B c}{2}+\left (-\frac {3 A c}{2}-\frac {5 B c}{2}\right ) \sin (e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{3 c} \\ & = -\frac {2 a (3 A+5 B) \cos (e+f x)}{3 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a B \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 c f}+(2 a (A+B)) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {2 a (3 A+5 B) \cos (e+f x)}{3 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a B \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 c f}-\frac {(4 a (A+B)) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{f} \\ & = \frac {2 \sqrt {2} a (A+B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a (3 A+5 B) \cos (e+f x)}{3 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a B \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 c f} \\ \end{align*}
Time = 2.94 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.36 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (6 \sqrt {2} (A+B) \arctan \left (\frac {\sqrt {-c (1+\sin (e+f x))}}{\sqrt {2} \sqrt {c}}\right ) \sqrt {-c (1+\sin (e+f x))}+\sqrt {c} (6 A+9 B-B \cos (2 (e+f x))+2 (3 A+5 B) \sin (e+f x))\right )}{3 \sqrt {c} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)}} \]
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Time = 1.88 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.30
method | result | size |
default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, a \left (3 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) A +3 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) B -B \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}-3 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, A c -3 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, B c \right )}{3 c^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(159\) |
parts | \(-\frac {a A \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}\, \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {B a \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (3 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-2 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}\right )}{3 c^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {a \left (A +B \right ) \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \left (\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-2 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\right )}{c \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(268\) |
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (105) = 210\).
Time = 0.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.08 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\frac {3 \, \sqrt {2} {\left ({\left (A + B\right )} a c \cos \left (f x + e\right ) - {\left (A + B\right )} a c \sin \left (f x + e\right ) + {\left (A + B\right )} a c\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} + 2 \, {\left (B a \cos \left (f x + e\right )^{2} - {\left (3 \, A + 4 \, B\right )} a \cos \left (f x + e\right ) - {\left (3 \, A + 5 \, B\right )} a - {\left (B a \cos \left (f x + e\right ) + {\left (3 \, A + 5 \, B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\left (c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f\right )}} \]
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\[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=a \left (\int \frac {A}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {A \sin {\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin {\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin ^{2}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx\right ) \]
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\[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (105) = 210\).
Time = 0.33 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.46 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\frac {3 \, \sqrt {2} {\left (A a \sqrt {c} + B a \sqrt {c}\right )} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {4 \, \sqrt {2} {\left (3 \, A a \sqrt {c} + 5 \, B a \sqrt {c} - \frac {6 \, A a \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - \frac {6 \, B a \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {3 \, A a \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {9 \, B a \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )}}{c {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{3 \, f} \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (a+a\,\sin \left (e+f\,x\right )\right )}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]
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