Integrand size = 38, antiderivative size = 91 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx=\frac {(A+B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} a \sqrt {c} f}-\frac {(A-B) \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a c f} \]
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Time = 0.21 (sec) , antiderivative size = 91, 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, 2728, 212} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx=\frac {(A+B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} a \sqrt {c} f}-\frac {(A-B) \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a c f} \]
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Rule 212
Rule 2728
Rule 2934
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^2(e+f x) (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx}{a c} \\ & = -\frac {(A-B) \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a c f}+\frac {(A+B) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{2 a} \\ & = -\frac {(A-B) \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a c f}-\frac {(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 )}{a f} \\ & = \frac {(A+B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} a \sqrt {c} f}-\frac {(A-B) \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a c f} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.54 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-A+B-(1+i) \sqrt [4]{-1} (A+B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )}{a f (1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.88 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {\left (\sin \left (f x +e \right )-1\right ) \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, A +\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, B -2 \sqrt {c}\, A +2 \sqrt {c}\, B \right )}{2 a \sqrt {c}\, \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(130\) |
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Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.78 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\sqrt {2} {\left (A + B\right )} \sqrt {c} \cos \left (f x + e\right ) \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, \sqrt {-c \sin \left (f x + e\right ) + c} {\left (A - B\right )}}{4 \, a c f \cos \left (f x + e\right )} \]
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\[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\int \frac {A}{\sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin {\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx}{a} \]
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\[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {B \sin \left (f x + e\right ) + A}{{\left (a \sin \left (f x + e\right ) + a\right )} \sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.63 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\frac {\sqrt {2} {\left (A \sqrt {c} + B \sqrt {c}\right )} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{a c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {4 \, \sqrt {2} {\left (A \sqrt {c} - B \sqrt {c}\right )}}{a c {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{4 \, f} \]
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Timed out. \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {A+B\,\sin \left (e+f\,x\right )}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]
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