Integrand size = 35, antiderivative size = 355 \[ \int \frac {(g+h x) \sqrt {a-a \sin (e+f x)}}{\sqrt {c+c \sin (e+f x)}} \, dx=-\frac {i a (g+h x)^2 \cos (e+f x)}{2 h \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {2 i a (g+h x) \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a (g+h x) \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a h \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {i a h \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {i a h \cos (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \]
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Time = 0.64 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4700, 6873, 12, 6874, 4266, 2317, 2438, 3800, 2221} \[ \int \frac {(g+h x) \sqrt {a-a \sin (e+f x)}}{\sqrt {c+c \sin (e+f x)}} \, dx=-\frac {2 i a (g+h x) \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {i a h \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \cos (e+f x)}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {i a h \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \cos (e+f x)}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {i a h \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right ) \cos (e+f x)}{2 f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {i a (g+h x)^2 \cos (e+f x)}{2 h \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {a (g+h x) \log \left (1+e^{2 i (e+f x)}\right ) \cos (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}} \]
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4266
Rule 4700
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \int (g+h x) \sec (e+f x) (a-a \sin (e+f x)) \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \int a (g+h x) \sec (e+f x) (1-\sin (e+f x)) \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int (g+h x) \sec (e+f x) (1-\sin (e+f x)) \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int ((g+h x) \sec (e+f x)-(g+h x) \tan (e+f x)) \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = \frac {(a \cos (e+f x)) \int (g+h x) \sec (e+f x) \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \int (g+h x) \tan (e+f x) \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {i a (g+h x)^2 \cos (e+f x)}{2 h \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {2 i a (g+h x) \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(2 i a \cos (e+f x)) \int \frac {e^{2 i (e+f x)} (g+h x)}{1+e^{2 i (e+f x)}} \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a h \cos (e+f x)) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a h \cos (e+f x)) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {i a (g+h x)^2 \cos (e+f x)}{2 h \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {2 i a (g+h x) \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a (g+h x) \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(i a h \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(i a h \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a h \cos (e+f x)) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {i a (g+h x)^2 \cos (e+f x)}{2 h \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {2 i a (g+h x) \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a (g+h x) \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a h \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {i a h \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(i a h \cos (e+f x)) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ & = -\frac {i a (g+h x)^2 \cos (e+f x)}{2 h \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {2 i a (g+h x) \arctan \left (e^{i (e+f x)}\right ) \cos (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a (g+h x) \cos (e+f x) \log \left (1+e^{2 i (e+f x)}\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a h \cos (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {i a h \cos (e+f x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {i a h \cos (e+f x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}} \\ \end{align*}
Time = 2.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.43 \[ \int \frac {(g+h x) \sqrt {a-a \sin (e+f x)}}{\sqrt {c+c \sin (e+f x)}} \, dx=\frac {\left (i+e^{i (e+f x)}\right ) \left (f \left (f x (2 g+h x)-4 i (g+h x) \log \left (1+i e^{-i (e+f x)}\right )\right )+4 h \operatorname {PolyLog}\left (2,-i e^{-i (e+f x)}\right )\right ) \sqrt {a-a \sin (e+f x)}}{\sqrt {2} \left (-i+e^{i (e+f x)}\right ) \sqrt {-i c e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2} f^2} \]
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\[\int \frac {\left (h x +g \right ) \sqrt {a -\sin \left (f x +e \right ) a}}{\sqrt {c +c \sin \left (f x +e \right )}}d x\]
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Exception generated. \[ \int \frac {(g+h x) \sqrt {a-a \sin (e+f x)}}{\sqrt {c+c \sin (e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(g+h x) \sqrt {a-a \sin (e+f x)}}{\sqrt {c+c \sin (e+f x)}} \, dx=\int \frac {\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )} \left (g + h x\right )}{\sqrt {c \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {(g+h x) \sqrt {a-a \sin (e+f x)}}{\sqrt {c+c \sin (e+f x)}} \, dx=\int { \frac {{\left (h x + g\right )} \sqrt {-a \sin \left (f x + e\right ) + a}}{\sqrt {c \sin \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {(g+h x) \sqrt {a-a \sin (e+f x)}}{\sqrt {c+c \sin (e+f x)}} \, dx=\int { \frac {{\left (h x + g\right )} \sqrt {-a \sin \left (f x + e\right ) + a}}{\sqrt {c \sin \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {(g+h x) \sqrt {a-a \sin (e+f x)}}{\sqrt {c+c \sin (e+f x)}} \, dx=\int \frac {\left (g+h\,x\right )\,\sqrt {a-a\,\sin \left (e+f\,x\right )}}{\sqrt {c+c\,\sin \left (e+f\,x\right )}} \,d x \]
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