Integrand size = 40, antiderivative size = 99 \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {a (A+B) \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {a B \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3050, 2816, 2746, 31, 2817} \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {a (A+B) \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {a B \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \]
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Rule 31
Rule 2746
Rule 2816
Rule 2817
Rule 3050
Rubi steps \begin{align*} \text {integral}& = (A+B) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{3/2}} \, dx-\frac {B \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c} \\ & = \frac {a (A+B) \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {(a B \cos (e+f x)) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {a (A+B) \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {(a B \cos (e+f x)) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {a (A+B) \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {a B \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.57 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (A+B-i B f x+2 B \log \left (i-e^{i (e+f x)}\right )+i B \left (f x+2 i \log \left (i-e^{i (e+f x)}\right )\right ) \sin (e+f x)\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^{3/2}} \]
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Time = 2.76 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.40
method | result | size |
default | \(\frac {\sec \left (f x +e \right ) \left (-2 B \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+B \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+A \sin \left (f x +e \right )+B \sin \left (f x +e \right )+2 B \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-B \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{c f \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(139\) |
parts | \(\frac {A \tan \left (f x +e \right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{f c \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}-\frac {B \sec \left (f x +e \right ) \left (2 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )-\ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )-\sin \left (f x +e \right )-2 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+\ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{f c \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(168\) |
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\[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (A + B \sin {\left (e + f x \right )}\right )}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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none
Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {{\left (4 \, B \sqrt {c} \log \left ({\left | \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + \frac {A \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}\right )} \sqrt {a}}{2 \, c^{2} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
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Timed out. \[ \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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