Integrand size = 39, antiderivative size = 254 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 \sqrt {c+d} \sqrt {g} \sqrt {\frac {c (1-\csc (e+f x))}{c+d}} \sqrt {\frac {c (1+\csc (e+f x))}{c-d}} \operatorname {EllipticPi}\left (\frac {c+d}{d},\arcsin \left (\frac {\sqrt {g} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {g \sin (e+f x)}}\right ),-\frac {c+d}{c-d}\right ) \tan (e+f x)}{b f}+\frac {2 (b c-a d) \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {d+c \csc (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 c}{c+d}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{b (a+b) f \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.34 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3008, 2888, 3016} \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\frac {2 (b c-a d) \tan (e+f x) \sqrt {-\cot ^2(e+f x)} \sqrt {g \sin (e+f x)} \sqrt {\frac {c \csc (e+f x)+d}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 c}{c+d}\right )}{b f (a+b) \sqrt {c+d \sin (e+f x)}}+\frac {2 \sqrt {g} \sqrt {c+d} \tan (e+f x) \sqrt {\frac {c (1-\csc (e+f x))}{c+d}} \sqrt {\frac {c (\csc (e+f x)+1)}{c-d}} \operatorname {EllipticPi}\left (\frac {c+d}{d},\arcsin \left (\frac {\sqrt {g} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {g \sin (e+f x)}}\right ),-\frac {c+d}{c-d}\right )}{b f} \]
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Rule 2888
Rule 3008
Rule 3016
Rubi steps \begin{align*} \text {integral}& = \frac {d \int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{b}-\frac {(-b c+a d) \int \frac {\sqrt {g \sin (e+f x)}}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b} \\ & = \frac {2 \sqrt {c+d} \sqrt {g} \sqrt {\frac {c (1-\csc (e+f x))}{c+d}} \sqrt {\frac {c (1+\csc (e+f x))}{c-d}} \operatorname {EllipticPi}\left (\frac {c+d}{d},\arcsin \left (\frac {\sqrt {g} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {g \sin (e+f x)}}\right ),-\frac {c+d}{c-d}\right ) \tan (e+f x)}{b f}+\frac {2 (b c-a d) \sqrt {-\cot ^2(e+f x)} \sqrt {\frac {d+c \csc (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\csc (e+f x)}}{\sqrt {2}}\right ),\frac {2 c}{c+d}\right ) \sqrt {g \sin (e+f x)} \tan (e+f x)}{b (a+b) f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 32.30 (sec) , antiderivative size = 23019, normalized size of antiderivative = 90.63 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\text {Result too large to show} \]
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Result contains complex when optimal does not.
Time = 2.69 (sec) , antiderivative size = 5489, normalized size of antiderivative = 21.61
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Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {g \sin {\left (e + f x \right )}} \sqrt {c + d \sin {\left (e + f x \right )}}}{a + b \sin {\left (e + f x \right )}}\, dx \]
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\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c} \sqrt {g \sin \left (f x + e\right )}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int { \frac {\sqrt {d \sin \left (f x + e\right ) + c} \sqrt {g \sin \left (f x + e\right )}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx=\int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}{a+b\,\sin \left (e+f\,x\right )} \,d x \]
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