Integrand size = 35, antiderivative size = 391 \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=-\frac {2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} f}+\frac {2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} f} \]
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Time = 0.37 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3028, 2890, 3024} \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\frac {2 \sqrt {c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f \sqrt {a+b}}-\frac {2 \sqrt {c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f \sqrt {a+b}} \]
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Rule 2890
Rule 3024
Rule 3028
Rubi steps \begin{align*} \text {integral}& = c \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx+d \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx \\ & = -\frac {2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} f}+\frac {2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} f} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.70 \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=-\frac {2 \sqrt {c+d} \left (\operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )-\operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (-1+\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} f} \]
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Result contains complex when optimal does not.
Time = 7.63 (sec) , antiderivative size = 242134, normalized size of antiderivative = 619.27
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Timed out. \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\text {Timed out} \]
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\[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int \frac {\sqrt {a + b \sin {\left (e + f x \right )}} \sqrt {c + d \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )}}\, dx \]
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\[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{\sin \left (f x + e\right )} \,d x } \]
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\[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{\sin \left (f x + e\right )} \,d x } \]
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Timed out. \[ \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx=\int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )} \,d x \]
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