Integrand size = 35, antiderivative size = 398 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{a \sqrt {a+b} c f}-\frac {2 b \sqrt {a+b} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{a \sqrt {c+d} (b c-a d) f} \]
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Time = 0.40 (sec) , antiderivative size = 398, 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 = {3026, 2897, 3024} \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 b \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{a f \sqrt {c+d} (b c-a d)}-\frac {2 \sqrt {c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{a c f \sqrt {a+b}} \]
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Rule 2897
Rule 3024
Rule 3026
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{a}-\frac {b \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{a} \\ & = -\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))}{a \sqrt {a+b} c f}-\frac {2 b \sqrt {a+b} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \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))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{a \sqrt {c+d} (b c-a d) f} \\ \end{align*}
Time = 1.62 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.94 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 \sec (e+f x) \left (-\frac {(c+d) \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {\frac {(b c-a d) (-1+\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{c}-\frac {b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(-b c+a d) (-1+\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {\frac {(-b c+a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b c-a d}\right )}{a \sqrt {a+b} \sqrt {c+d} f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(24289\) vs. \(2(368)=736\).
Time = 7.12 (sec) , antiderivative size = 24290, normalized size of antiderivative = 61.03
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Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a + b \sin {\left (e + f x \right )}} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}\, dx \]
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\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )} \,d x } \]
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\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )} \,d x } \]
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Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
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