Optimal. Leaf size=398 \[ -\frac{2 b \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{a f \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{a c f \sqrt{a+b}} \]
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Rubi [A] time = 0.55485, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {2947, 2818, 2945} \[ -\frac{2 b \sqrt{a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{a f \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{a c f \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 2947
Rule 2818
Rule 2945
Rubi steps
\begin{align*} \int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx &=\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} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\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} F\left (\sin ^{-1}\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*}
Mathematica [A] time = 2.33965, size = 374, normalized size = 0.94 \[ \frac{2 \sec (e+f x) \left (-\frac{b (a+b) (c+d \sin (e+f x)) \sqrt{\frac{(a d-b c) (\sin (e+f x)-1)}{(a+b) (c+d \sin (e+f x))}} \sqrt{\frac{(a d-b c) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{b c-a d}-\frac{(c+d) (a+b \sin (e+f x)) \sqrt{\frac{(b c-a d) (\sin (e+f x)-1)}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{c}\right )}{a f \sqrt{a+b} \sqrt{c+d}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.73, size = 26864, normalized size = 67.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b \sin \left (f x + e\right ) + a} \sqrt{d \sin \left (f x + e\right ) + c}}{{\left (b c + a d\right )} \cos \left (f x + e\right )^{2} - b c - a d +{\left (b d \cos \left (f x + e\right )^{2} - a c - b d\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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