Optimal. Leaf size=254 \[ \frac{2 (b c-a d) \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{c \csc (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 c}{c+d}\right )}{b f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 \sqrt{g} \sqrt{c+d} \tan (e+f x) \sqrt{\frac{c (1-\csc (e+f x))}{c+d}} \sqrt{\frac{c (\csc (e+f x)+1)}{c-d}} \Pi \left (\frac{c+d}{d};\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{g \sin (e+f x)}}\right )|-\frac{c+d}{c-d}\right )}{b f} \]
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Rubi [A] time = 0.504676, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2929, 2809, 2937} \[ \frac{2 (b c-a d) \tan (e+f x) \sqrt{-\cot ^2(e+f x)} \sqrt{g \sin (e+f x)} \sqrt{\frac{c \csc (e+f x)+d}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\csc (e+f x)}}{\sqrt{2}}\right )|\frac{2 c}{c+d}\right )}{b f (a+b) \sqrt{c+d \sin (e+f x)}}+\frac{2 \sqrt{g} \sqrt{c+d} \tan (e+f x) \sqrt{\frac{c (1-\csc (e+f x))}{c+d}} \sqrt{\frac{c (\csc (e+f x)+1)}{c-d}} \Pi \left (\frac{c+d}{d};\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{g \sin (e+f x)}}\right )|-\frac{c+d}{c-d}\right )}{b f} \]
Antiderivative was successfully verified.
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Rule 2929
Rule 2809
Rule 2937
Rubi steps
\begin{align*} \int \frac{\sqrt{g \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx &=\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}} \Pi \left (\frac{c+d}{d};\sin ^{-1}\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}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\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*}
Mathematica [C] time = 30.2847, size = 23019, normalized size = 90.63 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.457, size = 6211, normalized size = 24.5 \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{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\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 \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{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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