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 = 0.50, antiderivative size = 254, normalized size of antiderivative = 1.00, 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 2809
Rule 2929
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*}
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Mathematica [C] time = 30.55, size = 23019, normalized size = 90.63 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \sin \left (f x + e\right ) + c} \sqrt {g \sin \left (f x + e\right )}}{b \sin \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.99, size = 6052, normalized size = 23.83 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \sin \left (f x + e\right ) + c} \sqrt {g \sin \left (f x + e\right )}}{b \sin \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}{a+b\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {g \sin {\left (e + f x \right )}} \sqrt {c + d \sin {\left (e + f x \right )}}}{a + b \sin {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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