Optimal. Leaf size=267 \[ \frac {g \sqrt {\frac {\sin (e+f x)}{\sin (e+f x)+1}} \sqrt {a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac {\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac {a-b}{a+b}\right )}{c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}+\frac {2 \sqrt {g} \sec (e+f x) \sqrt {\frac {a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt {\frac {a (\sin (e+f x)+1)}{a+b \sin (e+f x)}} (a+b \sin (e+f x)) \Pi \left (\frac {b}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {a+b \sin (e+f x)}}\right )|-\frac {a-b}{a+b}\right )}{c f \sqrt {a+b}} \]
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Rubi [A] time = 0.50, antiderivative size = 267, 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 = {2928, 2811, 2932} \[ \frac {g \sqrt {\frac {\sin (e+f x)}{\sin (e+f x)+1}} \sqrt {a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac {\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac {a-b}{a+b}\right )}{c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}+\frac {2 \sqrt {g} \sec (e+f x) \sqrt {\frac {a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt {\frac {a (\sin (e+f x)+1)}{a+b \sin (e+f x)}} (a+b \sin (e+f x)) \Pi \left (\frac {b}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {a+b \sin (e+f x)}}\right )|-\frac {a-b}{a+b}\right )}{c f \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 2811
Rule 2928
Rule 2932
Rubi steps
\begin {align*} \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx &=-\left (g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c+c \sin (e+f x))} \, dx\right )+\frac {g \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {g \sin (e+f x)}} \, dx}{c}\\ &=\frac {2 \sqrt {g} \Pi \left (\frac {b}{a+b};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {g \sin (e+f x)}}{\sqrt {g} \sqrt {a+b \sin (e+f x)}}\right )|-\frac {a-b}{a+b}\right ) \sec (e+f x) \sqrt {\frac {a (1-\sin (e+f x))}{a+b \sin (e+f x)}} \sqrt {\frac {a (1+\sin (e+f x))}{a+b \sin (e+f x)}} (a+b \sin (e+f x))}{\sqrt {a+b} c f}+\frac {g E\left (\sin ^{-1}\left (\frac {\cos (e+f x)}{1+\sin (e+f x)}\right )|-\frac {a-b}{a+b}\right ) \sqrt {\frac {\sin (e+f x)}{1+\sin (e+f x)}} \sqrt {a+b \sin (e+f x)}}{c f \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}}\\ \end {align*}
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Mathematica [C] time = 34.48, size = 13199, normalized size = 49.43 \[ \text {Result too large to show} \]
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 \[ \int \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{c \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.39, size = 22962, normalized size = 86.00 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{c \sin \left (f x + e\right ) + c}\,{d x} \]
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 \[ \int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {a+b\,\sin \left (e+f\,x\right )}}{c+c\,\sin \left (e+f\,x\right )} \,d x \]
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 \[ \frac {\int \frac {\sqrt {g \sin {\left (e + f x \right )}} \sqrt {a + b \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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