Optimal. Leaf size=252 \[ \frac{g \sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac{a-b}{a+b}\right )}{c f (a-b) \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}-\frac{2 \sqrt{g} \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{c f (a-b)} \]
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Rubi [A] time = 0.511218, antiderivative size = 252, 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 = {2936, 2816, 2932} \[ \frac{g \sqrt{\frac{\sin (e+f x)}{\sin (e+f x)+1}} \sqrt{a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac{\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac{a-b}{a+b}\right )}{c f (a-b) \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}-\frac{2 \sqrt{g} \sqrt{a+b} \tan (e+f x) \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right )}{c f (a-b)} \]
Antiderivative was successfully verified.
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Rule 2936
Rule 2816
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 &=-\frac{g \int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+c \sin (e+f x))} \, dx}{a-b}+\frac{(a g) \int \frac{1}{\sqrt{g \sin (e+f x)} \sqrt{a+b \sin (e+f x)}} \, dx}{(a-b) c}\\ &=\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)}}{(a-b) c f \sqrt{g \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a+b) (1+\sin (e+f x))}}}-\frac{2 \sqrt{a+b} \sqrt{g} \sqrt{\frac{a (1-\csc (e+f x))}{a+b}} \sqrt{\frac{a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{g} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{g \sin (e+f x)}}\right )|-\frac{a+b}{a-b}\right ) \tan (e+f x)}{(a-b) c f}\\ \end{align*}
Mathematica [B] time = 33.5851, size = 5708, normalized size = 22.65 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.442, size = 6817, normalized size = 27.1 \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{g \sin \left (f x + e\right )}}{\sqrt{b \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\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{g \sin \left (f x + e\right )}}{b c \cos \left (f x + e\right )^{2} -{\left (a + b\right )} c \sin \left (f x + e\right ) -{\left (a + b\right )} c}, 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*} \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 )} + \sqrt{a + b \sin{\left (e + f x \right )}}}\, dx}{c} \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{g \sin \left (f x + e\right )}}{\sqrt{b \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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