Optimal. Leaf size=252 \[ \frac {g \sqrt {\frac {\sin (e+f x)}{\sin (e+f x)+1}} \sqrt {a+b \sin (e+f x)} E\left (\sin ^{-1}\left (\frac {\cos (e+f x)}{\sin (e+f x)+1}\right )|-\frac {a-b}{a+b}\right )}{c f (a-b) \sqrt {g \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{(a+b) (\sin (e+f x)+1)}}}-\frac {2 \sqrt {g} \sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {g} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {g \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{c f (a-b)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.51, antiderivative size = 252, 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 = {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.
[In]
[Out]
Rule 2816
Rule 2932
Rule 2936
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 = 33.62, size = 5708, normalized size = 22.65 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{b c \cos \left (f x + e\right )^{2} - {\left (a + b\right )} c \sin \left (f x + e\right ) - {\left (a + b\right )} c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {g \sin \left (f x + e\right )}}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.91, size = 6817, normalized size = 27.05 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {g \sin \left (f x + e\right )}}{\sqrt {b \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {g\,\sin \left (e+f\,x\right )}}{\sqrt {a+b\,\sin \left (e+f\,x\right )}\,\left (c+c\,\sin \left (e+f\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {g \sin {\left (e + f x \right )}}}{\sqrt {a + b \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + \sqrt {a + b \sin {\left (e + f x \right )}}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________