Optimal. Leaf size=43 \[ \frac {2 \sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}{c f g} \]
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Rubi [A] time = 0.20, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2930, 12, 30} \[ \frac {2 \sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}{c f g} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2930
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {g \sin (e+f x)} (c-c \sin (e+f x))} \, dx &=-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{c g x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{f}\\ &=-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{c f g}\\ &=\frac {2 \sec (e+f x) \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c f g}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 40, normalized size = 0.93 \[ \frac {2 \tan (e+f x) \sqrt {a (\sin (e+f x)+1)}}{c f \sqrt {g \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 41, normalized size = 0.95 \[ \frac {2 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {g \sin \left (f x + e\right )}}{c f g \cos \left (f x + e\right )} \]
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 \[ \int -\frac {\sqrt {a \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) - c\right )} \sqrt {g \sin \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 45, normalized size = 1.05 \[ \frac {2 \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sin \left (f x +e \right )}{c f \sqrt {g \sin \left (f x +e \right )}\, \cos \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 309, normalized size = 7.19 \[ -\frac {\frac {4 \, {\left ({\left (\frac {3 \, \sqrt {2} \sqrt {a} \sqrt {g} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sqrt {2} \sqrt {a} \sqrt {g} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \frac {2 \, {\left (\frac {3 \, \sqrt {2} \sqrt {a} \sqrt {g} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sqrt {2} \sqrt {a} \sqrt {g} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{\sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}\right )}}{c g - \frac {c g \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \frac {2 \, \sqrt {2} \sqrt {a} \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{\frac {3}{2}} + \frac {3 \, \sqrt {2} \sqrt {a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}{c \sqrt {g}} - \frac {2 \, \sqrt {2} \sqrt {a} \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{\frac {3}{2}} - \frac {3 \, \sqrt {2} \sqrt {a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}{c \sqrt {g}}}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.94, size = 52, normalized size = 1.21 \[ \frac {2\,\sin \left (2\,e+2\,f\,x\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}}{c\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,\sqrt {g\,\sin \left (e+f\,x\right )}} \]
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 \[ - \frac {\int \frac {\sqrt {a \sin {\left (e + f x \right )} + a}}{\sqrt {g \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} - \sqrt {g \sin {\left (e + f x \right )}}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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