Optimal. Leaf size=62 \[ \frac {2 \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2663, 2661} \[ \frac {2 \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2661
Rule 2663
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sin (e+f x)}} \, dx &=\frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}} \, dx}{\sqrt {a+b \sin (e+f x)}}\\ &=\frac {2 F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.98 \[ -\frac {2 \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \sin \left (f x + e\right ) + a}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 126, normalized size = 2.03 \[ \frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (f x +e \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )}{b \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {1}{\sqrt {b \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.91, size = 55, normalized size = 0.89 \[ -\frac {2\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |\frac {2\,b}{a+b}\right )\,\sqrt {\frac {a+b\,\sin \left (e+f\,x\right )}{a+b}}}{f\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \]
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 \[ \int \frac {1}{\sqrt {a + b \sin {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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