Optimal. Leaf size=76 \[ \frac {\sqrt {2} \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}} F\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{d \sqrt {2 a+b \sin (2 c+2 d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2666, 2663, 2661} \[ \frac {\sqrt {2} \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}} F\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{d \sqrt {2 a+b \sin (2 c+2 d x)}} \]
Antiderivative was successfully verified.
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Rule 2661
Rule 2663
Rule 2666
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \cos (c+d x) \sin (c+d x)}} \, dx &=\int \frac {1}{\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \, dx\\ &=\frac {\sqrt {\frac {a+\frac {1}{2} b \sin (2 c+2 d x)}{a+\frac {b}{2}}} \int \frac {1}{\sqrt {\frac {a}{a+\frac {b}{2}}+\frac {b \sin (2 c+2 d x)}{2 \left (a+\frac {b}{2}\right )}}} \, dx}{\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}}\\ &=\frac {\sqrt {2} F\left (c-\frac {\pi }{4}+d x|\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}{d \sqrt {2 a+b \sin (2 c+2 d x)}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 70, normalized size = 0.92 \[ \frac {\sqrt {\frac {2 a+b \sin (2 (c+d x))}{2 a+b}} F\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{d \sqrt {a+\frac {1}{2} b \sin (2 (c+d x))}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 165, normalized size = 2.17 \[ \frac {2 \left (2 a -b \right ) \sqrt {\frac {2 a +b \sin \left (2 d x +2 c \right )}{2 a -b}}\, \sqrt {-\frac {\left (\sin \left (2 d x +2 c \right )-1\right ) b}{2 a +b}}\, \sqrt {-\frac {\left (1+\sin \left (2 d x +2 c \right )\right ) b}{2 a -b}}\, \EllipticF \left (\sqrt {\frac {2 a +b \sin \left (2 d x +2 c \right )}{2 a -b}}, \sqrt {\frac {2 a -b}{2 a +b}}\right )}{b \cos \left (2 d x +2 c \right ) \sqrt {4 a +2 b \sin \left (2 d x +2 c \right )}\, d} \]
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 \[ \int \frac {1}{\sqrt {b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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