Optimal. Leaf size=53 \[ \frac {\sqrt {\sin (2 a+2 b x)} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2573, 2641} \[ \frac {\sqrt {\sin (2 a+2 b x)} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{b \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2573
Rule 2641
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \, dx &=\frac {\sqrt {\sin (2 a+2 b x)} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}\\ &=\frac {F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)}}{b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 65, normalized size = 1.23 \[ \frac {2 \cos ^2(a+b x)^{3/4} \tan (a+b x) \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\sin ^2(a+b x)\right )}{b \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}{c d \cos \left (b x + a\right ) \sin \left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 151, normalized size = 2.85 \[ -\frac {\EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \left (\sin ^{2}\left (b x +a \right )\right ) \sqrt {2}}{b \sqrt {c \sin \left (b x +a \right )}\, \left (-1+\cos \left (b x +a \right )\right ) \sqrt {d \cos \left (b x +a \right )}} \]
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 \[ \int \frac {1}{\sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {d\,\cos \left (a+b\,x\right )}\,\sqrt {c\,\sin \left (a+b\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {c \sin {\left (a + b x \right )}} \sqrt {d \cos {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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