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 = 0.0554808, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, 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*}
Mathematica [C] time = 0.0573399, 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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Maple [B] time = 0.089, size = 151, normalized size = 2.9 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{b \left ( -1+\cos \left ( bx+a \right ) \right ) }\sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{d\cos \left ( bx+a \right ) }}}{\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \sin{\left (a + b x \right )}} \sqrt{d \cos{\left (a + b x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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