Optimal. Leaf size=53 \[ \frac{\sqrt{\sin (2 a+2 b x)} \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right ) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{b} \]
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Rubi [A] time = 0.0927454, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2630, 2573, 2641} \[ \frac{\sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 2630
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \, dx &=\left (\sqrt{c \cos (a+b x)} \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{d \sin (a+b x)}\right ) \int \frac{1}{\sqrt{c \cos (a+b x)} \sqrt{d \sin (a+b x)}} \, dx\\ &=\left (\sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{\sin (2 a+2 b x)}\right ) \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=\frac{\sqrt{d \csc (a+b x)} F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{c \sec (a+b x)} \sqrt{\sin (2 a+2 b x)}}{b}\\ \end{align*}
Mathematica [C] time = 0.670832, size = 68, normalized size = 1.28 \[ \frac{\tan ^3(a+b x) \left (-\cot ^2(a+b x)\right )^{7/4} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{3}{2},\csc ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.196, size = 157, normalized size = 3. \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{b \left ( -1+\cos \left ( bx+a \right ) \right ) }\sqrt{{\frac{c}{\cos \left ( bx+a \right ) }}}\sqrt{{\frac{d}{\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 ) }{\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 ) } \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 \sqrt{d \csc \left (b x + a\right )} \sqrt{c \sec \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 (\sqrt{d \csc \left (b x + a\right )} \sqrt{c \sec \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 \sqrt{c \sec{\left (a + b x \right )}} \sqrt{d \csc{\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 \sqrt{d \csc \left (b x + a\right )} \sqrt{c \sec \left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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