Optimal. Leaf size=24 \[ -\frac{\sqrt{\sin (2 a+2 b x)} \csc (a+b x)}{b} \]
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Rubi [A] time = 0.0238121, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {4292} \[ -\frac{\sqrt{\sin (2 a+2 b x)} \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 4292
Rubi steps
\begin{align*} \int \frac{\csc (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx &=-\frac{\csc (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}\\ \end{align*}
Mathematica [A] time = 0.0467407, size = 23, normalized size = 0.96 \[ -\frac{\sqrt{\sin (2 (a+b x))} \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.546, size = 308, normalized size = 12.8 \begin{align*}{\frac{1}{b}\sqrt{-{\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \left ( \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}-1 \right ) ^{-1}}} \left ( 2\,\sqrt{\tan \left ( 1/2\,bx+a/2 \right ) \left ( \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }\sqrt{\tan \left ( 1/2\,bx+a/2 \right ) +1}\sqrt{-2\,\tan \left ( 1/2\,bx+a/2 \right ) +2}\sqrt{-\tan \left ( 1/2\,bx+a/2 \right ) }{\it EllipticE} \left ( \sqrt{\tan \left ( 1/2\,bx+a/2 \right ) +1},1/2\,\sqrt{2} \right ) -\sqrt{\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \left ( \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}-1 \right ) }\sqrt{\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1}\sqrt{-2\,\tan \left ( 1/2\,bx+a/2 \right ) +2}\sqrt{-\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) }{\it EllipticF} \left ( \sqrt{\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1},{\frac{\sqrt{2}}{2}} \right ) +\sqrt{ \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) } \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}-\sqrt{ \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) } \right ) \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{ \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{bx}{2}}+{\frac{a}{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 \frac{\csc \left (b x + a\right )}{\sqrt{\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.491203, size = 103, normalized size = 4.29 \begin{align*} -\frac{\sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + \sin \left (b x + a\right )}{b \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\csc \left (b x + a\right )}{\sqrt{\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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