Optimal. Leaf size=28 \[ -\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0281631, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {4292} \[ -\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4292
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx &=-\frac{\csc ^3(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0493045, size = 27, normalized size = 0.96 \[ -\frac{\sin ^{\frac{3}{2}}(2 (a+b x)) \csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 3.405, size = 192, normalized size = 6.9 \begin{align*}{\frac{1}{3\,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 ( \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}-1 \right ) \left ( 4\,\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 EllipticF} \left ( \sqrt{\tan \left ( 1/2\,bx+a/2 \right ) +1},1/2\,\sqrt{2} \right ) \tan \left ( 1/2\,bx+a/2 \right ) + \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{4}-1 \right ) \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\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 ) }}}{\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 \csc \left (b x + a\right )^{3} \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 [B] time = 0.487701, size = 140, normalized size = 5. \begin{align*} \frac{2 \,{\left (\sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} \cos \left (b x + a\right ) + \cos \left (b x + a\right )^{2} - 1\right )}}{3 \,{\left (b \cos \left (b x + a\right )^{2} - b\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 \csc \left (b x + a\right )^{3} \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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