Optimal. Leaf size=81 \[ \frac{\sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{2 b}-\frac{\log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{2 b} \]
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Rubi [A] time = 0.0722475, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4308, 4301, 4306} \[ \frac{\sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{2 b}-\frac{\log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 4308
Rule 4301
Rule 4306
Rubi steps
\begin{align*} \int \csc (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx &=2 \int \cos (a+b x) \sqrt{\sin (2 a+2 b x)} \, dx\\ &=\frac{\sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}+\int \frac{\sin (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{2 b}-\frac{\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}\right )}{2 b}+\frac{\sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}\\ \end{align*}
Mathematica [A] time = 0.0863831, size = 70, normalized size = 0.86 \[ -\frac{-2 \sin (a+b x) \sqrt{\sin (2 (a+b x))}+\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left (\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.805, size = 362, normalized size = 4.5 \begin{align*} 4\,{\frac{ \left ( \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( 2\,\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 ( 1/2\,bx+a/2 \right ) \left ( \tan \left ( 1/2\,bx+a/2 \right ) -1 \right ) \left ( \tan \left ( 1/2\,bx+a/2 \right ) +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 EllipticF} \left ( \sqrt{\tan \left ( 1/2\,bx+a/2 \right ) +1},1/2\,\sqrt{2} \right ) \sqrt{\tan \left ( 1/2\,bx+a/2 \right ) \left ( \tan \left ( 1/2\,bx+a/2 \right ) -1 \right ) \left ( \tan \left ( 1/2\,bx+a/2 \right ) +1 \right ) }+2\,\sqrt{ \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}-\tan \left ( 1/2\,bx+a/2 \right ) } \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{2} \right ) }{b\sqrt{\tan \left ( 1/2\,bx+a/2 \right ) \left ( \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }\sqrt{ \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}-\tan \left ( 1/2\,bx+a/2 \right ) }\sqrt{\tan \left ( 1/2\,bx+a/2 \right ) \left ( \tan \left ( 1/2\,bx+a/2 \right ) -1 \right ) \left ( \tan \left ( 1/2\,bx+a/2 \right ) +1 \right ) }}\sqrt{-{\frac{\tan \left ( 1/2\,bx+a/2 \right ) }{ \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \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 ) \sin \left (2 \, b x + 2 \, a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.541782, size = 734, normalized size = 9.06 \begin{align*} \frac{8 \, \sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} \sin \left (b x + a\right ) + 2 \, \arctan \left (-\frac{\sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )}{\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 2 \, \arctan \left (-\frac{2 \, \sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) + \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt{2}{\left (4 \, \cos \left (b x + a\right )^{3} -{\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{8 \, b} \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 ) \sin \left (2 \, b x + 2 \, a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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