Optimal. Leaf size=53 \[ \frac{4 \sin (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}}-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.0642377, antiderivative size = 53, 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, 4303, 4292} \[ \frac{4 \sin (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}}-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
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Rule 4308
Rule 4303
Rule 4292
Rubi steps
\begin{align*} \int \frac{\csc (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx &=2 \int \frac{\cos (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{4}{3} \int \frac{\sin (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{4 \sin (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.100315, size = 43, normalized size = 0.81 \[ \frac{\sqrt{\sin (2 (a+b x))} \left (\frac{1}{2} \sec (a+b x)-\frac{1}{6} \cot (a+b x) \csc (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 2.469, size = 194, normalized size = 3.7 \begin{align*} -{\frac{1}{12\,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 ( 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 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 \frac{\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 [A] time = 0.498337, size = 192, normalized size = 3.62 \begin{align*} \frac{4 \, \cos \left (b x + a\right )^{3} + \sqrt{2}{\left (4 \, \cos \left (b x + a\right )^{2} - 3\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} - 4 \, \cos \left (b x + a\right )}{6 \,{\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\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 )}{\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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