Optimal. Leaf size=81 \[ \frac{32 \sin (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}-\frac{16 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{7 b \sqrt{\sin (2 a+2 b x)}} \]
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Rubi [A] time = 0.0918782, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4300, 4308, 4303, 4292} \[ \frac{32 \sin (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}-\frac{16 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{7 b \sqrt{\sin (2 a+2 b x)}} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 4308
Rule 4303
Rule 4292
Rubi steps
\begin{align*} \int \frac{\csc ^3(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx &=-\frac{\csc ^3(a+b x)}{7 b \sqrt{\sin (2 a+2 b x)}}+\frac{8}{7} \int \frac{\csc (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\csc ^3(a+b x)}{7 b \sqrt{\sin (2 a+2 b x)}}+\frac{16}{7} \int \frac{\cos (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{16 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{7 b \sqrt{\sin (2 a+2 b x)}}+\frac{32}{21} \int \frac{\sin (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{16 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{7 b \sqrt{\sin (2 a+2 b x)}}+\frac{32 \sin (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.116, size = 55, normalized size = 0.68 \[ \frac{\sqrt{\sin (2 (a+b x))} (-12 \cos (2 (a+b x))+4 \cos (4 (a+b x))+5) \csc ^4(a+b x) \sec (a+b x)}{42 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 9.663, size = 222, normalized size = 2.7 \begin{align*} -{\frac{1}{336\,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 ( -3\, \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{8}+16\,\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 ) \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}-2\, \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}+2\, \left ( \tan \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+3 \right ) \left ( \tan \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-3}{\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 )^{3}}{\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 = 0.506828, size = 281, normalized size = 3.47 \begin{align*} \frac{32 \, \cos \left (b x + a\right )^{5} - 64 \, \cos \left (b x + a\right )^{3} + \sqrt{2}{\left (32 \, \cos \left (b x + a\right )^{4} - 56 \, \cos \left (b x + a\right )^{2} + 21\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )}{42 \,{\left (b \cos \left (b x + a\right )^{5} - 2 \, 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 )^{3}}{\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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