Optimal. Leaf size=107 \[ \frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{64 \cos (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
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Rubi [A] time = 0.115507, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4300, 4308, 4303, 4304, 4291} \[ \frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{64 \cos (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 4308
Rule 4303
Rule 4304
Rule 4291
Rubi steps
\begin{align*} \int \frac{\csc ^3(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx &=-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{4}{3} \int \frac{\csc (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{8}{3} \int \frac{\cos (a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{32}{15} \int \frac{\sin (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{64}{45} \int \frac{\cos (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{64 \cos (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [A] time = 0.091329, size = 62, normalized size = 0.58 \[ -\frac{\sqrt{\sin (2 (a+b x))} \left (5 \csc ^5(a+b x)+17 \csc ^3(a+b x)+113 \csc (a+b x)-15 \tan (a+b x) \sec (a+b x)\right )}{180 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 18.636, size = 560, normalized size = 5.2 \begin{align*} \text{result too large to display} \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.527352, size = 356, normalized size = 3.33 \begin{align*} -\frac{\sqrt{2}{\left (128 \, \cos \left (b x + a\right )^{6} - 288 \, \cos \left (b x + a\right )^{4} + 180 \, \cos \left (b x + a\right )^{2} - 15\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 128 \,{\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{180 \,{\left (b \cos \left (b x + a\right )^{6} - 2 \, b \cos \left (b x + a\right )^{4} + b \cos \left (b x + a\right )^{2}\right )} \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 )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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