Optimal. Leaf size=27 \[ \frac{8 \sin (a+b x)}{b}-\frac{8 \sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0389608, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4288, 2633} \[ \frac{8 \sin (a+b x)}{b}-\frac{8 \sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2633
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^3(a+b x) \, dx\\ &=-\frac{8 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac{8 \sin (a+b x)}{b}-\frac{8 \sin ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0114353, size = 28, normalized size = 1.04 \[ 8 \left (\frac{\sin (a+b x)}{b}-\frac{\sin ^3(a+b x)}{3 b}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 22, normalized size = 0.8 \begin{align*}{\frac{ \left ( 16+8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06617, size = 31, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (\sin \left (3 \, b x + 3 \, a\right ) + 9 \, \sin \left (b x + a\right )\right )}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.473609, size = 55, normalized size = 2.04 \begin{align*} \frac{8 \,{\left (\cos \left (b x + a\right )^{2} + 2\right )} \sin \left (b x + a\right )}{3 \, 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 [A] time = 1.35142, size = 30, normalized size = 1.11 \begin{align*} -\frac{8 \,{\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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