Optimal. Leaf size=31 \[ \frac{8 \sin ^3(a+b x)}{3 b}-\frac{8 \sin ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.0513627, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4288, 2564, 14} \[ \frac{8 \sin ^3(a+b x)}{3 b}-\frac{8 \sin ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2564
Rule 14
Rubi steps
\begin{align*} \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^3(a+b x) \sin ^2(a+b x) \, dx\\ &=\frac{8 \operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{8 \operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{8 \sin ^3(a+b x)}{3 b}-\frac{8 \sin ^5(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0475814, size = 28, normalized size = 0.9 \[ \frac{8 \left (5 \sin ^3(a+b x)-3 \sin ^5(a+b x)\right )}{15 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 41, normalized size = 1.3 \begin{align*} 8\,{\frac{-1/5\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}+1/15\, \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20096, size = 49, normalized size = 1.58 \begin{align*} -\frac{3 \, \sin \left (5 \, b x + 5 \, a\right ) + 5 \, \sin \left (3 \, b x + 3 \, a\right ) - 30 \, \sin \left (b x + a\right )}{30 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.477091, size = 84, normalized size = 2.71 \begin{align*} -\frac{8 \,{\left (3 \, \cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{15 \, 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 = 1.34838, size = 35, normalized size = 1.13 \begin{align*} -\frac{8 \,{\left (3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}\right )}}{15 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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