Optimal. Leaf size=15 \[ -\frac{4 \cos ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0372745, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4288, 2565, 30} \[ -\frac{4 \cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2565
Rule 30
Rubi steps
\begin{align*} \int \csc (a+b x) \sin ^2(2 a+2 b x) \, dx &=4 \int \cos ^2(a+b x) \sin (a+b x) \, dx\\ &=-\frac{4 \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{4 \cos ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0068125, size = 15, normalized size = 1. \[ -\frac{4 \cos ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 14, normalized size = 0.9 \begin{align*} -{\frac{4\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17229, size = 31, normalized size = 2.07 \begin{align*} -\frac{\cos \left (3 \, b x + 3 \, a\right ) + 3 \, \cos \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.471947, size = 31, normalized size = 2.07 \begin{align*} -\frac{4 \, \cos \left (b x + a\right )^{3}}{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 [B] time = 1.26339, size = 70, normalized size = 4.67 \begin{align*} \frac{8 \,{\left (\frac{3 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}}{3 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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