Optimal. Leaf size=13 \[ -\frac{\csc (a+b x)}{4 b} \]
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Rubi [A] time = 0.0345041, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4287, 2606, 8} \[ -\frac{\csc (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 4287
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \cos ^3(a+b x) \csc ^2(2 a+2 b x) \, dx &=\frac{1}{4} \int \cot (a+b x) \csc (a+b x) \, dx\\ &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,\csc (a+b x))}{4 b}\\ &=-\frac{\csc (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0122181, size = 13, normalized size = 1. \[ -\frac{\csc (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{4\,b\sin \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06123, size = 113, normalized size = 8.69 \begin{align*} -\frac{\cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + \sin \left (b x + a\right )}{2 \,{\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.458795, size = 31, normalized size = 2.38 \begin{align*} -\frac{1}{4 \, b \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 [A] time = 1.23168, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{4 \, b \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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