Optimal. Leaf size=11 \[ -\frac{2 \csc (a+b x)}{b} \]
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Rubi [A] time = 0.0255341, antiderivative size = 11, 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, 2606, 8} \[ -\frac{2 \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cot (a+b x) \csc (a+b x) \, dx\\ &=-\frac{2 \operatorname{Subst}(\int 1 \, dx,x,\csc (a+b x))}{b}\\ &=-\frac{2 \csc (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0103081, size = 11, normalized size = 1. \[ -\frac{2 \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 14, normalized size = 1.3 \begin{align*} -2\,{\frac{1}{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 = 1.03779, size = 113, normalized size = 10.27 \begin{align*} -\frac{4 \,{\left (\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 )\right )}}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.453849, size = 28, normalized size = 2.55 \begin{align*} -\frac{2}{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 = 1.2038, size = 18, normalized size = 1.64 \begin{align*} -\frac{2}{b \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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