Optimal. Leaf size=15 \[ -\frac{\csc ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0184351, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2606, 30} \[ -\frac{\csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \cot (a+b x) \csc ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac{\csc ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0096359, size = 15, normalized size = 1. \[ -\frac{\csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 14, normalized size = 0.9 \begin{align*} -{\frac{1}{3\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973422, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{3 \, b \sin \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07392, size = 58, normalized size = 3.87 \begin{align*} \frac{1}{3 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.47669, size = 24, normalized size = 1.6 \begin{align*} \begin{cases} - \frac{1}{3 b \sin ^{3}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos{\left (a \right )}}{\sin ^{4}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11615, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{3 \, b \sin \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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