Optimal. Leaf size=15 \[ -\frac{\csc ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0178605, 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} \[ -\frac{\csc ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \cot (a+b x) \csc ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}(\int x \, dx,x,\csc (a+b x))}{b}\\ &=-\frac{\csc ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0121944, size = 15, normalized size = 1. \[ -\frac{\csc ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 14, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961035, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{2 \, b \sin \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48954, size = 38, normalized size = 2.53 \begin{align*} \frac{1}{2 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.976703, size = 24, normalized size = 1.6 \begin{align*} \begin{cases} - \frac{1}{2 b \sin ^{2}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos{\left (a \right )}}{\sin ^{3}{\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 = 1.13086, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{2 \, b \sin \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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