Optimal. Leaf size=11 \[ -\frac{\csc (a+b x)}{b} \]
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Rubi [A] time = 0.0093198, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2606, 8} \[ -\frac{\csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \cot (a+b x) \csc (a+b x) \, dx &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,\csc (a+b x))}{b}\\ &=-\frac{\csc (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0083335, size = 11, normalized size = 1. \[ -\frac{\csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 14, normalized size = 1.3 \begin{align*} -{\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 [A] time = 0.994688, size = 18, normalized size = 1.64 \begin{align*} -\frac{1}{b \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81136, size = 28, normalized size = 2.55 \begin{align*} -\frac{1}{b \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 = 0.684761, size = 20, normalized size = 1.82 \begin{align*} \begin{cases} - \frac{1}{b \sin{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos{\left (a \right )}}{\sin ^{2}{\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 = 1.15179, size = 18, normalized size = 1.64 \begin{align*} -\frac{1}{b \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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