Optimal. Leaf size=12 \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{b} \]
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Rubi [A] time = 0.0042156, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3770} \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \csc (a+b x) \, dx &=-\frac{\tanh ^{-1}(\cos (a+b x))}{b}\\ \end{align*}
Mathematica [B] time = 0.0181638, size = 38, normalized size = 3.17 \[ \frac{\log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\log \left (\cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 20, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( \csc \left ( bx+a \right ) +\cot \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05517, size = 26, normalized size = 2.17 \begin{align*} -\frac{\log \left (\cot \left (b x + a\right ) + \csc \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.473559, size = 93, normalized size = 7.75 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.51486, size = 37, normalized size = 3.08 \begin{align*} \begin{cases} - \frac{\log{\left (\cot{\left (a + b x \right )} + \csc{\left (a + b x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x \left (\cot{\left (a \right )} \csc{\left (a \right )} + \csc ^{2}{\left (a \right )}\right )}{\cot{\left (a \right )} + \csc{\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 = 1.30093, size = 20, normalized size = 1.67 \begin{align*} \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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