Optimal. Leaf size=11 \[ \frac{\log (\sin (a+b x))}{b} \]
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Rubi [A] time = 0.0038209, antiderivative size = 11, 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 = {3475} \[ \frac{\log (\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin{align*} \int \cot (a+b x) \, dx &=\frac{\log (\sin (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0097907, size = 19, normalized size = 1.73 \[ \frac{\log (\tan (a+b x))+\log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 29, normalized size = 2.6 \begin{align*} -{\frac{\ln \left ( 1+ \left ( \tan \left ( bx+a \right ) \right ) ^{2} \right ) }{2\,b}}+{\frac{\ln \left ( \tan \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 = 0.963702, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (\sin \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 = 1.76488, size = 65, normalized size = 5.91 \begin{align*} \frac{\log \left (\frac{\tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} + 1}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.33813, size = 29, normalized size = 2.64 \begin{align*} \begin{cases} - \frac{\log{\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} + \frac{\log{\left (\tan{\left (a + b x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\tan{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13425, size = 76, normalized size = 6.91 \begin{align*} \frac{\log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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