Optimal. Leaf size=5 \[ -\tanh ^{-1}(\cos (x)) \]
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Rubi [A] time = 0.0022161, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3770} \[ -\tanh ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \csc (x) \, dx &=-\tanh ^{-1}(\cos (x))\\ \end{align*}
Mathematica [B] time = 0.0025744, size = 17, normalized size = 3.4 \[ \log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 9, normalized size = 1.8 \begin{align*} -\ln \left ( \csc \left ( x \right ) +\cot \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.930988, size = 11, normalized size = 2.2 \begin{align*} -\log \left (\cot \left (x\right ) + \csc \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15113, size = 77, normalized size = 15.4 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.093246, size = 15, normalized size = 3. \begin{align*} \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{2} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07252, size = 8, normalized size = 1.6 \begin{align*} \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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