Optimal. Leaf size=7 \[ \log (1-\cos (x)) \]
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Rubi [A] time = 0.013242, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2667, 31} \[ \log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 2667
Rule 31
Rubi steps
\begin{align*} \int (1+\cos (x)) \csc (x) \, dx &=-\operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\cos (x)\right )\\ &=\log (1-\cos (x))\\ \end{align*}
Mathematica [B] time = 0.0054386, size = 20, normalized size = 2.86 \[ \log \left (\sin \left (\frac{x}{2}\right )\right )+\log (\sin (x))-\log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 6, normalized size = 0.9 \begin{align*} \ln \left ( \cos \left ( x \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928996, size = 7, normalized size = 1. \begin{align*} \log \left (\cos \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14859, size = 32, normalized size = 4.57 \begin{align*} \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 = 2.02163, size = 12, normalized size = 1.71 \begin{align*} - \log{\left (\cot{\left (x \right )} + \csc{\left (x \right )} \right )} + \log{\left (\sin{\left (x \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05414, size = 9, normalized size = 1.29 \begin{align*} \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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