Optimal. Leaf size=7 \[ -\log (\cos (x)+1) \]
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Rubi [A] time = 0.0269082, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3160, 2667, 31} \[ -\log (\cos (x)+1) \]
Antiderivative was successfully verified.
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Rule 3160
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\cot (x)+\csc (x)} \, dx &=\int \frac{\sin (x)}{1+\cos (x)} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\cos (x)\right )\\ &=-\log (1+\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0119092, size = 9, normalized size = 1.29 \[ -2 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 8, normalized size = 1.1 \begin{align*} -\ln \left ( 1+\cos \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.45563, size = 19, normalized size = 2.71 \begin{align*} \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13351, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cot{\left (x \right )} + \csc{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1338, 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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