Optimal. Leaf size=7 \[ x-\tanh ^{-1}(\cos (x)) \]
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Rubi [A] time = 0.09, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4391, 2839, 3770, 8} \[ x-\tanh ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2839
Rule 3770
Rule 4391
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\sec (x)-\tan (x)} \, dx &=\int \frac {\cos (x) \cot (x)}{1-\sin (x)} \, dx\\ &=\int 1 \, dx+\int \csc (x) \, dx\\ &=x-\tanh ^{-1}(\cos (x))\\ \end {align*}
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Mathematica [B] time = 0.02, size = 18, normalized size = 2.57 \[ x+\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 20, normalized size = 2.86 \[ x - \frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 8, normalized size = 1.14 \[ x + \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 8, normalized size = 1.14 \[ \ln \left (\tan \left (\frac {x}{2}\right )\right )+x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 23, normalized size = 3.29 \[ 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) + \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 23, normalized size = 3.29 \[ \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-2\,\mathrm {atan}\left (\frac {8}{4\,\mathrm {tan}\left (\frac {x}{2}\right )-4}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot {\relax (x )}}{- \tan {\relax (x )} + \sec {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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