Optimal. Leaf size=10 \[ \cos (x)+\log (1-\cos (x)) \]
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Rubi [A] time = 0.07, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4392, 2833, 43} \[ \cos (x)+\log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2833
Rule 4392
Rubi steps
\begin {align*} \int \frac {\cos (x)}{-\cot (x)+\csc (x)} \, dx &=\int \frac {\cos (x) \sin (x)}{1-\cos (x)} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {x}{1+x} \, dx,x,-\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,-\cos (x)\right )\\ &=\cos (x)+\log (1-\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 2.00 \[ 2 \log \left (\sin \left (\frac {x}{2}\right )\right )-2 \sin ^2\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 10, normalized size = 1.00 \[ \cos \relax (x) + \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 10, normalized size = 1.00 \[ \cos \relax (x) + \log \left (-\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 9, normalized size = 0.90 \[ \cos \relax (x )+\ln \left (\cos \relax (x )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 46, normalized size = 4.60 \[ \frac {2}{\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1} + 2 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) - \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 31, normalized size = 3.10 \[ 2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \]
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 \[ - \int \frac {\cos {\relax (x )}}{\cot {\relax (x )} - \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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