Optimal. Leaf size=13 \[ \log (1-\cos (x))-\log (\cos (x)) \]
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Rubi [A] time = 0.06, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4392, 2707, 36, 29, 31} \[ \log (1-\cos (x))-\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2707
Rule 4392
Rubi steps
\begin {align*} \int \frac {\sec (x)}{-\cot (x)+\csc (x)} \, dx &=\int \frac {\tan (x)}{1-\cos (x)} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,-\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-\cos (x)\right )+\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,-\cos (x)\right )\\ &=\log (1-\cos (x))-\log (\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.92 \[ 2 \log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (1-2 \sin ^2\left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 15, normalized size = 1.15 \[ -\log \left (-\cos \relax (x)\right ) + \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 14, normalized size = 1.08 \[ \log \left (-\cos \relax (x) + 1\right ) - \log \left ({\left | \cos \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 6, normalized size = 0.46 \[ \ln \left (-1+\sec \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 41, normalized size = 3.15 \[ -\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) - \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} - 1\right ) + 2 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 19, normalized size = 1.46 \[ 2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2-1\right ) \]
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 \[ - \int \frac {\sec {\relax (x )}}{\cot {\relax (x )} - \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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