Optimal. Leaf size=23 \[ \frac {\log (\sin (x))}{a}-\frac {\log (a \cos (x)+b \sin (x))}{a} \]
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Rubi [A] time = 0.07, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3101, 3475, 3133} \[ \frac {\log (\sin (x))}{a}-\frac {\log (a \cos (x)+b \sin (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3101
Rule 3133
Rule 3475
Rubi steps
\begin {align*} \int \frac {\csc (x)}{a \cos (x)+b \sin (x)} \, dx &=\frac {\int \cot (x) \, dx}{a}-\frac {\int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a}\\ &=\frac {\log (\sin (x))}{a}-\frac {\log (a \cos (x)+b \sin (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 20, normalized size = 0.87 \[ \frac {\log (\sin (x))-\log (a \cos (x)+b \sin (x))}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 44, normalized size = 1.91 \[ -\frac {\log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right ) - \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.72, size = 22, normalized size = 0.96 \[ -\frac {\log \left ({\left | b \tan \relax (x) + a \right |}\right )}{a} + \frac {\log \left ({\left | \tan \relax (x) \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 21, normalized size = 0.91 \[ -\frac {\ln \left (a +b \tan \relax (x )\right )}{a}+\frac {\ln \left (\tan \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 48, normalized size = 2.09 \[ -\frac {\log \left (-a - \frac {2 \, b \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}{a} + \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 32, normalized size = 1.39 \[ -\frac {\ln \left (-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a\right )-\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a} \]
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 \[ \int \frac {\csc {\relax (x )}}{a \cos {\relax (x )} + b \sin {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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