Optimal. Leaf size=12 \[ -\frac {\log (a+b \cot (x))}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 31} \[ -\frac {\log (a+b \cot (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3506
Rubi steps
\begin {align*} \int \frac {\csc ^2(x)}{a+b \cot (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cot (x)\right )}{b}\\ &=-\frac {\log (a+b \cot (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 20, normalized size = 1.67 \[ \frac {\log (\sin (x))-\log (a \sin (x)+b \cos (x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 45, normalized size = 3.75 \[ -\frac {\log \left (2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}\right ) - \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 22, normalized size = 1.83 \[ -\frac {\log \left ({\left | a \tan \relax (x) + b \right |}\right )}{b} + \frac {\log \left ({\left | \tan \relax (x) \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 13, normalized size = 1.08 \[ -\frac {\ln \left (a +b \cot \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 12, normalized size = 1.00 \[ -\frac {\log \left (b \cot \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 16, normalized size = 1.33 \[ -\frac {2\,\mathrm {atanh}\left (\frac {2\,a\,\mathrm {tan}\relax (x)}{b}+1\right )}{b} \]
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 \[ \int \frac {\csc ^{2}{\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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