Optimal. Leaf size=12 \[ -\frac{\log (a+b \cot (x))}{b} \]
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Rubi [A] time = 0.0411302, antiderivative size = 12, normalized size of antiderivative = 1., 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 3506
Rule 31
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*}
Mathematica [A] time = 0.0586226, 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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Maple [A] time = 0.025, size = 13, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( a+b\cot \left ( x \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960791, size = 16, normalized size = 1.33 \begin{align*} -\frac{\log \left (b \cot \left (x\right ) + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.267, size = 123, normalized size = 10.25 \begin{align*} -\frac{\log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right ) - \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (x \right )}}{a + b \cot{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12864, size = 30, normalized size = 2.5 \begin{align*} -\frac{\log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{b} + \frac{\log \left ({\left | \tan \left (x\right ) \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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