Optimal. Leaf size=19 \[ \frac{\log (a+b \csc (x))}{a}+\frac{\log (\sin (x))}{a} \]
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Rubi [A] time = 0.024461, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3885, 36, 29, 31} \[ \frac{\log (a+b \csc (x))}{a}+\frac{\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot (x)}{a+b \csc (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x (a+x)} \, dx,x,b \csc (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,b \csc (x)\right )}{a}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \csc (x)\right )}{a}\\ &=\frac{\log (a+b \csc (x))}{a}+\frac{\log (\sin (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0083844, size = 11, normalized size = 0.58 \[ \frac{\log (a \sin (x)+b)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 21, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\csc \left ( x \right ) \right ) }{a}}-{\frac{\ln \left ( \csc \left ( x \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950783, size = 15, normalized size = 0.79 \begin{align*} \frac{\log \left (a \sin \left (x\right ) + b\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.489073, size = 28, normalized size = 1.47 \begin{align*} \frac{\log \left (a \sin \left (x\right ) + b\right )}{a} \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{\cot{\left (x \right )}}{a + b \csc{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20945, size = 16, normalized size = 0.84 \begin{align*} \frac{\log \left ({\left | a \sin \left (x\right ) + b \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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