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 = 0.0698094, antiderivative size = 23, normalized size of antiderivative = 1., 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 3475
Rule 3133
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*}
Mathematica [A] time = 0.0453793, 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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Maple [A] time = 0.076, size = 21, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( a+b\tan \left ( x \right ) \right ) }{a}}+{\frac{\ln \left ( \tan \left ( x \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14257, size = 65, normalized size = 2.83 \begin{align*} -\frac{\log \left (-a - \frac{2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a} + \frac{\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.503741, size = 123, normalized size = 5.35 \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} + b^{2}\right ) - \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right )}{2 \, 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{\csc{\left (x \right )}}{a \cos{\left (x \right )} + b \sin{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18594, size = 30, normalized size = 1.3 \begin{align*} -\frac{\log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a} + \frac{\log \left ({\left | \tan \left (x\right ) \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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