Optimal. Leaf size=9 \[ \frac{\log (\sin (x)+1)}{a} \]
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Rubi [A] time = 0.0217797, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3879, 31} \[ \frac{\log (\sin (x)+1)}{a} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot (x)}{a+a \csc (x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+a x} \, dx,x,\sin (x)\right )\\ &=\frac{\log (1+\sin (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0085864, size = 9, normalized size = 1. \[ \frac{\log (\sin (x)+1)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 19, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( 1+\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 = 0.980983, size = 12, normalized size = 1.33 \begin{align*} \frac{\log \left (\sin \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 = 0.480226, size = 26, normalized size = 2.89 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\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*} \frac{\int \frac{\cot{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29816, size = 12, normalized size = 1.33 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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