Optimal. Leaf size=16 \[ -\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a} \]
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Rubi [A] time = 0.0415825, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3879, 43} \[ -\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int \frac{\cot ^3(x)}{a+a \csc (x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-a x}{x^2} \, dx,x,\sin (x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{a}{x}\right ) \, dx,x,\sin (x)\right )}{a^2}\\ &=-\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0203576, size = 11, normalized size = 0.69 \[ -\frac{\csc (x)+\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 16, normalized size = 1. \begin{align*} -{\frac{\csc \left ( x \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 = 1.04973, size = 24, normalized size = 1.5 \begin{align*} -\frac{\log \left (\sin \left (x\right )\right )}{a} - \frac{1}{a \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.491916, size = 57, normalized size = 3.56 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) + 1}{a \sin \left (x\right )} \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 ^{3}{\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 = 1.35568, size = 26, normalized size = 1.62 \begin{align*} -\frac{\log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} - \frac{1}{a \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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