Optimal. Leaf size=16 \[ -\frac {\csc (x)}{a}-\frac {\log (\sin (x))}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 43
Rule 3879
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*}
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Mathematica [A] time = 0.02, size = 11, normalized size = 0.69 \[ -\frac {\csc (x)+\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 19, normalized size = 1.19 \[ -\frac {\log \left (\frac {1}{2} \, \sin \relax (x)\right ) \sin \relax (x) + 1}{a \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 14, normalized size = 0.88 \[ -\frac {\frac {1}{\sin \relax (x)} + \log \left ({\left | \sin \relax (x) \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 16, normalized size = 1.00 \[ -\frac {\csc \relax (x )}{a}+\frac {\ln \left (\csc \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 18, normalized size = 1.12 \[ -\frac {\log \left (\sin \relax (x)\right )}{a} - \frac {1}{a \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 36, normalized size = 2.25 \[ -\frac {\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )+\frac {1}{2\,\mathrm {tan}\left (\frac {x}{2}\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{3}{\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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