Optimal. Leaf size=14 \[ -\frac{1}{2} \cot ^2(x)-\log (\sin (x)) \]
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Rubi [A] time = 0.0074278, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3473, 3475} \[ -\frac{1}{2} \cot ^2(x)-\log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \cot ^3(x) \, dx &=-\frac{1}{2} \cot ^2(x)-\int \cot (x) \, dx\\ &=-\frac{1}{2} \cot ^2(x)-\log (\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0030025, size = 14, normalized size = 1. \[ -\frac{1}{2} \csc ^2(x)-\log (\sin (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 17, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}+{\frac{\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.930429, size = 19, normalized size = 1.36 \begin{align*} -\frac{1}{2 \, \sin \left (x\right )^{2}} - \frac{1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97195, size = 90, normalized size = 6.43 \begin{align*} -\frac{{\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) - 2}{2 \,{\left (\cos \left (2 \, x\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.087574, size = 14, normalized size = 1. \begin{align*} - \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2 \sin ^{2}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06149, size = 30, normalized size = 2.14 \begin{align*} \frac{1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} - \frac{1}{2} \, \log \left (-\cos \left (x\right )^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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