Optimal. Leaf size=14 \[ -x-\frac{4}{3} \cot \left (\frac{3 x}{4}\right ) \]
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Rubi [A] time = 0.0062183, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ -x-\frac{4}{3} \cot \left (\frac{3 x}{4}\right ) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \cot ^2\left (\frac{3 x}{4}\right ) \, dx &=-\frac{4}{3} \cot \left (\frac{3 x}{4}\right )-\int 1 \, dx\\ &=-x-\frac{4}{3} \cot \left (\frac{3 x}{4}\right )\\ \end{align*}
Mathematica [C] time = 0.013271, size = 28, normalized size = 2. \[ -\frac{4}{3} \cot \left (\frac{3 x}{4}\right ) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2\left (\frac{3 x}{4}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 14, normalized size = 1. \begin{align*} -{\frac{4}{3}\cot \left ({\frac{3\,x}{4}} \right ) }+{\frac{2\,\pi }{3}}-x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46041, size = 16, normalized size = 1.14 \begin{align*} -x - \frac{4}{3 \, \tan \left (\frac{3}{4} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01595, size = 72, normalized size = 5.14 \begin{align*} -\frac{3 \, x \sin \left (\frac{3}{2} \, x\right ) + 4 \, \cos \left (\frac{3}{2} \, x\right ) + 4}{3 \, \sin \left (\frac{3}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.068754, size = 19, normalized size = 1.36 \begin{align*} - x - \frac{4 \cos{\left (\frac{3 x}{4} \right )}}{3 \sin{\left (\frac{3 x}{4} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0636, size = 24, normalized size = 1.71 \begin{align*} -x - \frac{2}{3 \, \tan \left (\frac{3}{8} \, x\right )} + \frac{2}{3} \, \tan \left (\frac{3}{8} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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