Optimal. Leaf size=32 \[ x-\cot ^3\left (\frac{x}{3}+\frac{\pi }{4}\right )+3 \cot \left (\frac{x}{3}+\frac{\pi }{4}\right ) \]
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Rubi [A] time = 0.0139363, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3473, 8} \[ x-\cot ^3\left (\frac{x}{3}+\frac{\pi }{4}\right )+3 \cot \left (\frac{x}{3}+\frac{\pi }{4}\right ) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \cot ^4\left (\frac{\pi }{4}+\frac{x}{3}\right ) \, dx &=-\cot ^3\left (\frac{\pi }{4}+\frac{x}{3}\right )-\int \tan ^2\left (\frac{\pi }{4}-\frac{x}{3}\right ) \, dx\\ &=3 \cot \left (\frac{\pi }{4}+\frac{x}{3}\right )-\cot ^3\left (\frac{\pi }{4}+\frac{x}{3}\right )+\int 1 \, dx\\ &=x+3 \cot \left (\frac{\pi }{4}+\frac{x}{3}\right )-\cot ^3\left (\frac{\pi }{4}+\frac{x}{3}\right )\\ \end{align*}
Mathematica [C] time = 0.0315336, size = 40, normalized size = 1.25 \[ -\cot ^3\left (\frac{x}{3}+\frac{\pi }{4}\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\tan ^2\left (\frac{x}{3}+\frac{\pi }{4}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.9 \begin{align*} - \left ( \cot \left ({\frac{\pi }{4}}+{\frac{x}{3}} \right ) \right ) ^{3}+3\,\cot \left ( \pi /4+x/3 \right ) -{\frac{3\,\pi }{4}}+x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42652, size = 41, normalized size = 1.28 \begin{align*} \frac{3}{4} \, \pi + x + \frac{3 \, \tan \left (\frac{1}{4} \, \pi + \frac{1}{3} \, x\right )^{2} - 1}{\tan \left (\frac{1}{4} \, \pi + \frac{1}{3} \, x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68143, size = 207, normalized size = 6.47 \begin{align*} \frac{4 \, \cos \left (\frac{1}{2} \, \pi + \frac{2}{3} \, x\right )^{2} +{\left (x \cos \left (\frac{1}{2} \, \pi + \frac{2}{3} \, x\right ) - x\right )} \sin \left (\frac{1}{2} \, \pi + \frac{2}{3} \, x\right ) + 2 \, \cos \left (\frac{1}{2} \, \pi + \frac{2}{3} \, x\right ) - 2}{{\left (\cos \left (\frac{1}{2} \, \pi + \frac{2}{3} \, x\right ) - 1\right )} \sin \left (\frac{1}{2} \, \pi + \frac{2}{3} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.194499, size = 20, normalized size = 0.62 \begin{align*} x - \cot ^{3}{\left (\frac{x}{3} + \frac{\pi }{4} \right )} + 3 \cot{\left (\frac{x}{3} + \frac{\pi }{4} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09137, size = 72, normalized size = 2.25 \begin{align*} \frac{3}{4} \, \pi + \frac{1}{8} \, \tan \left (\frac{1}{8} \, \pi + \frac{1}{6} \, x\right )^{3} + x + \frac{15 \, \tan \left (\frac{1}{8} \, \pi + \frac{1}{6} \, x\right )^{2} - 1}{8 \, \tan \left (\frac{1}{8} \, \pi + \frac{1}{6} \, x\right )^{3}} - \frac{15}{8} \, \tan \left (\frac{1}{8} \, \pi + \frac{1}{6} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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