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 = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 8
Rule 3473
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*}
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Mathematica [C] time = 0.02, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.70, size = 70, normalized size = 2.19 \[ \frac {4 \, \cos \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right )^{2} + {\left (x \cos \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right ) - x\right )} \sin \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right ) + 2 \, \cos \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right ) - 2}{{\left (\cos \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right ) - 1\right )} \sin \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.12, size = 53, normalized size = 1.66 \[ \frac {3}{4} \, \pi + \frac {1}{8} \, \tan \left (\frac {1}{8} \, \pi + \frac {1}{6} \, x\right )^{3} + x + \frac {15 \, \tan \left (\frac {1}{8} \, \pi + \frac {1}{6} \, x\right )^{2} - 1}{8 \, \tan \left (\frac {1}{8} \, \pi + \frac {1}{6} \, x\right )^{3}} - \frac {15}{8} \, \tan \left (\frac {1}{8} \, \pi + \frac {1}{6} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 1.19
method | result | size |
derivativedivides | \(-\left (\cot ^{3}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )+3 \cot \left (\frac {\pi }{4}+\frac {x}{3}\right )-\frac {3 \pi }{2}+3 \,\mathrm {arccot}\left (\cot \left (\frac {\pi }{4}+\frac {x}{3}\right )\right )\) | \(38\) |
default | \(-\left (\cot ^{3}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )+3 \cot \left (\frac {\pi }{4}+\frac {x}{3}\right )-\frac {3 \pi }{2}+3 \,\mathrm {arccot}\left (\cot \left (\frac {\pi }{4}+\frac {x}{3}\right )\right )\) | \(38\) |
norman | \(\frac {-1+x \left (\tan ^{3}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )+3 \left (\tan ^{2}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )}{\tan \left (\frac {\pi }{4}+\frac {x}{3}\right )^{3}}\) | \(38\) |
risch | \(x +\frac {4 i \left (-3 \,{\mathrm e}^{\frac {4 i x}{3}}-3 i {\mathrm e}^{\frac {2 i x}{3}}+2\right )}{\left ({\mathrm e}^{\frac {i \left (3 \pi +4 x \right )}{6}}-1\right )^{3}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 30, normalized size = 0.94 \[ \frac {3}{4} \, \pi + x + \frac {3 \, \tan \left (\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )^{2} - 1}{\tan \left (\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 24, normalized size = 0.75 \[ -{\mathrm {cot}\left (\frac {\Pi }{4}+\frac {x}{3}\right )}^3+3\,\mathrm {cot}\left (\frac {\Pi }{4}+\frac {x}{3}\right )+x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 20, normalized size = 0.62 \[ x - \cot ^{3}{\left (\frac {x}{3} + \frac {\pi }{4} \right )} + 3 \cot {\left (\frac {x}{3} + \frac {\pi }{4} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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