Optimal. Leaf size=12 \[ x-\frac {1}{3} \cot ^3(x)+\cot (x) \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ x-\frac {1}{3} \cot ^3(x)+\cot (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \cot ^4(x) \, dx &=-\frac {1}{3} \cot ^3(x)-\int \cot ^2(x) \, dx\\ &=\cot (x)-\frac {\cot ^3(x)}{3}+\int 1 \, dx\\ &=x+\cot (x)-\frac {\cot ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.50 \[ x+\frac {4 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 48, normalized size = 4.00 \[ \frac {4 \, \cos \left (2 \, x\right )^{2} + 3 \, {\left (x \cos \left (2 \, x\right ) - x\right )} \sin \left (2 \, x\right ) + 2 \, \cos \left (2 \, x\right ) - 2}{3 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.02, size = 34, normalized size = 2.83 \[ \frac {1}{24} \, \tan \left (\frac {1}{2} \, x\right )^{3} + x + \frac {15 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{24 \, \tan \left (\frac {1}{2} \, x\right )^{3}} - \frac {5}{8} \, \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 1.17 \[ -\frac {\left (\cot ^{3}\relax (x )\right )}{3}+x +\cot \relax (x )-\frac {\pi }{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 16, normalized size = 1.33 \[ x + \frac {3 \, \tan \relax (x)^{2} - 1}{3 \, \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 10, normalized size = 0.83 \[ -\frac {{\mathrm {cot}\relax (x)}^3}{3}+\mathrm {cot}\relax (x)+x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 19, normalized size = 1.58 \[ x + \frac {\cos {\relax (x )}}{\sin {\relax (x )}} - \frac {\cos ^{3}{\relax (x )}}{3 \sin ^{3}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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