Optimal. Leaf size=8 \[ -x-\cot (x) \]
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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3554, 8}
\begin {gather*} -x-\cot (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rubi steps
\begin {align*} \int \cot ^2(x) \, dx &=-\cot (x)-\int 1 \, dx\\ &=-x-\cot (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} -x-\cot (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 14, normalized size = 1.75
method | result | size |
norman | \(\frac {-1-x \tan \left (x \right )}{\tan \left (x \right )}\) | \(13\) |
derivativedivides | \(-\cot \left (x \right )+\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (x \right )\right )\) | \(14\) |
default | \(-\cot \left (x \right )+\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (x \right )\right )\) | \(14\) |
risch | \(-x -\frac {2 i}{{\mathrm e}^{2 i x}-1}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.44, size = 10, normalized size = 1.25 \begin {gather*} -x - \frac {1}{\tan \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs.
\(2 (8) = 16\).
time = 1.21, size = 20, normalized size = 2.50 \begin {gather*} -\frac {x \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} - x - \frac {\cos {\left (x \right )}}{\sin {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs.
\(2 (8) = 16\).
time = 0.46, size = 18, normalized size = 2.25 \begin {gather*} -x - \frac {1}{2 \, \tan \left (\frac {1}{2} \, x\right )} + \frac {1}{2} \, \tan \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 8, normalized size = 1.00 \begin {gather*} -x-\mathrm {cot}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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