Optimal. Leaf size=4 \[ -\cot (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3254, 3852, 8}
\begin {gather*} -\cot (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3254
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{1-\cos ^2(x)} \, dx &=\int \csc ^2(x) \, dx\\ &=-\text {Subst}(\int 1 \, dx,x,\cot (x))\\ &=-\cot (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 4, normalized size = 1.00 \begin {gather*} -\cot (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 7, normalized size = 1.75
method | result | size |
default | \(-\frac {1}{\tan \left (x \right )}\) | \(7\) |
risch | \(-\frac {2 i}{{\mathrm e}^{2 i x}-1}\) | \(13\) |
norman | \(\frac {-\frac {1}{2}+\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}}{\tan \left (\frac {x}{2}\right )}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 6, normalized size = 1.50 \begin {gather*} -\frac {1}{\tan \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 8, normalized size = 2.00 \begin {gather*} -\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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs.
\(2 (3) = 6\).
time = 0.20, size = 14, normalized size = 3.50 \begin {gather*} \frac {\tan {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tan {\left (\frac {x}{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 6, normalized size = 1.50 \begin {gather*} -\frac {1}{\tan \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.24, size = 4, normalized size = 1.00 \begin {gather*} -\mathrm {cot}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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