Optimal. Leaf size=8 \[ -x-2 \cot (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3250, 3254,
3852, 8} \begin {gather*} -x-2 \cot (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3250
Rule 3254
Rule 3852
Rubi steps
\begin {align*} \int \frac {1+\cos ^2(x)}{1-\cos ^2(x)} \, dx &=-x+2 \int \frac {1}{1-\cos ^2(x)} \, dx\\ &=-x+2 \int \csc ^2(x) \, dx\\ &=-x-2 \text {Subst}(\int 1 \, dx,x,\cot (x))\\ &=-x-2 \cot (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} -x-2 \cot (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 13, normalized size = 1.62
method | result | size |
default | \(-\frac {2}{\tan \left (x \right )}-\arctan \left (\tan \left (x \right )\right )\) | \(13\) |
risch | \(-x -\frac {4 i}{{\mathrm e}^{2 i x}-1}\) | \(17\) |
norman | \(\frac {-1+\tan ^{4}\left (\frac {x}{2}\right )+\tan ^{6}\left (\frac {x}{2}\right )-\left (\tan ^{2}\left (\frac {x}{2}\right )\right )-x \tan \left (\frac {x}{2}\right )-2 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2} \tan \left (\frac {x}{2}\right )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.65, size = 10, normalized size = 1.25 \begin {gather*} -x - \frac {2}{\tan \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 15, normalized size = 1.88 \begin {gather*} -\frac {x \sin \left (x\right ) + 2 \, \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 [A]
time = 0.38, size = 12, normalized size = 1.50 \begin {gather*} - x + \tan {\left (\frac {x}{2} \right )} - \frac {1}{\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 = 1.10, size = 16, normalized size = 2.00 \begin {gather*} -x - \frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \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 = 0.21, size = 8, normalized size = 1.00 \begin {gather*} -x-2\,\mathrm {cot}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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