Optimal. Leaf size=5 \[ -\tanh ^{-1}(\cos (x)) \]
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Rubi [A]
time = 0.00, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3855}
\begin {gather*} -\tanh ^{-1}(\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rubi steps
\begin {align*} \int \csc (x) \, dx &=-\tanh ^{-1}(\cos (x))\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(17\) vs. \(2(5)=10\).
time = 0.00, size = 17, normalized size = 3.40 \begin {gather*} -\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(19\) vs. \(2(5)=10\).
time = 1.80, size = 15, normalized size = 3.00 \begin {gather*} -\frac {\text {Log}\left [1+\text {Cos}\left [x\right ]\right ]}{2}+\frac {\text {Log}\left [-1+\text {Cos}\left [x\right ]\right ]}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 9, normalized size = 1.80
method | result | size |
norman | \(\ln \left (\tan \left (\frac {x}{2}\right )\right )\) | \(6\) |
lookup | \(-\ln \left (\csc \left (x \right )+\cot \left (x \right )\right )\) | \(9\) |
default | \(-\ln \left (\csc \left (x \right )+\cot \left (x \right )\right )\) | \(9\) |
risch | \(-\ln \left (1+{\mathrm e}^{i x}\right )+\ln \left ({\mathrm e}^{i x}-1\right )\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 8, normalized size = 1.60 \begin {gather*} -\log \left (\cot \left (x\right ) + \csc \left (x\right )\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. 19 vs.
\(2 (5) = 10\).
time = 0.34, size = 19, normalized size = 3.80 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\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. 15 vs.
\(2 (5) = 10\)
time = 0.05, size = 15, normalized size = 3.00 \begin {gather*} \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{2} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 11, normalized size = 2.20 \begin {gather*} \frac {2}{2} \ln \left |\tan \left (\frac {x}{2}\right )\right | \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 5, normalized size = 1.00 \begin {gather*} \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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