Optimal. Leaf size=11 \[ -\frac {3}{4} \tanh ^{-1}\left (\cos \left (\frac {x}{3}\right )\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 3770} \[ -\frac {3}{4} \tanh ^{-1}\left (\cos \left (\frac {x}{3}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{4} \csc \left (\frac {x}{3}\right ) \, dx &=\frac {1}{4} \int \csc \left (\frac {x}{3}\right ) \, dx\\ &=-\frac {3}{4} \tanh ^{-1}\left (\cos \left (\frac {x}{3}\right )\right )\\ \end {align*}
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Mathematica [B] time = 0.01, size = 23, normalized size = 2.09 \[ \frac {1}{4} \left (3 \log \left (\sin \left (\frac {x}{6}\right )\right )-3 \log \left (\cos \left (\frac {x}{6}\right )\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{4} \csc \left (\frac {x}{3}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.96, size = 23, normalized size = 2.09 \[ -\frac {3}{8} \, \log \left (\frac {1}{2} \, \cos \left (\frac {1}{3} \, x\right ) + \frac {1}{2}\right ) + \frac {3}{8} \, \log \left (-\frac {1}{2} \, \cos \left (\frac {1}{3} \, x\right ) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.85, size = 23, normalized size = 2.09 \[ -\frac {3}{8} \, \log \left (3 \, \cos \left (\frac {1}{3} \, x\right ) + 3\right ) + \frac {3}{8} \, \log \left (-3 \, \cos \left (\frac {1}{3} \, x\right ) + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 8, normalized size = 0.73
method | result | size |
norman | \(\frac {3 \ln \left (\tan \left (\frac {x}{6}\right )\right )}{4}\) | \(8\) |
derivativedivides | \(\frac {3 \ln \left (\csc \left (\frac {x}{3}\right )-\cot \left (\frac {x}{3}\right )\right )}{4}\) | \(15\) |
default | \(\frac {3 \ln \left (\csc \left (\frac {x}{3}\right )-\cot \left (\frac {x}{3}\right )\right )}{4}\) | \(15\) |
risch | \(-\frac {3 \ln \left ({\mathrm e}^{\frac {i x}{3}}+1\right )}{4}+\frac {3 \ln \left ({\mathrm e}^{\frac {i x}{3}}-1\right )}{4}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 19, normalized size = 1.73 \[ -\frac {3}{8} \, \log \left (\cos \left (\frac {1}{3} \, x\right ) + 1\right ) + \frac {3}{8} \, \log \left (\cos \left (\frac {1}{3} \, x\right ) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 7, normalized size = 0.64 \[ \frac {3\,\ln \left (\mathrm {tan}\left (\frac {x}{6}\right )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 22, normalized size = 2.00 \[ \frac {3 \log {\left (\cos {\left (\frac {x}{3} \right )} - 1 \right )}}{8} - \frac {3 \log {\left (\cos {\left (\frac {x}{3} \right )} + 1 \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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