Optimal. Leaf size=9 \[ \log \left (1+\tan \left (\frac {x}{2}\right )\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3203, 31}
\begin {gather*} \log \left (\tan \left (\frac {x}{2}\right )+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 3203
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=2 \text {Subst}\left (\int \frac {1}{2+2 x} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\log \left (1+\tan \left (\frac {x}{2}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(24\) vs. \(2(9)=18\).
time = 0.01, size = 24, normalized size = 2.67 \begin {gather*} -\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 8, normalized size = 0.89
method | result | size |
default | \(\ln \left (1+\tan \left (\frac {x}{2}\right )\right )\) | \(8\) |
parallelrisch | \(\ln \left (1+\tan \left (\frac {x}{2}\right )\right )\) | \(8\) |
norman | \(\ln \left (2+2 \tan \left (\frac {x}{2}\right )\right )\) | \(10\) |
risch | \(\ln \left (i+{\mathrm e}^{i x}\right )-\ln \left ({\mathrm e}^{i x}+1\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 12, normalized size = 1.33 \begin {gather*} \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\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. 17 vs.
\(2 (7) = 14\).
time = 0.60, size = 17, normalized size = 1.89 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 7, normalized size = 0.78 \begin {gather*} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 8, normalized size = 0.89 \begin {gather*} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 7, normalized size = 0.78 \begin {gather*} \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 19, normalized size = 2.11 \begin {gather*} -2 \ln \left (\frac {\tan \left (\frac {x}{2}\right )}{2}+\frac {1}{2}-\frac {\sin \left (x \right )}{2 \cos \left (x \right )}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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