Optimal. Leaf size=15 \[ -\frac {1}{2} \tanh ^{-1}\left (\sin \left (\frac {\pi }{4}+2 x\right )\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3855}
\begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sin \left (2 x+\frac {\pi }{4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rubi steps
\begin {align*} \int -\sec \left (\frac {\pi }{4}+2 x\right ) \, dx &=-\frac {1}{2} \tanh ^{-1}\left (\sin \left (\frac {\pi }{4}+2 x\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sin \left (\frac {\pi }{4}+2 x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 21, normalized size = 1.40
method | result | size |
derivativedivides | \(-\frac {\ln \left (\sec \left (\frac {\pi }{4}+2 x \right )+\tan \left (\frac {\pi }{4}+2 x \right )\right )}{2}\) | \(21\) |
default | \(-\frac {\ln \left (\sec \left (\frac {\pi }{4}+2 x \right )+\tan \left (\frac {\pi }{4}+2 x \right )\right )}{2}\) | \(21\) |
norman | \(\frac {\ln \left (\tan \left (\frac {\pi }{8}+x \right )-1\right )}{2}-\frac {\ln \left (\tan \left (\frac {\pi }{8}+x \right )+1\right )}{2}\) | \(24\) |
risch | \(\frac {\ln \left ({\mathrm e}^{\frac {i \left (\pi +8 x \right )}{4}}-i\right )}{2}-\frac {\ln \left ({\mathrm e}^{\frac {i \left (\pi +8 x \right )}{4}}+i\right )}{2}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (11) = 22\).
time = 2.07, size = 27, normalized size = 1.80 \begin {gather*} -\frac {1}{4} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 2 \, x\right ) - 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. 29 vs.
\(2 (11) = 22\).
time = 0.73, size = 29, normalized size = 1.93 \begin {gather*} -\frac {1}{4} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac {1}{4} \, \log \left (-\sin \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 22, normalized size = 1.47 \begin {gather*} \frac {\log {\left (\tan {\left (x + \frac {\pi }{8} \right )} - 1 \right )}}{2} - \frac {\log {\left (\tan {\left (x + \frac {\pi }{8} \right )} + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (11) = 22\).
time = 0.89, size = 29, normalized size = 1.93 \begin {gather*} -\frac {1}{4} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac {1}{4} \, \log \left (-\sin \left (\frac {1}{4} \, \pi + 2 \, x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 24, normalized size = 1.60 \begin {gather*} -\frac {\ln \left (\frac {\sin \left (\frac {\Pi }{4}+2\,x\right )+1}{\cos \left (\frac {\Pi }{4}+2\,x\right )}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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