Optimal. Leaf size=27 \[ \frac {x}{2}-\frac {\cos (x)}{2}-\log \left (\cos \left (\frac {x}{2}+\frac {\pi }{4}\right )\right ) \]
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Rubi [F] time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \cos ^2\left (\frac {x}{2}\right ) \tan \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \cos ^2\left (\frac {x}{2}\right ) \tan \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx &=\int \cos ^2\left (\frac {x}{2}\right ) \tan \left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx\\ \end {align*}
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Mathematica [A] time = 0.17, size = 24, normalized size = 0.89 \[ \frac {1}{2} \left (x-\cos (x)-\log (\cos (x))+2 \tanh ^{-1}\left (\cot \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.58, size = 27, normalized size = 1.00 \[ -\cos \left (\frac {1}{2} \, x\right )^{2} + \frac {1}{2} \, x - \frac {1}{2} \, \log \left (-2 \, \cos \left (\frac {1}{2} \, x\right ) \sin \left (\frac {1}{2} \, x\right ) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.12, size = 93, normalized size = 3.44 \[ \frac {x \tan \left (\frac {1}{2} \, x\right )^{2} - \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right )^{2} + x - \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) - 1}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 22, normalized size = 0.81 \[ \frac {x}{2}-\frac {\cos \relax (x )}{2}+\frac {\ln \left (\sec \relax (x )+\tan \relax (x )\right )}{2}-\frac {\ln \left (\cos \relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 74, normalized size = 2.74 \[ \frac {2 \, x \cos \relax (x)^{2} + 2 \, x \sin \relax (x)^{2} - \cos \left (2 \, x\right ) \cos \relax (x) - 2 \, {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2}\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) - \sin \left (2 \, x\right ) \sin \relax (x) - \cos \relax (x)}{4 \, {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 38, normalized size = 1.41 \[ -2\,\ln \left ({\mathrm {e}}^{\frac {\Pi \,1{}\mathrm {i}}{2}}\,{\mathrm {e}}^{x\,1{}\mathrm {i}}+1\right )\,{\sin \left (\frac {\Pi }{4}\right )}^2+x\,{\mathrm {e}}^{\frac {\Pi \,1{}\mathrm {i}}{4}}\,\sin \left (\frac {\Pi }{4}\right )-\frac {\cos \relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{2}{\left (\frac {x}{2} \right )} \tan {\left (\frac {x}{2} + \frac {\pi }{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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