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 = 0.0625198, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 0.166814, 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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Maple [A] time = 0.14, size = 22, normalized size = 0.8 \begin{align*}{\frac{x}{2}}-{\frac{\cos \left ( x \right ) }{2}}+{\frac{\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{2}}-{\frac{\ln \left ( \cos \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.44345, size = 100, normalized size = 3.7 \begin{align*} \frac{2 \, x \cos \left (x\right )^{2} + 2 \, x \sin \left (x\right )^{2} - \cos \left (2 \, x\right ) \cos \left (x\right ) - 2 \,{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2}\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) - \sin \left (2 \, x\right ) \sin \left (x\right ) - \cos \left (x\right )}{4 \,{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06683, size = 85, normalized size = 3.15 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos ^{2}{\left (\frac{x}{2} \right )} \tan{\left (\frac{x}{2} + \frac{\pi }{4} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1435, size = 126, normalized size = 4.67 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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