Optimal. Leaf size=8 \[ x \tan (x)+\log (\cos (x)) \]
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Rubi [A] time = 0.0177246, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4184, 3475} \[ x \tan (x)+\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x \sec ^2(x) \, dx &=x \tan (x)-\int \tan (x) \, dx\\ &=\log (\cos (x))+x \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0053494, size = 8, normalized size = 1. \[ x \tan (x)+\log (\cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 9, normalized size = 1.1 \begin{align*} \ln \left ( \cos \left ( x \right ) \right ) +x\tan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46013, size = 100, normalized size = 12.5 \begin{align*} \frac{{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + 4 \, x \sin \left (2 \, x\right )}{2 \,{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14376, size = 55, normalized size = 6.88 \begin{align*} \frac{\cos \left (x\right ) \log \left (-\cos \left (x\right )\right ) + x \sin \left (x\right )}{\cos \left (x\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 x \sec ^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13281, size = 139, normalized size = 17.38 \begin{align*} \frac{\log \left (\frac{4 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}}{\tan \left (\frac{1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac{1}{2} \, x\right )^{2} - 4 \, x \tan \left (\frac{1}{2} \, x\right ) - \log \left (\frac{4 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}}{\tan \left (\frac{1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )}{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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