Optimal. Leaf size=8 \[ x \tan (x)+\log (\cos (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 3475
Rule 4184
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*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 1.00 \[ x \tan (x)+\log (\cos (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 18, normalized size = 2.25 \[ \frac {\cos \relax (x) \log \left (-\cos \relax (x)\right ) + x \sin \relax (x)}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 103, normalized size = 12.88 \[ \frac {\log \left (\frac {4 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, x \tan \left (\frac {1}{2} \, x\right ) - \log \left (\frac {4 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 9, normalized size = 1.12 \[ \ln \left (\cos \relax (x )\right )+x \tan \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 74, normalized size = 9.25 \[ \frac {{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + 4 \, x \sin \left (2 \, x\right )}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 8, normalized size = 1.00 \[ \ln \left (\cos \relax (x)\right )+x\,\mathrm {tan}\relax (x) \]
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 \[ \int x \sec ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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