Optimal. Leaf size=10 \[ x \sec (x)-\tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 10, 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 = {3757, 3770} \[ x \sec (x)-\tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3757
Rule 3770
Rubi steps
\begin {align*} \int x \sec (x) \tan (x) \, dx &=x \sec (x)-\int \sec (x) \, dx\\ &=-\tanh ^{-1}(\sin (x))+x \sec (x)\\ \end {align*}
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Mathematica [B] time = 0.01, size = 37, normalized size = 3.70 \[ x \sec (x)+\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 29, normalized size = 2.90 \[ -\frac {\cos \relax (x) \log \left (\sin \relax (x) + 1\right ) - \cos \relax (x) \log \left (-\sin \relax (x) + 1\right ) - 2 \, x}{2 \, \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 150, normalized size = 15.00 \[ -\frac {2 \, x \tan \left (\frac {1}{2} \, x\right )^{2} + \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{2} - \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, x - \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) + \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, x\right ) + 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 16, normalized size = 1.60 \[ -\ln \left (\sec \relax (x )+\tan \relax (x )\right )+\frac {x}{\cos \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.28, size = 121, normalized size = 12.10 \[ \frac {4 \, x \cos \left (2 \, x\right ) \cos \relax (x) + 4 \, x \sin \left (2 \, x\right ) \sin \relax (x) + 4 \, x \cos \relax (x) - {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) + {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right )}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 19, normalized size = 1.90 \[ \frac {x}{\cos \relax (x)}+\mathrm {atan}\left (\cos \relax (x)+\sin \relax (x)\,1{}\mathrm {i}\right )\,2{}\mathrm {i} \]
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 \[ \int x \tan {\relax (x )} \sec {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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