Optimal. Leaf size=15 \[ -\frac {x^2}{2}+x \tan (x)+\log (\cos (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3720, 3475, 30} \[ -\frac {x^2}{2}+x \tan (x)+\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 30
Rule 3475
Rule 3720
Rubi steps
\begin {align*} \int x \tan ^2(x) \, dx &=x \tan (x)-\int x \, dx-\int \tan (x) \, dx\\ &=-\frac {x^2}{2}+\log (\cos (x))+x \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 1.00 \[ -\frac {x^2}{2}+x \tan (x)+\log (\cos (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \tan ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.83, size = 21, normalized size = 1.40 \[ -\frac {1}{2} \, x^{2} + x \tan \relax (x) + \frac {1}{2} \, \log \left (\frac {1}{\tan \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 23, normalized size = 1.53 \[ -\frac {1}{2} \, x^{2} + x \tan \relax (x) + \frac {1}{2} \, \log \left (\frac {4}{\tan \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 20, normalized size = 1.33
method | result | size |
norman | \(x \tan \relax (x )-\frac {x^{2}}{2}-\frac {\ln \left (1+\tan ^{2}\relax (x )\right )}{2}\) | \(20\) |
risch | \(-\frac {x^{2}}{2}-2 i x +\frac {2 i x}{1+{\mathrm e}^{2 i x}}+\ln \left (1+{\mathrm e}^{2 i x}\right )\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.99, size = 107, normalized size = 7.13 \[ -\frac {x^{2} \cos \left (2 \, x\right )^{2} + x^{2} \sin \left (2 \, x\right )^{2} + 2 \, x^{2} \cos \left (2 \, x\right ) + x^{2} - {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) - 4 \, x \sin \left (2 \, x\right )}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 13, normalized size = 0.87 \[ \ln \left (\cos \relax (x)\right )+x\,\mathrm {tan}\relax (x)-\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 19, normalized size = 1.27 \[ - \frac {x^{2}}{2} + x \tan {\relax (x )} - \frac {\log {\left (\tan ^{2}{\relax (x )} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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