Optimal. Leaf size=11 \[ x \sec ^2(x)-\tan (x) \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {12, 3757, 3767, 8} \[ x \sec ^2(x)-\tan (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 3757
Rule 3767
Rubi steps
\begin {align*} \int 2 x \sec ^2(x) \tan (x) \, dx &=2 \int x \sec ^2(x) \tan (x) \, dx\\ &=x \sec ^2(x)-\int \sec ^2(x) \, dx\\ &=x \sec ^2(x)+\operatorname {Subst}(\int 1 \, dx,x,-\tan (x))\\ &=x \sec ^2(x)-\tan (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.64 \[ 2 \left (\frac {1}{2} x \sec ^2(x)-\frac {\tan (x)}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 15, normalized size = 1.36 \[ -\frac {\cos \relax (x) \sin \relax (x) - x}{\cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 52, normalized size = 4.73 \[ \frac {x \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, x \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{3} + x - 2 \, \tan \left (\frac {1}{2} \, x\right )}{\tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 12, normalized size = 1.09 \[ \frac {x}{\cos \relax (x )^{2}}-\tan \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 133, normalized size = 12.09 \[ \frac {2 \, {\left (4 \, x \cos \left (2 \, x\right )^{2} + 4 \, x \sin \left (2 \, x\right )^{2} + {\left (2 \, x \cos \left (2 \, x\right ) + \sin \left (2 \, x\right )\right )} \cos \left (4 \, x\right ) + 2 \, x \cos \left (2 \, x\right ) + {\left (2 \, x \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1\right )} \sin \left (4 \, x\right ) - \sin \left (2 \, x\right )\right )}}{2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.09, size = 16, normalized size = 1.45 \[ \frac {2\,x-\sin \left (2\,x\right )}{2\,{\cos \relax (x)}^2} \]
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 \[ 2 \int x \tan {\relax (x )} \sec ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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