Optimal. Leaf size=4 \[ x+\tan (x) \]
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Rubi [A] time = 0.06, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3657, 3473, 8} \[ x+\tan (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3657
Rubi steps
\begin {align*} \int \frac {\sec ^2(x) \left (2+\tan ^2(x)\right )}{1+\tan ^2(x)} \, dx &=\int \left (2+\tan ^2(x)\right ) \, dx\\ &=2 x+\int \tan ^2(x) \, dx\\ &=2 x+\tan (x)-\int 1 \, dx\\ &=x+\tan (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 4, normalized size = 1.00 \[ x+\tan (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 12, normalized size = 3.00 \[ \frac {x \cos \relax (x) + \sin \relax (x)}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 4, normalized size = 1.00 \[ x + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 5, normalized size = 1.25 \[ x +\tan \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 4, normalized size = 1.00 \[ x + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.00, size = 4, normalized size = 1.00 \[ x+\mathrm {tan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.74, size = 27, normalized size = 6.75 \[ \frac {x \sec ^{2}{\relax (x )}}{\tan ^{2}{\relax (x )} + 1} + \frac {\tan {\relax (x )} \sec ^{2}{\relax (x )}}{\tan ^{2}{\relax (x )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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