Optimal. Leaf size=4 \[ x+\tan (x) \]
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Rubi [A] time = 0.0629814, antiderivative size = 4, normalized size of antiderivative = 1., 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 3657
Rule 3473
Rule 8
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*}
Mathematica [A] time = 0.0052156, size = 4, normalized size = 1. \[ x+\tan (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 5, normalized size = 1.3 \begin{align*} x+\tan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49087, size = 5, normalized size = 1.25 \begin{align*} x + \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12041, size = 38, normalized size = 9.5 \begin{align*} \frac{x \cos \left (x\right ) + \sin \left (x\right )}{\cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.923632, size = 27, normalized size = 6.75 \begin{align*} \frac{x \sec ^{2}{\left (x \right )}}{\tan ^{2}{\left (x \right )} + 1} + \frac{\tan{\left (x \right )} \sec ^{2}{\left (x \right )}}{\tan ^{2}{\left (x \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08675, size = 5, normalized size = 1.25 \begin{align*} x + \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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