Optimal. Leaf size=14 \[ \frac{x}{2}+\frac{1}{2} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0153262, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3657, 2635, 8} \[ \frac{x}{2}+\frac{1}{2} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 3657
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{1+\tan ^2(x)} \, dx &=\int \cos ^2(x) \, dx\\ &=\frac{1}{2} \cos (x) \sin (x)+\frac{\int 1 \, dx}{2}\\ &=\frac{x}{2}+\frac{1}{2} \cos (x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0025164, size = 14, normalized size = 1. \[ \frac{x}{2}+\frac{1}{4} \sin (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 17, normalized size = 1.2 \begin{align*}{\frac{\tan \left ( x \right ) }{2\, \left ( \tan \left ( x \right ) \right ) ^{2}+2}}+{\frac{x}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40717, size = 22, normalized size = 1.57 \begin{align*} \frac{1}{2} \, x + \frac{\tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28108, size = 62, normalized size = 4.43 \begin{align*} \frac{x \tan \left (x\right )^{2} + x + \tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.383732, size = 36, normalized size = 2.57 \begin{align*} \frac{x \tan ^{2}{\left (x \right )}}{2 \tan ^{2}{\left (x \right )} + 2} + \frac{x}{2 \tan ^{2}{\left (x \right )} + 2} + \frac{\tan{\left (x \right )}}{2 \tan ^{2}{\left (x \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05259, size = 22, normalized size = 1.57 \begin{align*} \frac{1}{2} \, x + \frac{\tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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