Optimal. Leaf size=27 \[ \frac{x}{3}-\frac{1}{3} \tan ^{-1}\left (\frac{2 \sin (x) \cos (x)}{2 \cos ^2(x)+1}\right ) \]
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Rubi [A] time = 0.0292568, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3675, 203} \[ \frac{x}{3}-\frac{1}{3} \tan ^{-1}\left (\frac{2 \sin (x) \cos (x)}{2 \cos ^2(x)+1}\right ) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec ^2(x)}{9+\tan ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{9+x^2} \, dx,x,\tan (x)\right )\\ &=\frac{x}{3}-\frac{1}{3} \tan ^{-1}\left (\frac{2 \cos (x) \sin (x)}{1+2 \cos ^2(x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0239948, size = 9, normalized size = 0.33 \[ -\frac{1}{3} \tan ^{-1}(3 \cot (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 8, normalized size = 0.3 \begin{align*}{\frac{1}{3}\arctan \left ({\frac{\tan \left ( x \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44673, size = 9, normalized size = 0.33 \begin{align*} \frac{1}{3} \, \arctan \left (\frac{1}{3} \, \tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13205, size = 70, normalized size = 2.59 \begin{align*} -\frac{1}{6} \, \arctan \left (\frac{10 \, \cos \left (x\right )^{2} - 1}{6 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (x \right )}}{\tan ^{2}{\left (x \right )} + 9}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08678, size = 9, normalized size = 0.33 \begin{align*} \frac{1}{3} \, \arctan \left (\frac{1}{3} \, \tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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