Optimal. Leaf size=48 \[ \frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sin (3 x+2) \cos (3 x+2)}{\cos ^2(3 x+2)+\sqrt{2}+1}\right )}{3 \sqrt{2}} \]
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Rubi [A] time = 0.026868, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {203} \[ \frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sin (3 x+2) \cos (3 x+2)}{\cos ^2(3 x+2)+\sqrt{2}+1}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 203
Rubi steps
\begin{align*} \int \frac{1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{2+x^2} \, dx,x,\tan (2+3 x)\right )\\ &=\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\cos (2+3 x) \sin (2+3 x)}{1+\sqrt{2}+\cos ^2(2+3 x)}\right )}{3 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0205361, size = 22, normalized size = 0.46 \[ \frac{\tan ^{-1}\left (\frac{\tan (3 x+2)}{\sqrt{2}}\right )}{3 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 18, normalized size = 0.4 \begin{align*}{\frac{\sqrt{2}}{6}\arctan \left ({\frac{\tan \left ( 2+3\,x \right ) \sqrt{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57235, size = 23, normalized size = 0.48 \begin{align*} \frac{1}{6} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \tan \left (3 \, x + 2\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79003, size = 124, normalized size = 2.58 \begin{align*} -\frac{1}{12} \, \sqrt{2} \arctan \left (\frac{3 \, \sqrt{2} \cos \left (3 \, x + 2\right )^{2} - \sqrt{2}}{4 \, \cos \left (3 \, x + 2\right ) \sin \left (3 \, x + 2\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.2753, size = 76, normalized size = 1.58 \begin{align*} \frac{\sqrt{2} \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{3 x}{2} + 1 \right )} - 1 \right )} + \pi \left \lfloor{\frac{\frac{3 x}{2} - \frac{\pi }{2} + 1}{\pi }}\right \rfloor \right )}{6} + \frac{\sqrt{2} \left (\operatorname{atan}{\left (\sqrt{2} \tan{\left (\frac{3 x}{2} + 1 \right )} + 1 \right )} + \pi \left \lfloor{\frac{\frac{3 x}{2} - \frac{\pi }{2} + 1}{\pi }}\right \rfloor \right )}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13776, size = 77, normalized size = 1.6 \begin{align*} \frac{1}{6} \, \sqrt{2}{\left (3 \, x + \arctan \left (-\frac{\sqrt{2} \sin \left (6 \, x + 4\right ) - \sin \left (6 \, x + 4\right )}{\sqrt{2} \cos \left (6 \, x + 4\right ) + \sqrt{2} - \cos \left (6 \, x + 4\right ) + 1}\right ) + 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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