Optimal. Leaf size=22 \[ \frac{\sin \left (\sqrt{3} \sqrt{x^2+2}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.189288, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {6715, 3432, 15, 2637} \[ \frac{\sin \left (\sqrt{3} \sqrt{x^2+2}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 3432
Rule 15
Rule 2637
Rubi steps
\begin{align*} \int \frac{x \cos \left (\sqrt{3} \sqrt{2+x^2}\right )}{\sqrt{2+x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\cos \left (\sqrt{3} \sqrt{2+x}\right )}{\sqrt{2+x}} \, dx,x,x^2\right )\\ &=\operatorname{Subst}\left (\int \frac{x \cos \left (\sqrt{3} x\right )}{\sqrt{x^2}} \, dx,x,\sqrt{2+x^2}\right )\\ &=1 \operatorname{Subst}\left (\int \cos \left (\sqrt{3} x\right ) \, dx,x,\sqrt{2+x^2}\right )\\ &=\frac{\sin \left (\sqrt{3} \sqrt{2+x^2}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0526874, size = 22, normalized size = 1. \[ \frac{\sin \left (\sqrt{3} \sqrt{x^2+2}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 18, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{3}\sin \left ( \sqrt{3}\sqrt{{x}^{2}+2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59016, size = 23, normalized size = 1.05 \begin{align*} \frac{1}{3} \, \sqrt{3} \sin \left (\sqrt{3} \sqrt{x^{2} + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.33569, size = 112, normalized size = 5.09 \begin{align*} \frac{2 \, \sqrt{3} \tan \left (\frac{1}{2} \, \sqrt{3} \sqrt{x^{2} + 2}\right )}{3 \,{\left (\tan \left (\frac{1}{2} \, \sqrt{3} \sqrt{x^{2} + 2}\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.58456, size = 20, normalized size = 0.91 \begin{align*} \frac{\sqrt{3} \sin{\left (\sqrt{3} \sqrt{x^{2} + 2} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09903, size = 23, normalized size = 1.05 \begin{align*} \frac{1}{3} \, \sqrt{3} \sin \left (\sqrt{3} \sqrt{x^{2} + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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