Optimal. Leaf size=22 \[ \frac {\sin \left (\sqrt {3} \sqrt {x^2+2}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.19, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 15
Rule 2637
Rule 3432
Rule 6715
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*}
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Mathematica [A] time = 0.06, size = 22, normalized size = 1.00 \[ \frac {\sin \left (\sqrt {3} \sqrt {x^2+2}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 37, normalized size = 1.68 \[ \frac {2 \, \sqrt {3} \tan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x^{2} + 2}\right )}{3 \, {\left (\tan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x^{2} + 2}\right )^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 17, normalized size = 0.77 \[ \frac {1}{3} \, \sqrt {3} \sin \left (\sqrt {3} \sqrt {x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 18, normalized size = 0.82 \[ \frac {\sin \left (\sqrt {3}\, \sqrt {x^{2}+2}\right ) \sqrt {3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 17, normalized size = 0.77 \[ \frac {1}{3} \, \sqrt {3} \sin \left (\sqrt {3} \sqrt {x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 15, normalized size = 0.68 \[ \frac {\sqrt {3}\,\sin \left (\sqrt {3\,x^2+6}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 20, normalized size = 0.91 \[ \frac {\sqrt {3} \sin {\left (\sqrt {3} \sqrt {x^{2} + 2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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