Integrand size = 27, antiderivative size = 22 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sin \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {3}} \]
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6847, 3513, 15, 2717} \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sin \left (\sqrt {3} \sqrt {x^2+2}\right )}{\sqrt {3}} \]
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Rule 15
Rule 2717
Rule 3513
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\cos \left (\sqrt {3} \sqrt {2+x}\right )}{\sqrt {2+x}} \, dx,x,x^2\right ) \\ & = \text {Subst}\left (\int \frac {x \cos \left (\sqrt {3} x\right )}{\sqrt {x^2}} \, dx,x,\sqrt {2+x^2}\right ) \\ & = 1 \text {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*}
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sin \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {3}} \]
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Time = 0.57 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\sin \left (\sqrt {3}\, \sqrt {x^{2}+2}\right ) \sqrt {3}}{3}\) | \(18\) |
default | \(\frac {\sin \left (\sqrt {3}\, \sqrt {x^{2}+2}\right ) \sqrt {3}}{3}\) | \(18\) |
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {2 \, \sqrt {3} \tan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x^{2} + 2}\right )}{3 \, {\left (\tan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x^{2} + 2}\right )^{2} + 1\right )}} \]
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Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sqrt {3} \sin {\left (\sqrt {3} \sqrt {x^{2} + 2} \right )}}{3} \]
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none
Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \sin \left (\sqrt {3} \sqrt {x^{2} + 2}\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \sin \left (\sqrt {3} \sqrt {x^{2} + 2}\right ) \]
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Time = 26.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {x \cos \left (\sqrt {3} \sqrt {2+x^2}\right )}{\sqrt {2+x^2}} \, dx=\frac {\sqrt {3}\,\sin \left (\sqrt {3\,x^2+6}\right )}{3} \]
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