Integrand size = 21, antiderivative size = 10 \[ \int \frac {x \cos \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sin \left (\sqrt {1+x^2}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6847, 3513, 15, 2717} \[ \int \frac {x \cos \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sin \left (\sqrt {x^2+1}\right ) \]
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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 {1+x}\right )}{\sqrt {1+x}} \, dx,x,x^2\right ) \\ & = \text {Subst}\left (\int \frac {x \cos (x)}{\sqrt {x^2}} \, dx,x,\sqrt {1+x^2}\right ) \\ & = 1 \text {Subst}\left (\int \cos (x) \, dx,x,\sqrt {1+x^2}\right ) \\ & = \sin \left (\sqrt {1+x^2}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x \cos \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sin \left (\sqrt {1+x^2}\right ) \]
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Time = 0.52 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\sin \left (\sqrt {x^{2}+1}\right )\) | \(9\) |
default | \(\sin \left (\sqrt {x^{2}+1}\right )\) | \(9\) |
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none
Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x \cos \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sin \left (\sqrt {x^{2} + 1}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x \cos \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sin {\left (\sqrt {x^{2} + 1} \right )} \]
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none
Time = 0.21 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x \cos \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sin \left (\sqrt {x^{2} + 1}\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x \cos \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sin \left (\sqrt {x^{2} + 1}\right ) \]
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Time = 25.94 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {x \cos \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sin \left (\sqrt {x^2+1}\right ) \]
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