Integrand size = 16, antiderivative size = 10 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \cos \left (\sqrt {1+x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3512, 15, 2718} \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \cos \left (\sqrt {x+1}\right ) \]
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Rule 15
Rule 2718
Rule 3512
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \sin (x)}{\sqrt {x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = 2 \text {Subst}\left (\int \sin (x) \, dx,x,\sqrt {1+x}\right ) \\ & = -2 \cos \left (\sqrt {1+x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \cos \left (\sqrt {1+x}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(-2 \cos \left (\sqrt {1+x}\right )\) | \(9\) |
default | \(-2 \cos \left (\sqrt {1+x}\right )\) | \(9\) |
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none
Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \, \cos \left (\sqrt {x + 1}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=- 2 \cos {\left (\sqrt {x + 1} \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \, \cos \left (\sqrt {x + 1}\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2 \, \cos \left (\sqrt {x + 1}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\sin \left (\sqrt {1+x}\right )}{\sqrt {1+x}} \, dx=-2\,\cos \left (\sqrt {x+1}\right ) \]
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