Integrand size = 37, antiderivative size = 24 \[ \int \frac {(-1+2 x) \cos \left (\sqrt {6+3 (-1+2 x)^2}\right )}{\sqrt {6+3 (-1+2 x)^2}} \, dx=\frac {1}{6} \sin \left (\sqrt {3} \sqrt {2+(-1+2 x)^2}\right ) \]
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Time = 0.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6847, 3513, 15, 2717} \[ \int \frac {(-1+2 x) \cos \left (\sqrt {6+3 (-1+2 x)^2}\right )}{\sqrt {6+3 (-1+2 x)^2}} \, dx=\frac {1}{6} \sin \left (\sqrt {3} \sqrt {(2 x-1)^2+2}\right ) \]
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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 {x \cos \left (\sqrt {6+3 x^2}\right )}{\sqrt {6+3 x^2}} \, dx,x,-1+2 x\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {\cos \left (\sqrt {6+3 x}\right )}{\sqrt {6+3 x}} \, dx,x,(-1+2 x)^2\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {x \cos (x)}{\sqrt {x^2}} \, dx,x,\sqrt {3} \sqrt {2+(-1+2 x)^2}\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \cos (x) \, dx,x,\sqrt {3} \sqrt {2+(-1+2 x)^2}\right ) \\ & = \frac {1}{6} \sin \left (\sqrt {3} \sqrt {2+(-1+2 x)^2}\right ) \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {(-1+2 x) \cos \left (\sqrt {6+3 (-1+2 x)^2}\right )}{\sqrt {6+3 (-1+2 x)^2}} \, dx=\frac {1}{6} \sin \left (\sqrt {6+3 (1-2 x)^2}\right ) \]
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Time = 4.67 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67
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none
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {(-1+2 x) \cos \left (\sqrt {6+3 (-1+2 x)^2}\right )}{\sqrt {6+3 (-1+2 x)^2}} \, dx=\frac {1}{6} \, \sin \left (\sqrt {12 \, x^{2} - 12 \, x + 9}\right ) \]
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Time = 1.98 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {(-1+2 x) \cos \left (\sqrt {6+3 (-1+2 x)^2}\right )}{\sqrt {6+3 (-1+2 x)^2}} \, dx=\frac {\sin {\left (\sqrt {3 \left (2 x - 1\right )^{2} + 6} \right )}}{6} \]
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Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {(-1+2 x) \cos \left (\sqrt {6+3 (-1+2 x)^2}\right )}{\sqrt {6+3 (-1+2 x)^2}} \, dx=\frac {1}{6} \, \sin \left (\sqrt {3 \, {\left (2 \, x - 1\right )}^{2} + 6}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {(-1+2 x) \cos \left (\sqrt {6+3 (-1+2 x)^2}\right )}{\sqrt {6+3 (-1+2 x)^2}} \, dx=\frac {1}{6} \, \sin \left (\sqrt {3} \sqrt {4 \, x^{2} - 4 \, x + 3}\right ) \]
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Time = 26.75 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {(-1+2 x) \cos \left (\sqrt {6+3 (-1+2 x)^2}\right )}{\sqrt {6+3 (-1+2 x)^2}} \, dx=\frac {\sin \left (\sqrt {3\,{\left (2\,x-1\right )}^2+6}\right )}{6} \]
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