Integrand size = 17, antiderivative size = 56 \[ \int \frac {\sqrt {2+x^2}}{1+4 x} \, dx=\frac {\sqrt {2+x^2}}{4}-\frac {1}{16} \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{16} \sqrt {33} \text {arctanh}\left (\frac {8-x}{\sqrt {33} \sqrt {2+x^2}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {749, 858, 221, 739, 212} \[ \int \frac {\sqrt {2+x^2}}{1+4 x} \, dx=-\frac {1}{16} \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{16} \sqrt {33} \text {arctanh}\left (\frac {8-x}{\sqrt {33} \sqrt {x^2+2}}\right )+\frac {\sqrt {x^2+2}}{4} \]
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Rule 212
Rule 221
Rule 739
Rule 749
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2+x^2}}{4}+\frac {1}{4} \int \frac {8-x}{(1+4 x) \sqrt {2+x^2}} \, dx \\ & = \frac {\sqrt {2+x^2}}{4}-\frac {1}{16} \int \frac {1}{\sqrt {2+x^2}} \, dx+\frac {33}{16} \int \frac {1}{(1+4 x) \sqrt {2+x^2}} \, dx \\ & = \frac {\sqrt {2+x^2}}{4}-\frac {1}{16} \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {33}{16} \text {Subst}\left (\int \frac {1}{33-x^2} \, dx,x,\frac {8-x}{\sqrt {2+x^2}}\right ) \\ & = \frac {\sqrt {2+x^2}}{4}-\frac {1}{16} \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{16} \sqrt {33} \tanh ^{-1}\left (\frac {8-x}{\sqrt {33} \sqrt {2+x^2}}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {2+x^2}}{1+4 x} \, dx=\frac {1}{16} \left (4 \sqrt {2+x^2}+2 \sqrt {33} \text {arctanh}\left (\frac {1+4 x-4 \sqrt {2+x^2}}{\sqrt {33}}\right )+\log \left (-x+\sqrt {2+x^2}\right )\right ) \]
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Time = 1.98 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {\sqrt {x^{2}+2}}{4}-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {2}\, x}{2}\right )}{16}-\frac {\sqrt {33}\, \operatorname {arctanh}\left (\frac {8 \left (4-\frac {x}{2}\right ) \sqrt {33}}{33 \sqrt {16 \left (x +\frac {1}{4}\right )^{2}-8 x +31}}\right )}{16}\) | \(50\) |
default | \(\frac {\sqrt {16 \left (x +\frac {1}{4}\right )^{2}-8 x +31}}{16}-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {2}\, x}{2}\right )}{16}-\frac {\sqrt {33}\, \operatorname {arctanh}\left (\frac {8 \left (4-\frac {x}{2}\right ) \sqrt {33}}{33 \sqrt {16 \left (x +\frac {1}{4}\right )^{2}-8 x +31}}\right )}{16}\) | \(57\) |
trager | \(\frac {\sqrt {x^{2}+2}}{4}-\frac {\ln \left (x +\sqrt {x^{2}+2}\right )}{16}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-33\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-33\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-33\right )+33 \sqrt {x^{2}+2}}{1+4 x}\right )}{16}\) | \(66\) |
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Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {2+x^2}}{1+4 x} \, dx=\frac {1}{16} \, \sqrt {33} \log \left (-\frac {\sqrt {33} {\left (x - 8\right )} + \sqrt {x^{2} + 2} {\left (\sqrt {33} + 33\right )} + x - 8}{4 \, x + 1}\right ) + \frac {1}{4} \, \sqrt {x^{2} + 2} + \frac {1}{16} \, \log \left (-x + \sqrt {x^{2} + 2}\right ) \]
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\[ \int \frac {\sqrt {2+x^2}}{1+4 x} \, dx=\int \frac {\sqrt {x^{2} + 2}}{4 x + 1}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {2+x^2}}{1+4 x} \, dx=\frac {1}{16} \, \sqrt {33} \operatorname {arsinh}\left (\frac {\sqrt {2} x}{2 \, {\left | 4 \, x + 1 \right |}} - \frac {4 \, \sqrt {2}}{{\left | 4 \, x + 1 \right |}}\right ) + \frac {1}{4} \, \sqrt {x^{2} + 2} - \frac {1}{16} \, \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {2+x^2}}{1+4 x} \, dx=\frac {1}{16} \, \sqrt {33} \log \left (\frac {{\left | -4 \, x - \sqrt {33} + 4 \, \sqrt {x^{2} + 2} - 1 \right |}}{{\left | -4 \, x + \sqrt {33} + 4 \, \sqrt {x^{2} + 2} - 1 \right |}}\right ) + \frac {1}{4} \, \sqrt {x^{2} + 2} + \frac {1}{16} \, \log \left (-x + \sqrt {x^{2} + 2}\right ) \]
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Time = 9.48 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {2+x^2}}{1+4 x} \, dx=\frac {\sqrt {x^2+2}}{4}-\frac {\mathrm {asinh}\left (\frac {\sqrt {2}\,x}{2}\right )}{16}+\frac {\sqrt {33}\,\left (132\,\ln \left (x+\frac {1}{4}\right )-132\,\ln \left (x-\sqrt {33}\,\sqrt {x^2+2}-8\right )\right )}{2112} \]
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