Integrand size = 29, antiderivative size = 80 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x-3 \sqrt {1+x+x^2}+\frac {5}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right )+4 \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right )-\text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right )+\log (x)-4 \log (1+x) \]
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Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6874, 748, 857, 633, 221, 738, 212, 6872} \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=\frac {5}{2} \text {arcsinh}\left (\frac {2 x+1}{\sqrt {3}}\right )+4 \text {arctanh}\left (\frac {1-x}{2 \sqrt {x^2+x+1}}\right )-\text {arctanh}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )-3 \sqrt {x^2+x+1}+x+\log (x)-4 \log (x+1) \]
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Rule 212
Rule 221
Rule 633
Rule 738
Rule 748
Rule 857
Rule 6872
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 x}{-1+\sqrt {1+x+x^2}}+\frac {\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}}\right ) \, dx \\ & = -\left (3 \int \frac {x}{-1+\sqrt {1+x+x^2}} \, dx\right )+\int \frac {\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx \\ & = -\left (3 \int \left (\frac {1}{1+x}+\frac {\sqrt {1+x+x^2}}{1+x}\right ) \, dx\right )+\int \left (1+\frac {1}{-1+\sqrt {1+x+x^2}}\right ) \, dx \\ & = x-3 \log (1+x)-3 \int \frac {\sqrt {1+x+x^2}}{1+x} \, dx+\int \frac {1}{-1+\sqrt {1+x+x^2}} \, dx \\ & = x-3 \sqrt {1+x+x^2}-3 \log (1+x)+\frac {3}{2} \int \frac {-1+x}{(1+x) \sqrt {1+x+x^2}} \, dx+\int \left (\frac {1}{-1-x}+\frac {1}{x}+\frac {\sqrt {1+x+x^2}}{x}-\frac {\sqrt {1+x+x^2}}{1+x}\right ) \, dx \\ & = x-3 \sqrt {1+x+x^2}+\log (x)-4 \log (1+x)+\frac {3}{2} \int \frac {1}{\sqrt {1+x+x^2}} \, dx-3 \int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx+\int \frac {\sqrt {1+x+x^2}}{x} \, dx-\int \frac {\sqrt {1+x+x^2}}{1+x} \, dx \\ & = x-3 \sqrt {1+x+x^2}+\log (x)-4 \log (1+x)-\frac {1}{2} \int \frac {-2-x}{x \sqrt {1+x+x^2}} \, dx+\frac {1}{2} \int \frac {-1+x}{(1+x) \sqrt {1+x+x^2}} \, dx+6 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1-x}{\sqrt {1+x+x^2}}\right )+\frac {1}{2} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right ) \\ & = x-3 \sqrt {1+x+x^2}+\frac {3}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right )+3 \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right )+\log (x)-4 \log (1+x)+2 \left (\frac {1}{2} \int \frac {1}{\sqrt {1+x+x^2}} \, dx\right )+\int \frac {1}{x \sqrt {1+x+x^2}} \, dx-\int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx \\ & = x-3 \sqrt {1+x+x^2}+\frac {3}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right )+3 \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right )+\log (x)-4 \log (1+x)+2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1-x}{\sqrt {1+x+x^2}}\right )-2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2+x}{\sqrt {1+x+x^2}}\right )+2 \frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )}{2 \sqrt {3}} \\ & = x-3 \sqrt {1+x+x^2}+\frac {5}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right )+4 \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right )-\text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right )+\log (x)-4 \log (1+x) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x-3 \sqrt {1+x+x^2}-8 \log \left (-2-x+\sqrt {1+x+x^2}\right )+2 \log \left (-1-x+\sqrt {1+x+x^2}\right )+\frac {1}{2} \log \left (-1-2 x+2 \sqrt {1+x+x^2}\right ) \]
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Time = 0.44 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00
method | result | size |
default | \(\ln \left (x \right )-4 \ln \left (1+x \right )+x +\sqrt {x^{2}+x +1}+\frac {5 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}-\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )-4 \sqrt {\left (1+x \right )^{2}-x}+4 \,\operatorname {arctanh}\left (\frac {1-x}{2 \sqrt {\left (1+x \right )^{2}-x}}\right )\) | \(80\) |
trager | \(-1+x -3 \sqrt {x^{2}+x +1}+\frac {\ln \left (\frac {-8+3865870 x^{6}-96 x +1790544 x^{5}+8 \sqrt {x^{2}+x +1}+445596 x^{4}-1526 x^{3}-507 x^{2}+2392341 x^{10}+308624 x^{12}+8448 x^{14}+4224608 x^{9}+1008642 x^{11}+64992 x^{13}+5493060 x^{7}+92 x \sqrt {x^{2}+x +1}+5593140 x^{8}+512 x^{15}+458 x^{2} \sqrt {x^{2}+x +1}+512 \sqrt {x^{2}+x +1}\, x^{14}+8192 \sqrt {x^{2}+x +1}\, x^{13}+60704 \sqrt {x^{2}+x +1}\, x^{12}+275296 \sqrt {x^{2}+x +1}\, x^{11}+849754 \sqrt {x^{2}+x +1}\, x^{10}+1875388 \sqrt {x^{2}+x +1}\, x^{9}+3018000 \sqrt {x^{2}+x +1}\, x^{8}+3530640 \sqrt {x^{2}+x +1}\, x^{7}+2914860 \sqrt {x^{2}+x +1}\, x^{6}+1571080 \sqrt {x^{2}+x +1}\, x^{5}+450424 x^{4} \sqrt {x^{2}+x +1}+1264 x^{3} \sqrt {x^{2}+x +1}}{\left (1+x \right )^{16}}\right )}{2}\) | \(288\) |
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Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.24 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x - 3 \, \sqrt {x^{2} + x + 1} - 4 \, \log \left (x + 1\right ) + \log \left (x\right ) - \log \left (-x + \sqrt {x^{2} + x + 1} + 1\right ) + 4 \, \log \left (-x + \sqrt {x^{2} + x + 1}\right ) + \log \left (-x + \sqrt {x^{2} + x + 1} - 1\right ) - 4 \, \log \left (-x + \sqrt {x^{2} + x + 1} - 2\right ) - \frac {5}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
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\[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=- \int \frac {3 x}{\sqrt {x^{2} + x + 1} - 1}\, dx - \int \left (- \frac {\sqrt {x^{2} + x + 1}}{\sqrt {x^{2} + x + 1} - 1}\right )\, dx \]
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\[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=\int { -\frac {3 \, x - \sqrt {x^{2} + x + 1}}{\sqrt {x^{2} + x + 1} - 1} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.31 \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x - 3 \, \sqrt {x^{2} + x + 1} - \frac {5}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) - 4 \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) - \log \left ({\left | -x + \sqrt {x^{2} + x + 1} + 1 \right |}\right ) + 4 \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 1 \right |}\right ) - 4 \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 2 \right |}\right ) \]
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Timed out. \[ \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx=x-4\,\ln \left (x+1\right )+\ln \left (x\right )-\int \frac {\left (3\,x-1\right )\,\sqrt {x^2+x+1}}{x\,\left (x+1\right )} \,d x \]
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