Optimal. Leaf size=80 \[ -3 \sqrt {x^2+x+1}+4 \tanh ^{-1}\left (\frac {1-x}{2 \sqrt {x^2+x+1}}\right )-\tanh ^{-1}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )+x+\log (x)-4 \log (x+1)+\frac {5}{2} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.34, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6742, 734, 843, 619, 215, 724, 206, 6740} \[ -3 \sqrt {x^2+x+1}+4 \tanh ^{-1}\left (\frac {1-x}{2 \sqrt {x^2+x+1}}\right )-\tanh ^{-1}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )+x+\log (x)-4 \log (x+1)+\frac {5}{2} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 619
Rule 724
Rule 734
Rule 843
Rule 6740
Rule 6742
Rubi steps
\begin {align*} \int \frac {-3 x+\sqrt {1+x+x^2}}{-1+\sqrt {1+x+x^2}} \, dx &=\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 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1-x}{\sqrt {1+x+x^2}}\right )+\frac {1}{2} \sqrt {3} \operatorname {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} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+3 \tanh ^{-1}\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} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+3 \tanh ^{-1}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right )+\log (x)-4 \log (1+x)+2 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1-x}{\sqrt {1+x+x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2+x}{\sqrt {1+x+x^2}}\right )+2 \frac {\operatorname {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} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+4 \tanh ^{-1}\left (\frac {1-x}{2 \sqrt {1+x+x^2}}\right )-\tanh ^{-1}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right )+\log (x)-4 \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.12, size = 80, normalized size = 1.00 \[ -3 \sqrt {x^2+x+1}+4 \tanh ^{-1}\left (\frac {1-x}{2 \sqrt {x^2+x+1}}\right )-\tanh ^{-1}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )+x+\log (x)-4 \log (x+1)+\frac {5}{2} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 72, normalized size = 0.90 \[ -3 \sqrt {x^2+x+1}-8 \log \left (\sqrt {x^2+x+1}-x-2\right )+2 \log \left (\sqrt {x^2+x+1}-x-1\right )+\frac {1}{2} \log \left (2 \sqrt {x^2+x+1}-2 x-1\right )+x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 99, normalized size = 1.24 \[ x - 3 \, \sqrt {x^{2} + x + 1} - 4 \, \log \left (x + 1\right ) + \log \relax (x) - \log \left (-x + \sqrt {x^{2} + x + 1} + 1\right ) + 4 \, \log \left (-x + \sqrt {x^{2} + x + 1}\right ) + \log \left (-x + \sqrt {x^{2} + x + 1} - 1\right ) - 4 \, \log \left (-x + \sqrt {x^{2} + x + 1} - 2\right ) - \frac {5}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 105, normalized size = 1.31 \[ x - 3 \, \sqrt {x^{2} + x + 1} - \frac {5}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) - 4 \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) - \log \left ({\left | -x + \sqrt {x^{2} + x + 1} + 1 \right |}\right ) + 4 \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} \right |}\right ) + \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 1 \right |}\right ) - 4 \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 80, normalized size = 1.00
method | result | size |
default | \(-4 \ln \left (1+x \right )+\ln \relax (x )+x -4 \sqrt {\left (1+x \right )^{2}-x}+\frac {5 \arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{2}+4 \arctanh \left (\frac {1-x}{2 \sqrt {\left (1+x \right )^{2}-x}}\right )+\sqrt {x^{2}+x +1}-\arctanh \left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )\) | \(80\) |
trager | \(-1+x -3 \sqrt {x^{2}+x +1}+\frac {\ln \left (\frac {-8-96 x +5493060 x^{7}+1790544 x^{5}-1526 x^{3}+3865870 x^{6}+445596 x^{4}-507 x^{2}+4224608 x^{9}+8448 x^{14}+5593140 x^{8}+458 x^{2} \sqrt {x^{2}+x +1}+8 \sqrt {x^{2}+x +1}+2392341 x^{10}+308624 x^{12}+512 x^{15}+64992 x^{13}+1008642 x^{11}+92 \sqrt {x^{2}+x +1}\, x +512 \sqrt {x^{2}+x +1}\, x^{14}+8192 \sqrt {x^{2}+x +1}\, x^{13}+60704 \sqrt {x^{2}+x +1}\, x^{12}+275296 \sqrt {x^{2}+x +1}\, x^{11}+849754 \sqrt {x^{2}+x +1}\, x^{10}+1875388 \sqrt {x^{2}+x +1}\, x^{9}+3018000 \sqrt {x^{2}+x +1}\, x^{8}+3530640 \sqrt {x^{2}+x +1}\, x^{7}+2914860 \sqrt {x^{2}+x +1}\, x^{6}+1571080 \sqrt {x^{2}+x +1}\, x^{5}+450424 \sqrt {x^{2}+x +1}\, x^{4}+1264 \sqrt {x^{2}+x +1}\, x^{3}}{\left (1+x \right )^{16}}\right )}{2}\) | \(288\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3}{4} \, x^{2} + \frac {1}{2} \, x + \int -\frac {3 \, x^{3} + 2 \, x^{2} - x}{2 \, {\left (x^{2} + x - 2 \, \sqrt {x^{2} + x + 1} + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ x-4\,\ln \left (x+1\right )+\ln \relax (x)-\int \frac {\left (3\,x-1\right )\,\sqrt {x^2+x+1}}{x\,\left (x+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {3 x}{\sqrt {x^{2} + x + 1} - 1}\, dx - \int \left (- \frac {\sqrt {x^{2} + x + 1}}{\sqrt {x^{2} + x + 1} - 1}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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