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 = 0.343471, antiderivative size = 80, normalized size of antiderivative = 1., 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 6742
Rule 734
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rule 6740
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*}
Mathematica [A] time = 0.110335, size = 80, normalized size = 1. \[ -3 \sqrt{x^2+x+1}+4 \tanh ^{-1}\left (\frac{1-x}{2 \sqrt{x^2+x+1}}\right )-\tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right )+x+\log (x)-4 \log (x+1)+\frac{5}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 80, normalized size = 1. \begin{align*} x-4\,\ln \left ( 1+x \right ) +\ln \left ( x \right ) +\sqrt{{x}^{2}+x+1}+{\frac{5}{2}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) }-{\it Artanh} \left ({\frac{2+x}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \right ) -4\,\sqrt{ \left ( 1+x \right ) ^{2}-x}+4\,{\it Artanh} \left ( 1/2\,{\frac{1-x}{\sqrt{ \left ( 1+x \right ) ^{2}-x}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93481, size = 306, normalized size = 3.82 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{3 x}{\sqrt{x^{2} + x + 1} - 1}\, dx - \int - \frac{\sqrt{x^{2} + x + 1}}{\sqrt{x^{2} + x + 1} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14201, size = 142, normalized size = 1.78 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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