Optimal. Leaf size=95 \[ -\frac{\sqrt{x^2-x}}{2}+x \log \left (4 \sqrt{x^2-x}+4 x-1\right )-\frac{1}{16} \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )-\frac{7}{8} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-x}}\right )-\frac{x}{2}+\frac{1}{16} \log (8 x+1) \]
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Rubi [A] time = 0.162303, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {2537, 2533, 6742, 640, 620, 206, 734, 843, 724} \[ -\frac{\sqrt{x^2-x}}{2}+x \log \left (4 \sqrt{x^2-x}+4 x-1\right )-\frac{1}{16} \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )-\frac{7}{8} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-x}}\right )-\frac{x}{2}+\frac{1}{16} \log (8 x+1) \]
Antiderivative was successfully verified.
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Rule 2537
Rule 2533
Rule 6742
Rule 640
Rule 620
Rule 206
Rule 734
Rule 843
Rule 724
Rubi steps
\begin{align*} \int \log \left (-1+4 x+4 \sqrt{(-1+x) x}\right ) \, dx &=\int \log \left (-1+4 x+4 \sqrt{-x+x^2}\right ) \, dx\\ &=x \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+8 \int \frac{x}{-4 (1+2 x) \sqrt{-x+x^2}+8 \left (-x+x^2\right )} \, dx\\ &=x \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+8 \int \left (-\frac{1}{16}+\frac{1}{16 (1+8 x)}-\frac{x}{12 \sqrt{-x+x^2}}+\frac{\sqrt{-x+x^2}}{6 (1+8 x)}\right ) \, dx\\ &=-\frac{x}{2}+\frac{1}{16} \log (1+8 x)+x \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{2}{3} \int \frac{x}{\sqrt{-x+x^2}} \, dx+\frac{4}{3} \int \frac{\sqrt{-x+x^2}}{1+8 x} \, dx\\ &=-\frac{x}{2}-\frac{1}{2} \sqrt{-x+x^2}+\frac{1}{16} \log (1+8 x)+x \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{1}{12} \int \frac{-1+10 x}{(1+8 x) \sqrt{-x+x^2}} \, dx-\frac{1}{3} \int \frac{1}{\sqrt{-x+x^2}} \, dx\\ &=-\frac{x}{2}-\frac{1}{2} \sqrt{-x+x^2}+\frac{1}{16} \log (1+8 x)+x \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{5}{48} \int \frac{1}{\sqrt{-x+x^2}} \, dx+\frac{3}{16} \int \frac{1}{(1+8 x) \sqrt{-x+x^2}} \, dx-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )\\ &=-\frac{x}{2}-\frac{1}{2} \sqrt{-x+x^2}-\frac{2}{3} \tanh ^{-1}\left (\frac{x}{\sqrt{-x+x^2}}\right )+\frac{1}{16} \log (1+8 x)+x \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{5}{24} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )-\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{36-x^2} \, dx,x,\frac{1-10 x}{\sqrt{-x+x^2}}\right )\\ &=-\frac{x}{2}-\frac{1}{2} \sqrt{-x+x^2}-\frac{1}{16} \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{-x+x^2}}\right )-\frac{7}{8} \tanh ^{-1}\left (\frac{x}{\sqrt{-x+x^2}}\right )+\frac{1}{16} \log (1+8 x)+x \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0282042, size = 85, normalized size = 0.89 \[ \frac{1}{16} \left (-8 x-8 \sqrt{(x-1) x}+16 x \log \left (4 x+4 \sqrt{(x-1) x}-1\right )+2 \log (8 x+1)-7 \log \left (-2 x-2 \sqrt{(x-1) x}+1\right )-\log \left (-10 x+6 \sqrt{(x-1) x}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 80, normalized size = 0.8 \begin{align*} x\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ) -{\frac{7}{16}\ln \left ( -{\frac{1}{2}}+x+\sqrt{{x}^{2}-x} \right ) }-{\frac{1}{16}{\it Artanh} \left ({\frac{32}{3} \left ({\frac{1}{8}}-{\frac{5\,x}{4}} \right ){\frac{1}{\sqrt{64\, \left ( x+1/8 \right ) ^{2}-80\,x-1}}}} \right ) }-{\frac{1}{2}\sqrt{{x}^{2}-x}}-{\frac{x}{2}}+{\frac{\ln \left ( 1+8\,x \right ) }{16}} \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*} x \log \left (4 \, \sqrt{x - 1} \sqrt{x} + 4 \, x - 1\right ) - \frac{1}{2} \, x + \int \frac{2 \, x^{2} + x}{2 \,{\left (4 \, x^{3} - 5 \, x^{2} + 4 \,{\left (x^{\frac{5}{2}} - x^{\frac{3}{2}}\right )} \sqrt{x - 1} + x\right )}}\,{d x} - \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61002, size = 278, normalized size = 2.93 \begin{align*}{\left (x + 1\right )} \log \left (4 \, x + 4 \, \sqrt{x^{2} - x} - 1\right ) - \frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x^{2} - x} - \frac{7}{16} \, \log \left (8 \, x + 1\right ) + \frac{15}{16} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x} + 1\right ) - \frac{7}{16} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x} - 1\right ) + \frac{7}{16} \, \log \left (-4 \, x + 4 \, \sqrt{x^{2} - x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32307, size = 136, normalized size = 1.43 \begin{align*} x \log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right ) - \frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x^{2} - x} + \frac{1}{16} \, \log \left ({\left | 8 \, x + 1 \right |}\right ) + \frac{7}{16} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x} + 1 \right |}\right ) + \frac{1}{16} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x} - 1 \right |}\right ) - \frac{1}{16} \, \log \left ({\left | -4 \, x + 4 \, \sqrt{x^{2} - x} + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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