Optimal. Leaf size=127 \[ -\frac{x^2}{8}+\frac{1}{16} (1-2 x) \sqrt{x^2-x}-\frac{11 \sqrt{x^2-x}}{32}+\frac{1}{2} x^2 \log \left (4 \sqrt{x^2-x}+4 x-1\right )+\frac{1}{256} \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )-\frac{33}{128} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-x}}\right )+\frac{x}{32}-\frac{1}{256} \log (8 x+1) \]
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Rubi [A] time = 0.245347, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {2537, 2535, 6742, 640, 620, 206, 612, 734, 843, 724} \[ -\frac{x^2}{8}+\frac{1}{16} (1-2 x) \sqrt{x^2-x}-\frac{11 \sqrt{x^2-x}}{32}+\frac{1}{2} x^2 \log \left (4 \sqrt{x^2-x}+4 x-1\right )+\frac{1}{256} \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )-\frac{33}{128} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-x}}\right )+\frac{x}{32}-\frac{1}{256} \log (8 x+1) \]
Antiderivative was successfully verified.
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Rule 2537
Rule 2535
Rule 6742
Rule 640
Rule 620
Rule 206
Rule 612
Rule 734
Rule 843
Rule 724
Rubi steps
\begin{align*} \int x \log \left (-1+4 x+4 \sqrt{(-1+x) x}\right ) \, dx &=\int x \log \left (-1+4 x+4 \sqrt{-x+x^2}\right ) \, dx\\ &=\frac{1}{2} x^2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+4 \int \frac{x^2}{-4 (1+2 x) \sqrt{-x+x^2}+8 \left (-x+x^2\right )} \, dx\\ &=\frac{1}{2} x^2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+4 \int \left (\frac{1}{128}-\frac{x}{16}-\frac{1}{128 (1+8 x)}-\frac{x}{12 \sqrt{-x+x^2}}-\frac{1}{16} \sqrt{-x+x^2}+\frac{\sqrt{-x+x^2}}{48 (-1-8 x)}\right ) \, dx\\ &=\frac{x}{32}-\frac{x^2}{8}-\frac{1}{256} \log (1+8 x)+\frac{1}{2} x^2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{1}{12} \int \frac{\sqrt{-x+x^2}}{-1-8 x} \, dx-\frac{1}{4} \int \sqrt{-x+x^2} \, dx-\frac{1}{3} \int \frac{x}{\sqrt{-x+x^2}} \, dx\\ &=\frac{x}{32}-\frac{x^2}{8}-\frac{11}{32} \sqrt{-x+x^2}+\frac{1}{16} (1-2 x) \sqrt{-x+x^2}-\frac{1}{256} \log (1+8 x)+\frac{1}{2} x^2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{1}{192} \int \frac{1-10 x}{(-1-8 x) \sqrt{-x+x^2}} \, dx+\frac{1}{32} \int \frac{1}{\sqrt{-x+x^2}} \, dx-\frac{1}{6} \int \frac{1}{\sqrt{-x+x^2}} \, dx\\ &=\frac{x}{32}-\frac{x^2}{8}-\frac{11}{32} \sqrt{-x+x^2}+\frac{1}{16} (1-2 x) \sqrt{-x+x^2}-\frac{1}{256} \log (1+8 x)+\frac{1}{2} x^2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{5}{768} \int \frac{1}{\sqrt{-x+x^2}} \, dx+\frac{3}{256} \int \frac{1}{(-1-8 x) \sqrt{-x+x^2}} \, dx+\frac{1}{16} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )\\ &=\frac{x}{32}-\frac{x^2}{8}-\frac{11}{32} \sqrt{-x+x^2}+\frac{1}{16} (1-2 x) \sqrt{-x+x^2}-\frac{13}{48} \tanh ^{-1}\left (\frac{x}{\sqrt{-x+x^2}}\right )-\frac{1}{256} \log (1+8 x)+\frac{1}{2} x^2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{5}{384} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )-\frac{3}{128} \operatorname{Subst}\left (\int \frac{1}{36-x^2} \, dx,x,\frac{-1+10 x}{\sqrt{-x+x^2}}\right )\\ &=\frac{x}{32}-\frac{x^2}{8}-\frac{11}{32} \sqrt{-x+x^2}+\frac{1}{16} (1-2 x) \sqrt{-x+x^2}+\frac{1}{256} \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{-x+x^2}}\right )-\frac{33}{128} \tanh ^{-1}\left (\frac{x}{\sqrt{-x+x^2}}\right )-\frac{1}{256} \log (1+8 x)+\frac{1}{2} x^2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.269203, size = 102, normalized size = 0.8 \[ \frac{1}{256} \left (-32 x^2+128 x^2 \log \left (4 x+4 \sqrt{(x-1) x}-1\right )-32 \sqrt{(x-1) x} x+8 x-72 \sqrt{(x-1) x}-2 \log (8 x+1)-33 \log \left (-2 x-2 \sqrt{(x-1) x}+1\right )+\log \left (-10 x+6 \sqrt{(x-1) x}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.004, size = 0, normalized size = 0. \begin{align*} \int x\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ) \, dx \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*} \int x \log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.57492, size = 327, normalized size = 2.57 \begin{align*} -\frac{1}{8} \, x^{2} + \frac{1}{2} \,{\left (x^{2} - 1\right )} \log \left (4 \, x + 4 \, \sqrt{x^{2} - x} - 1\right ) - \frac{1}{32} \, \sqrt{x^{2} - x}{\left (4 \, x + 9\right )} + \frac{1}{32} \, x + \frac{63}{256} \, \log \left (8 \, x + 1\right ) - \frac{31}{256} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x} + 1\right ) + \frac{63}{256} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x} - 1\right ) - \frac{63}{256} \, \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 = 1.29707, size = 154, normalized size = 1.21 \begin{align*} \frac{1}{2} \, x^{2} \log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right ) - \frac{1}{8} \, x^{2} - \frac{1}{32} \, \sqrt{x^{2} - x}{\left (4 \, x + 9\right )} + \frac{1}{32} \, x - \frac{1}{256} \, \log \left ({\left | 8 \, x + 1 \right |}\right ) + \frac{33}{256} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x} + 1 \right |}\right ) - \frac{1}{256} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x} - 1 \right |}\right ) + \frac{1}{256} \, \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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