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 = 0.16, antiderivative size = 95, normalized size of antiderivative = 1.00, 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 206
Rule 620
Rule 640
Rule 724
Rule 734
Rule 843
Rule 2533
Rule 2537
Rule 6742
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.82, size = 101, normalized size = 1.06 \[ {\left (x + 1\right )} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - x} - \frac {7}{16} \, \log \left (8 \, x + 1\right ) + \frac {15}{16} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) - \frac {7}{16} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) + \frac {7}{16} \, \log \left (-4 \, x + 4 \, \sqrt {x^{2} - x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 101, normalized size = 1.06 \[ x \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) - \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - x} + \frac {1}{16} \, \log \left ({\left | 8 \, x + 1 \right |}\right ) + \frac {7}{16} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} + 1 \right |}\right ) + \frac {1}{16} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} - 1 \right |}\right ) - \frac {1}{16} \, \log \left ({\left | -4 \, x + 4 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 80, normalized size = 0.84 \[ x \ln \left (4 x -1+4 \sqrt {\left (x -1\right ) x}\right )-\frac {x}{2}-\frac {\arctanh \left (\frac {-\frac {40 x}{3}+\frac {4}{3}}{\sqrt {-80 x +64 \left (x +\frac {1}{8}\right )^{2}-1}}\right )}{16}+\frac {\ln \left (8 x +1\right )}{16}-\frac {7 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x}\right )}{16}-\frac {\sqrt {x^{2}-x}}{2} \]
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 \[ x \log \left (4 \, \sqrt {x - 1} \sqrt {x} + 4 \, x - 1\right ) - \frac {1}{2} \, x + \int \frac {2 \, x^{2} + x}{2 \, {\left (4 \, x^{3} - 5 \, x^{2} + 4 \, {\left (x^{\frac {5}{2}} - x^{\frac {3}{2}}\right )} \sqrt {x - 1} + x\right )}}\,{d x} - \frac {1}{2} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x} - 1\right ) \]
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 \[ \int \ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
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 \[ \int \log {\left (4 x + 4 \sqrt {x \left (x - 1\right )} - 1 \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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