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 = 0.25, antiderivative size = 127, normalized size of antiderivative = 1.00, 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 206
Rule 612
Rule 620
Rule 640
Rule 724
Rule 734
Rule 843
Rule 2535
Rule 2537
Rule 6742
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*}
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Mathematica [A] time = 0.29, size = 102, normalized size = 0.80 \[ \frac {1}{256} \left (-32 x^2+128 x^2 \log \left (4 x+4 \sqrt {(x-1) x}-1\right )-32 \sqrt {(x-1) x} x+8 x-72 \sqrt {(x-1) x}-2 \log (8 x+1)-33 \log \left (-2 x-2 \sqrt {(x-1) x}+1\right )+\log \left (-10 x+6 \sqrt {(x-1) x}+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 114, normalized size = 0.90 \[ -\frac {1}{8} \, x^{2} + \frac {1}{2} \, {\left (x^{2} - 1\right )} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - \frac {1}{32} \, \sqrt {x^{2} - x} {\left (4 \, x + 9\right )} + \frac {1}{32} \, x + \frac {63}{256} \, \log \left (8 \, x + 1\right ) - \frac {31}{256} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) + \frac {63}{256} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) - \frac {63}{256} \, \log \left (-4 \, x + 4 \, \sqrt {x^{2} - x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 114, normalized size = 0.90 \[ \frac {1}{2} \, x^{2} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) - \frac {1}{8} \, x^{2} - \frac {1}{32} \, \sqrt {x^{2} - x} {\left (4 \, x + 9\right )} + \frac {1}{32} \, x - \frac {1}{256} \, \log \left ({\left | 8 \, x + 1 \right |}\right ) + \frac {33}{256} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} + 1 \right |}\right ) - \frac {1}{256} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} - 1 \right |}\right ) + \frac {1}{256} \, \log \left ({\left | -4 \, x + 4 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x \ln \left (4 x -1+4 \sqrt {\left (x -1\right ) x}\right )\, dx \]
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 \[ \int x \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )\,{d x} \]
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 \[ \int x\,\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
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 \[ \int x \log {\left (4 x + 4 \sqrt {x^{2} - x} - 1 \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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