Optimal. Leaf size=76 \[ \frac {4 \sqrt {-x+x^2}}{x}+4 \tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {-x+x^2}}\right )+4 \log (x)-4 \log (1+8 x)-\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x} \]
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Rubi [A]
time = 0.17, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {2617, 2615,
6874, 654, 634, 212, 676, 678, 748, 857, 738} \begin {gather*} \frac {4 \sqrt {x^2-x}}{x}-\frac {\log \left (4 \sqrt {x^2-x}+4 x-1\right )}{x}+4 \tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {x^2-x}}\right )+4 \log (x)-4 \log (8 x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 654
Rule 676
Rule 678
Rule 738
Rule 748
Rule 857
Rule 2615
Rule 2617
Rule 6874
Rubi steps
\begin {align*} \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^2} \, dx &=\int \frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x^2} \, dx\\ &=-\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x}-8 \int \frac {1}{x \left (-4 (1+2 x) \sqrt {-x+x^2}+8 \left (-x+x^2\right )\right )} \, dx\\ &=-\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x}-8 \int \left (-\frac {1}{2 x}+\frac {4}{1+8 x}-\frac {x}{12 \sqrt {-x+x^2}}+\frac {\sqrt {-x+x^2}}{4 x^2}-\frac {5 \sqrt {-x+x^2}}{4 x}+\frac {32 \sqrt {-x+x^2}}{3 (1+8 x)}\right ) \, dx\\ &=4 \log (x)-4 \log (1+8 x)-\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x}+\frac {2}{3} \int \frac {x}{\sqrt {-x+x^2}} \, dx-2 \int \frac {\sqrt {-x+x^2}}{x^2} \, dx+10 \int \frac {\sqrt {-x+x^2}}{x} \, dx-\frac {256}{3} \int \frac {\sqrt {-x+x^2}}{1+8 x} \, dx\\ &=\frac {4 \sqrt {-x+x^2}}{x}+4 \log (x)-4 \log (1+8 x)-\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x}+\frac {1}{3} \int \frac {1}{\sqrt {-x+x^2}} \, dx-2 \int \frac {1}{\sqrt {-x+x^2}} \, dx-5 \int \frac {1}{\sqrt {-x+x^2}} \, dx+\frac {16}{3} \int \frac {-1+10 x}{(1+8 x) \sqrt {-x+x^2}} \, dx\\ &=\frac {4 \sqrt {-x+x^2}}{x}+4 \log (x)-4 \log (1+8 x)-\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x}+\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )-4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )+\frac {20}{3} \int \frac {1}{\sqrt {-x+x^2}} \, dx-10 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )-12 \int \frac {1}{(1+8 x) \sqrt {-x+x^2}} \, dx\\ &=\frac {4 \sqrt {-x+x^2}}{x}-\frac {40}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )+4 \log (x)-4 \log (1+8 x)-\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x}+\frac {40}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )+24 \text {Subst}\left (\int \frac {1}{36-x^2} \, dx,x,\frac {1-10 x}{\sqrt {-x+x^2}}\right )\\ &=\frac {4 \sqrt {-x+x^2}}{x}+4 \tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {-x+x^2}}\right )+4 \log (x)-4 \log (1+8 x)-\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 68, normalized size = 0.89 \begin {gather*} \frac {4 \sqrt {(-1+x) x}}{x}+4 \log (x)-8 \log (1+8 x)-\frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x}+4 \log \left (1-10 x+6 \sqrt {(-1+x) x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (-1+4 x +4 \sqrt {\left (-1+x \right ) x}\right )}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 115, normalized size = 1.51 \begin {gather*} -\frac {7 \, x \log \left (8 \, x + 1\right ) + 2 \, {\left (x + 1\right )} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - 8 \, x \log \left (x\right ) + x \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) + 7 \, x \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) - 7 \, x \log \left (-4 \, x + 4 \, \sqrt {x^{2} - x} + 1\right ) - 8 \, x - 8 \, \sqrt {x^{2} - x}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.68, size = 92, normalized size = 1.21 \begin {gather*} -\frac {\log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )}{x} + \frac {4}{x - \sqrt {x^{2} - x}} - 4 \, \log \left ({\left | 8 \, x + 1 \right |}\right ) + 4 \, \log \left ({\left | x \right |}\right ) - 4 \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} - 1 \right |}\right ) + 4 \, \log \left ({\left | -4 \, x + 4 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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