Optimal. Leaf size=76 \[ \frac{4 \sqrt{x^2-x}}{x}-\frac{\log \left (4 \sqrt{x^2-x}+4 x-1\right )}{x}+4 \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )+4 \log (x)-4 \log (8 x+1) \]
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Rubi [A] time = 0.262083, antiderivative size = 76, normalized size of antiderivative = 1., 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 = {2537, 2535, 6742, 640, 620, 206, 662, 664, 734, 843, 724} \[ \frac{4 \sqrt{x^2-x}}{x}-\frac{\log \left (4 \sqrt{x^2-x}+4 x-1\right )}{x}+4 \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )+4 \log (x)-4 \log (8 x+1) \]
Antiderivative was successfully verified.
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Rule 2537
Rule 2535
Rule 6742
Rule 640
Rule 620
Rule 206
Rule 662
Rule 664
Rule 734
Rule 843
Rule 724
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} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )-4 \operatorname{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 \operatorname{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} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )+24 \operatorname{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*}
Mathematica [A] time = 0.214096, size = 68, normalized size = 0.89 \[ \frac{4 \sqrt{(x-1) x}}{x}+4 \log (x)-8 \log (8 x+1)-\frac{\log \left (4 x+4 \sqrt{(x-1) x}-1\right )}{x}+4 \log \left (-10 x+6 \sqrt{(x-1) x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.008, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\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 \frac{\log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48659, size = 296, normalized size = 3.89 \begin{align*} -\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{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 = 1.34975, size = 124, normalized size = 1.63 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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