Optimal. Leaf size=101 \[ -\frac{2 \left (x^2-x\right )^{3/2}}{3 x^3}-\frac{10 \sqrt{x^2-x}}{x}-\frac{\log \left (4 \sqrt{x^2-x}+4 x-1\right )}{2 x^2}-16 \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )-\frac{2}{x}-16 \log (x)+16 \log (8 x+1) \]
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Rubi [A] time = 0.290585, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {2537, 2535, 6742, 640, 620, 206, 734, 843, 724, 650, 662, 664} \[ -\frac{2 \left (x^2-x\right )^{3/2}}{3 x^3}-\frac{10 \sqrt{x^2-x}}{x}-\frac{\log \left (4 \sqrt{x^2-x}+4 x-1\right )}{2 x^2}-16 \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )-\frac{2}{x}-16 \log (x)+16 \log (8 x+1) \]
Antiderivative was successfully verified.
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Rule 2537
Rule 2535
Rule 6742
Rule 640
Rule 620
Rule 206
Rule 734
Rule 843
Rule 724
Rule 650
Rule 662
Rule 664
Rubi steps
\begin{align*} \int \frac{\log \left (-1+4 x+4 \sqrt{(-1+x) x}\right )}{x^3} \, dx &=\int \frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{x^3} \, dx\\ &=-\frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{2 x^2}-4 \int \frac{1}{x^2 \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 )}{2 x^2}-4 \int \left (-\frac{1}{2 x^2}+\frac{4}{x}-\frac{32}{1+8 x}-\frac{x}{12 \sqrt{-x+x^2}}+\frac{256 \sqrt{-x+x^2}}{3 (-1-8 x)}+\frac{\sqrt{-x+x^2}}{4 x^3}-\frac{5 \sqrt{-x+x^2}}{4 x^2}+\frac{43 \sqrt{-x+x^2}}{4 x}\right ) \, dx\\ &=-\frac{2}{x}-16 \log (x)+16 \log (1+8 x)-\frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{2 x^2}+\frac{1}{3} \int \frac{x}{\sqrt{-x+x^2}} \, dx+5 \int \frac{\sqrt{-x+x^2}}{x^2} \, dx-43 \int \frac{\sqrt{-x+x^2}}{x} \, dx-\frac{1024}{3} \int \frac{\sqrt{-x+x^2}}{-1-8 x} \, dx-\int \frac{\sqrt{-x+x^2}}{x^3} \, dx\\ &=-\frac{2}{x}-\frac{10 \sqrt{-x+x^2}}{x}-\frac{2 \left (-x+x^2\right )^{3/2}}{3 x^3}-16 \log (x)+16 \log (1+8 x)-\frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{2 x^2}+\frac{1}{6} \int \frac{1}{\sqrt{-x+x^2}} \, dx+5 \int \frac{1}{\sqrt{-x+x^2}} \, dx-\frac{64}{3} \int \frac{1-10 x}{(-1-8 x) \sqrt{-x+x^2}} \, dx+\frac{43}{2} \int \frac{1}{\sqrt{-x+x^2}} \, dx\\ &=-\frac{2}{x}-\frac{10 \sqrt{-x+x^2}}{x}-\frac{2 \left (-x+x^2\right )^{3/2}}{3 x^3}-16 \log (x)+16 \log (1+8 x)-\frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{2 x^2}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )+10 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )-\frac{80}{3} \int \frac{1}{\sqrt{-x+x^2}} \, dx+43 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )-48 \int \frac{1}{(-1-8 x) \sqrt{-x+x^2}} \, dx\\ &=-\frac{2}{x}-\frac{10 \sqrt{-x+x^2}}{x}-\frac{2 \left (-x+x^2\right )^{3/2}}{3 x^3}+\frac{160}{3} \tanh ^{-1}\left (\frac{x}{\sqrt{-x+x^2}}\right )-16 \log (x)+16 \log (1+8 x)-\frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{2 x^2}-\frac{160}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-x+x^2}}\right )+96 \operatorname{Subst}\left (\int \frac{1}{36-x^2} \, dx,x,\frac{-1+10 x}{\sqrt{-x+x^2}}\right )\\ &=-\frac{2}{x}-\frac{10 \sqrt{-x+x^2}}{x}-\frac{2 \left (-x+x^2\right )^{3/2}}{3 x^3}-16 \tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{-x+x^2}}\right )-16 \log (x)+16 \log (1+8 x)-\frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.274751, size = 82, normalized size = 0.81 \[ -\frac{2 \sqrt{(x-1) x} (16 x-1)}{3 x^2}-\frac{\log \left (4 x+4 \sqrt{(x-1) x}-1\right )}{2 x^2}-\frac{2}{x}-16 \log (x)+32 \log (8 x+1)-16 \log \left (-10 x+6 \sqrt{(x-1) x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.006, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33293, size = 358, normalized size = 3.54 \begin{align*} \frac{189 \, x^{2} \log \left (8 \, x + 1\right ) - 192 \, x^{2} \log \left (x\right ) + 3 \, x^{2} \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x} + 1\right ) + 189 \, x^{2} \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x} - 1\right ) - 189 \, x^{2} \log \left (-4 \, x + 4 \, \sqrt{x^{2} - x} + 1\right ) - 128 \, x^{2} + 6 \,{\left (x^{2} - 1\right )} \log \left (4 \, x + 4 \, \sqrt{x^{2} - x} - 1\right ) - 8 \, \sqrt{x^{2} - x}{\left (16 \, x - 1\right )} - 24 \, x}{12 \, x^{2}} \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 = 1.40355, size = 176, normalized size = 1.74 \begin{align*} -\frac{2}{x} - \frac{\log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right )}{2 \, x^{2}} - \frac{2 \,{\left (18 \,{\left (x - \sqrt{x^{2} - x}\right )}^{2} - 3 \, x + 3 \, \sqrt{x^{2} - x} + 1\right )}}{3 \,{\left (x - \sqrt{x^{2} - x}\right )}^{3}} + 16 \, \log \left ({\left | 8 \, x + 1 \right |}\right ) - 16 \, \log \left ({\left | x \right |}\right ) + 16 \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x} - 1 \right |}\right ) - 16 \, \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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