Optimal. Leaf size=114 \[ -\frac{2 \log \left (4 \sqrt{x^2-x}+4 x-1\right )}{\sqrt{x}}-\frac{4 \sqrt{2} \sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{\sqrt{x-1} \sqrt{x}}-8 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x^2-x}}\right )+4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right ) \]
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Rubi [A] time = 0.308602, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {2537, 2535, 6733, 6742, 203, 1588, 2021, 2008, 1146, 444, 50, 63, 204} \[ -\frac{2 \log \left (4 \sqrt{x^2-x}+4 x-1\right )}{\sqrt{x}}-\frac{4 \sqrt{2} \sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{\sqrt{x-1} \sqrt{x}}-8 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x^2-x}}\right )+4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 2537
Rule 2535
Rule 6733
Rule 6742
Rule 203
Rule 1588
Rule 2021
Rule 2008
Rule 1146
Rule 444
Rule 50
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{\log \left (-1+4 x+4 \sqrt{(-1+x) x}\right )}{x^{3/2}} \, dx &=\int \frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{x^{3/2}} \, dx\\ &=-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}-16 \int \frac{1}{\sqrt{x} \left (-4 (1+2 x) \sqrt{-x+x^2}+8 \left (-x+x^2\right )\right )} \, dx\\ &=-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}-32 \operatorname{Subst}\left (\int \frac{1}{-4 \left (1+2 x^2\right ) \sqrt{-x^2+x^4}+8 \left (-x^2+x^4\right )} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}-32 \operatorname{Subst}\left (\int \left (-\frac{1}{2 \left (1+8 x^2\right )}-\frac{x^2}{12 \sqrt{-x^2+x^4}}+\frac{\sqrt{-x^2+x^4}}{4 x^2}+\frac{4 \sqrt{-x^2+x^4}}{3 \left (-1-8 x^2\right )}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}+\frac{8}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-x^2+x^4}} \, dx,x,\sqrt{x}\right )-8 \operatorname{Subst}\left (\int \frac{\sqrt{-x^2+x^4}}{x^2} \, dx,x,\sqrt{x}\right )+16 \operatorname{Subst}\left (\int \frac{1}{1+8 x^2} \, dx,x,\sqrt{x}\right )-\frac{128}{3} \operatorname{Subst}\left (\int \frac{\sqrt{-x^2+x^4}}{-1-8 x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{16 \sqrt{-x+x^2}}{3 \sqrt{x}}+4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}+8 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-x^2+x^4}} \, dx,x,\sqrt{x}\right )-\frac{\left (128 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{-1+x^2}}{-1-8 x^2} \, dx,x,\sqrt{x}\right )}{3 \sqrt{-1+x} \sqrt{x}}\\ &=-\frac{16 \sqrt{-x+x^2}}{3 \sqrt{x}}+4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}-8 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )-\frac{\left (64 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{-1+x}}{-1-8 x} \, dx,x,x\right )}{3 \sqrt{-1+x} \sqrt{x}}\\ &=4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-8 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}+\frac{\left (24 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{(-1-8 x) \sqrt{-1+x}} \, dx,x,x\right )}{\sqrt{-1+x} \sqrt{x}}\\ &=4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-8 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}+\frac{\left (48 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-9-8 x^2} \, dx,x,\sqrt{-1+x}\right )}{\sqrt{-1+x} \sqrt{x}}\\ &=-\frac{4 \sqrt{2} \sqrt{-x+x^2} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{-1+x}\right )}{\sqrt{-1+x} \sqrt{x}}+4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-8 \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}}\\ \end{align*}
Mathematica [C] time = 0.43421, size = 177, normalized size = 1.55 \[ -2 i \sqrt{2} \log \left (4 (8 x+1)^2\right )+i \sqrt{2} \log \left ((8 x+1) \left (-10 x-6 \sqrt{(x-1) x}+1\right )\right )-\frac{2 \log \left (4 x+4 \sqrt{(x-1) x}-1\right )}{\sqrt{x}}+i \sqrt{2} \log \left ((8 x+1) \left (-10 x+6 \sqrt{(x-1) x}+1\right )\right )+4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )+8 \tan ^{-1}\left (\frac{\sqrt{(x-1) x}}{\sqrt{x}}\right )-4 \sqrt{2} \tan ^{-1}\left (\frac{2 \sqrt{2} \sqrt{(x-1) x}}{3 \sqrt{x}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.006, size = 0, normalized size = 0. \begin{align*} \int{\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ){x}^{-{\frac{3}{2}}}}\, 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*} -\frac{2 \, \log \left (4 \, \sqrt{x - 1} \sqrt{x} + 4 \, x - 1\right )}{\sqrt{x}} - \frac{2}{\sqrt{x}} - \int \frac{2 \, x^{2} + x}{4 \, x^{\frac{9}{2}} - 5 \, x^{\frac{7}{2}} + x^{\frac{5}{2}} + 4 \,{\left (x^{4} - x^{3}\right )} \sqrt{x - 1}}\,{d x} - \log \left (\sqrt{x} + 1\right ) + \log \left (\sqrt{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28961, size = 240, normalized size = 2.11 \begin{align*} \frac{2 \,{\left (2 \, \sqrt{2} x \arctan \left (2 \, \sqrt{2} \sqrt{x}\right ) + 2 \, \sqrt{2} x \arctan \left (\frac{3 \, \sqrt{2} \sqrt{x}}{4 \, \sqrt{x^{2} - x}}\right ) - 4 \, x \arctan \left (\frac{\sqrt{x}}{\sqrt{x^{2} - x}}\right ) - \sqrt{x} \log \left (4 \, x + 4 \, \sqrt{x^{2} - x} - 1\right )\right )}}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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