Optimal. Leaf size=118 \[ -\frac{2 \sqrt{x^2-x}}{\sqrt{x}}+2 \sqrt{x} \log \left (4 \sqrt{x^2-x}+4 x-1\right )-\frac{\sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{\sqrt{2} \sqrt{x-1} \sqrt{x}}-2 \sqrt{x}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{\sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.385141, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {2537, 2535, 6733, 6742, 203, 1588, 1146, 444, 50, 63} \[ -\frac{2 \sqrt{x^2-x}}{\sqrt{x}}+2 \sqrt{x} \log \left (4 \sqrt{x^2-x}+4 x-1\right )-\frac{\sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{\sqrt{2} \sqrt{x-1} \sqrt{x}}-2 \sqrt{x}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2537
Rule 2535
Rule 6733
Rule 6742
Rule 203
Rule 1588
Rule 1146
Rule 444
Rule 50
Rule 63
Rubi steps
\begin{align*} \int \frac{\log \left (-1+4 x+4 \sqrt{(-1+x) x}\right )}{\sqrt{x}} \, dx &=\int \frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{\sqrt{x}} \, dx\\ &=2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+16 \int \frac{\sqrt{x}}{-4 (1+2 x) \sqrt{-x+x^2}+8 \left (-x+x^2\right )} \, dx\\ &=2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+32 \operatorname{Subst}\left (\int \frac{x^2}{-4 \left (1+2 x^2\right ) \sqrt{-x^2+x^4}+8 \left (-x^2+x^4\right )} \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+32 \operatorname{Subst}\left (\int \left (-\frac{1}{16}+\frac{1}{16 \left (1+8 x^2\right )}-\frac{x^2}{12 \sqrt{-x^2+x^4}}+\frac{\sqrt{-x^2+x^4}}{6 \left (1+8 x^2\right )}\right ) \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{x}+2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+2 \operatorname{Subst}\left (\int \frac{1}{1+8 x^2} \, dx,x,\sqrt{x}\right )-\frac{8}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-x^2+x^4}} \, dx,x,\sqrt{x}\right )+\frac{16}{3} \operatorname{Subst}\left (\int \frac{\sqrt{-x^2+x^4}}{1+8 x^2} \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{x}-\frac{8 \sqrt{-x+x^2}}{3 \sqrt{x}}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{\left (16 \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}}\\ &=-2 \sqrt{x}-\frac{8 \sqrt{-x+x^2}}{3 \sqrt{x}}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{\left (8 \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}}\\ &=-2 \sqrt{x}-\frac{2 \sqrt{-x+x^2}}{\sqrt{x}}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{\left (3 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} (1+8 x)} \, dx,x,x\right )}{\sqrt{-1+x} \sqrt{x}}\\ &=-2 \sqrt{x}-\frac{2 \sqrt{-x+x^2}}{\sqrt{x}}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{\left (6 \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}}\\ &=-2 \sqrt{x}-\frac{2 \sqrt{-x+x^2}}{\sqrt{x}}-\frac{\sqrt{-x+x^2} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{-1+x}\right )}{\sqrt{2} \sqrt{-1+x} \sqrt{x}}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+2 \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )\\ \end{align*}
Mathematica [C] time = 0.544953, size = 186, normalized size = 1.58 \[ \frac{1}{8} \left (-16 \sqrt{x}-\frac{16 \sqrt{(x-1) x}}{\sqrt{x}}-2 i \sqrt{2} \log \left (4 (8 x+1)^2\right )+i \sqrt{2} \log \left ((8 x+1) \left (-10 x-6 \sqrt{(x-1) x}+1\right )\right )+16 \sqrt{x} \log \left (4 x+4 \sqrt{(x-1) x}-1\right )+i \sqrt{2} \log \left ((8 x+1) \left (-10 x+6 \sqrt{(x-1) x}+1\right )\right )+4 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-4 \sqrt{2} \tan ^{-1}\left (\frac{2 \sqrt{2} \sqrt{(x-1) x}}{3 \sqrt{x}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.009, size = 0, normalized size = 0. \begin{align*} \int{\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ){\frac{1}{\sqrt{x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, \sqrt{x} \log \left (4 \, \sqrt{x - 1} \sqrt{x} + 4 \, x - 1\right ) - 4 \, \sqrt{x} + \int \frac{2 \, x^{2} + x}{4 \, x^{\frac{7}{2}} - 5 \, x^{\frac{5}{2}} + 4 \,{\left (x^{3} - x^{2}\right )} \sqrt{x - 1} + x^{\frac{3}{2}}}\,{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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.18175, size = 243, normalized size = 2.06 \begin{align*} \frac{\sqrt{2} x \arctan \left (2 \, \sqrt{2} \sqrt{x}\right ) + \sqrt{2} x \arctan \left (\frac{3 \, \sqrt{2} \sqrt{x}}{4 \, \sqrt{x^{2} - x}}\right ) + 4 \, x^{\frac{3}{2}} \log \left (4 \, x + 4 \, \sqrt{x^{2} - x} - 1\right ) - 4 \, x^{\frac{3}{2}} - 4 \, \sqrt{x^{2} - x} \sqrt{x}}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]