Optimal. Leaf size=37 \[ -\frac{1}{2} \sqrt{x^2-8 x} (4-x)-16 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-8 x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0066753, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {612, 620, 206} \[ -\frac{1}{2} \sqrt{x^2-8 x} (4-x)-16 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-8 x}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \sqrt{-8 x+x^2} \, dx &=-\frac{1}{2} (4-x) \sqrt{-8 x+x^2}-8 \int \frac{1}{\sqrt{-8 x+x^2}} \, dx\\ &=-\frac{1}{2} (4-x) \sqrt{-8 x+x^2}-16 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-8 x+x^2}}\right )\\ &=-\frac{1}{2} (4-x) \sqrt{-8 x+x^2}-16 \tanh ^{-1}\left (\frac{x}{\sqrt{-8 x+x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0376307, size = 48, normalized size = 1.3 \[ \frac{x \left (x^2-12 x+32\right )+32 \sqrt{-(x-8) x} \sin ^{-1}\left (\sqrt{1-\frac{x}{8}}\right )}{2 \sqrt{(x-8) x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.051, size = 33, normalized size = 0.9 \begin{align*}{\frac{2\,x-8}{4}\sqrt{{x}^{2}-8\,x}}-8\,\ln \left ( x-4+\sqrt{{x}^{2}-8\,x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.32576, size = 58, normalized size = 1.57 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} - 8 \, x} x - 2 \, \sqrt{x^{2} - 8 \, x} - 8 \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 8 \, x} - 8\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.3514, size = 85, normalized size = 2.3 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} - 8 \, x}{\left (x - 4\right )} + 8 \, \log \left (-x + \sqrt{x^{2} - 8 \, x} + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{2} - 8 x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.28506, size = 45, normalized size = 1.22 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} - 8 \, x}{\left (x - 4\right )} + 8 \, \log \left ({\left | -x + \sqrt{x^{2} - 8 \, x} + 4 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]