Integrand size = 34, antiderivative size = 131 \[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\frac {(-9+8 x) \sqrt {-x+x^2} \sqrt {-x \left (-x+\sqrt {-x+x^2}\right )}}{12 x}+\sqrt {x \left (x-\sqrt {-x+x^2}\right )} \left (\frac {1}{12} (-19+8 x)+\frac {3 \sqrt {x+\sqrt {-x+x^2}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{4 \sqrt {2} x}\right ) \]
1/12*(-9+8*x)*(x^2-x)^(1/2)*(-x*(-x+(x^2-x)^(1/2)))^(1/2)/x+(x*(x-(x^2-x)^ (1/2)))^(1/2)*(-19/12+2/3*x+3/8*2^(1/2)*(x+(x^2-x)^(1/2))^(1/2)*arctanh(2^ (1/2)*(x+(x^2-x)^(1/2))^(1/2))/x)
Time = 2.56 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=-\frac {\left (-x+\sqrt {(-1+x) x}\right ) \left (2 x \left (9+8 x^2-19 \sqrt {(-1+x) x}+x \left (-17+8 \sqrt {(-1+x) x}\right )\right )+9 \sqrt {2} \sqrt {(-1+x) x} \sqrt {x+\sqrt {(-1+x) x}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {(-1+x) x}}\right )\right )}{24 \sqrt {(-1+x) x} \sqrt {x \left (x-\sqrt {(-1+x) x}\right )}} \]
-1/24*((-x + Sqrt[(-1 + x)*x])*(2*x*(9 + 8*x^2 - 19*Sqrt[(-1 + x)*x] + x*( -17 + 8*Sqrt[(-1 + x)*x])) + 9*Sqrt[2]*Sqrt[(-1 + x)*x]*Sqrt[x + Sqrt[(-1 + x)*x]]*ArcTanh[Sqrt[2]*Sqrt[x + Sqrt[(-1 + x)*x]]]))/(Sqrt[(-1 + x)*x]*S qrt[x*(x - Sqrt[(-1 + x)*x])])
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {x^2-x}}{\sqrt {x^2-x \sqrt {x^2-x}}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x^2-x} \int \frac {\sqrt {x-1} \sqrt {x}}{\sqrt {x^2-x \sqrt {x^2-x}}}dx}{\sqrt {x-1} \sqrt {x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x^2-x} \int \frac {\sqrt {x-1} x}{\sqrt {x^2-x \sqrt {x^2-x}}}d\sqrt {x}}{\sqrt {x-1} \sqrt {x}}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {2 \sqrt {x^2-x} \int \frac {\sqrt {x-1} x}{\sqrt {x^2-x \sqrt {x^2-x}}}d\sqrt {x}}{\sqrt {x-1} \sqrt {x}}\) |
3.19.99.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
\[\int \frac {\sqrt {x^{2}-x}}{\sqrt {x^{2}-x \sqrt {x^{2}-x}}}d x\]
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\frac {9 \, \sqrt {2} x \log \left (-\frac {4 \, x^{2} + 2 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (\sqrt {2} x - \sqrt {2} \sqrt {x^{2} - x}\right )} - 4 \, \sqrt {x^{2} - x} x - x}{x}\right ) + 4 \, {\left (8 \, x^{2} + \sqrt {x^{2} - x} {\left (8 \, x - 9\right )} - 19 \, x\right )} \sqrt {x^{2} - \sqrt {x^{2} - x} x}}{48 \, x} \]
1/48*(9*sqrt(2)*x*log(-(4*x^2 + 2*sqrt(x^2 - sqrt(x^2 - x)*x)*(sqrt(2)*x - sqrt(2)*sqrt(x^2 - x)) - 4*sqrt(x^2 - x)*x - x)/x) + 4*(8*x^2 + sqrt(x^2 - x)*(8*x - 9) - 19*x)*sqrt(x^2 - sqrt(x^2 - x)*x))/x
\[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int \frac {\sqrt {x \left (x - 1\right )}}{\sqrt {x \left (x - \sqrt {x^{2} - x}\right )}}\, dx \]
\[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} - x}}{\sqrt {x^{2} - \sqrt {x^{2} - x} x}} \,d x } \]
\[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} - x}}{\sqrt {x^{2} - \sqrt {x^{2} - x} x}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int \frac {\sqrt {x^2-x}}{\sqrt {x^2-x\,\sqrt {x^2-x}}} \,d x \]