\(\int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx\) [1899]

Optimal result

Integrand size = 34, antiderivative size = 131 \[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\frac {(-9+8 x) \sqrt {-x+x^2} \sqrt {-x \left (-x+\sqrt {-x+x^2}\right )}}{12 x}+\sqrt {x \left (x-\sqrt {-x+x^2}\right )} \left (\frac {1}{12} (-19+8 x)+\frac {3 \sqrt {x+\sqrt {-x+x^2}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{4 \sqrt {2} x}\right ) \]

[Out]

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)

Rubi [F]

\[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx \]

[In]

Int[Sqrt[-x + x^2]/Sqrt[x^2 - x*Sqrt[-x + x^2]],x]

[Out]

(2*Sqrt[-x + x^2]*Defer[Subst][Defer[Int][(x^2*Sqrt[-1 + x^2])/Sqrt[x^4 - x^2*Sqrt[-x^2 + x^4]], x], x, Sqrt[x
]])/(Sqrt[-1 + x]*Sqrt[x])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x+x^2} \int \frac {\sqrt {-1+x} \sqrt {x}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx}{\sqrt {-1+x} \sqrt {x}} \\ & = \frac {\left (2 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-1+x^2}}{\sqrt {x^4-x^2 \sqrt {-x^2+x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-1+x} \sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.56 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=-\frac {\left (-x+\sqrt {(-1+x) x}\right ) \left (2 x \left (9+8 x^2-19 \sqrt {(-1+x) x}+x \left (-17+8 \sqrt {(-1+x) x}\right )\right )+9 \sqrt {2} \sqrt {(-1+x) x} \sqrt {x+\sqrt {(-1+x) x}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {(-1+x) x}}\right )\right )}{24 \sqrt {(-1+x) x} \sqrt {x \left (x-\sqrt {(-1+x) x}\right )}} \]

[In]

Integrate[Sqrt[-x + x^2]/Sqrt[x^2 - x*Sqrt[-x + x^2]],x]

[Out]

-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]
*Sqrt[x*(x - Sqrt[(-1 + x)*x])])

Maple [F]

\[\int \frac {\sqrt {x^{2}-x}}{\sqrt {x^{2}-x \sqrt {x^{2}-x}}}d x\]

[In]

int((x^2-x)^(1/2)/(x^2-x*(x^2-x)^(1/2))^(1/2),x)

[Out]

int((x^2-x)^(1/2)/(x^2-x*(x^2-x)^(1/2))^(1/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\frac {9 \, \sqrt {2} x \log \left (-\frac {4 \, x^{2} + 2 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (\sqrt {2} x - \sqrt {2} \sqrt {x^{2} - x}\right )} - 4 \, \sqrt {x^{2} - x} x - x}{x}\right ) + 4 \, {\left (8 \, x^{2} + \sqrt {x^{2} - x} {\left (8 \, x - 9\right )} - 19 \, x\right )} \sqrt {x^{2} - \sqrt {x^{2} - x} x}}{48 \, x} \]

[In]

integrate((x^2-x)^(1/2)/(x^2-x*(x^2-x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int \frac {\sqrt {x \left (x - 1\right )}}{\sqrt {x \left (x - \sqrt {x^{2} - x}\right )}}\, dx \]

[In]

integrate((x**2-x)**(1/2)/(x**2-x*(x**2-x)**(1/2))**(1/2),x)

[Out]

Integral(sqrt(x*(x - 1))/sqrt(x*(x - sqrt(x**2 - x))), x)

Maxima [F]

\[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} - x}}{\sqrt {x^{2} - \sqrt {x^{2} - x} x}} \,d x } \]

[In]

integrate((x^2-x)^(1/2)/(x^2-x*(x^2-x)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^2 - x)/sqrt(x^2 - sqrt(x^2 - x)*x), x)

Giac [F]

\[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} - x}}{\sqrt {x^{2} - \sqrt {x^{2} - x} x}} \,d x } \]

[In]

integrate((x^2-x)^(1/2)/(x^2-x*(x^2-x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x^2 - x)/sqrt(x^2 - sqrt(x^2 - x)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx=\int \frac {\sqrt {x^2-x}}{\sqrt {x^2-x\,\sqrt {x^2-x}}} \,d x \]

[In]

int((x^2 - x)^(1/2)/(x^2 - x*(x^2 - x)^(1/2))^(1/2),x)

[Out]

int((x^2 - x)^(1/2)/(x^2 - x*(x^2 - x)^(1/2))^(1/2), x)