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

Optimal. Leaf size=131 \[ \frac {(-9+8 x) \sqrt {-x+x^2} \sqrt {-x \left (-x+\sqrt {-x+x^2}\right )}}{12 x}+\sqrt {x \left (x-\sqrt {-x+x^2}\right )} \left (\frac {1}{12} (-19+8 x)+\frac {3 \sqrt {x+\sqrt {-x+x^2}} \tanh ^{-1}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{4 \sqrt {2} x}\right ) \]

[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)

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Rubi [F]
time = 1.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-x+x^2}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [A]
time = 2.53, size = 135, normalized size = 1.03 \begin {gather*} -\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}} \tanh ^{-1}\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 )}} \end {gather*}

Antiderivative was successfully verified.

[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])])

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-x}}{\sqrt {x^{2}-x \sqrt {x^{2}-x}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.35, size = 123, normalized size = 0.94 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x - 1\right )}}{\sqrt {x \left (x - \sqrt {x^{2} - x}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x^2-x}}{\sqrt {x^2-x\,\sqrt {x^2-x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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