3.1.15 \(\int \sqrt {-x+x^2} \, dx\) [15]

Optimal. Leaf size=39 \[ -\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right ) \]

[Out]

-1/4*arctanh(x/(x^2-x)^(1/2))-1/4*(1-2*x)*(x^2-x)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {626, 634, 212} \begin {gather*} -\frac {1}{4} \sqrt {x^2-x} (1-2 x)-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-x + x^2],x]

[Out]

-1/4*((1 - 2*x)*Sqrt[-x + x^2]) - ArcTanh[x/Sqrt[-x + x^2]]/4

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rubi steps

\begin {align*} \int \sqrt {-x+x^2} \, dx &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{8} \int \frac {1}{\sqrt {-x+x^2}} \, dx\\ &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )\\ &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 50, normalized size = 1.28 \begin {gather*} \frac {1}{4} \sqrt {(-1+x) x} \left (-1+2 x-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-1+x}}{-1+\sqrt {x}}\right )}{\sqrt {-1+x} \sqrt {x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-x + x^2],x]

[Out]

(Sqrt[(-1 + x)*x]*(-1 + 2*x - (2*ArcTanh[Sqrt[-1 + x]/(-1 + Sqrt[x])])/(Sqrt[-1 + x]*Sqrt[x])))/4

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Maple [A]
time = 0.41, size = 33, normalized size = 0.85

method	result	size
default	\(\frac {\left (2 x -1\right ) \sqrt {x^{2}-x}}{4}-\frac {\ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x}\right )}{8}\)	\(33\)
risch	\(\frac {\left (2 x -1\right ) x \left (x -1\right )}{4 \sqrt {x \left (x -1\right )}}-\frac {\ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x}\right )}{8}\)	\(35\)
trager	\(\left (\frac {x}{2}-\frac {1}{4}\right ) \sqrt {x^{2}-x}-\frac {\ln \left (2 x -1+2 \sqrt {x^{2}-x}\right )}{8}\)	\(36\)
meijerg	\(-\frac {i \sqrt {\mathrm {signum}\left (x -1\right )}\, \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \left (-6 x +3\right ) \sqrt {1-x}}{6}+\frac {i \sqrt {\pi }\, \arcsin \left (\sqrt {x}\right )}{2}\right )}{2 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x -1\right )}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(2*x-1)*(x^2-x)^(1/2)-1/8*ln(x-1/2+(x^2-x)^(1/2))

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Maxima [A]
time = 0.27, size = 43, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - x} x - \frac {1}{4} \, \sqrt {x^{2} - x} - \frac {1}{8} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x^2 - x)*x - 1/4*sqrt(x^2 - x) - 1/8*log(2*x + 2*sqrt(x^2 - x) - 1)

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Fricas [A]
time = 1.60, size = 36, normalized size = 0.92 \begin {gather*} \frac {1}{4} \, \sqrt {x^{2} - x} {\left (2 \, x - 1\right )} + \frac {1}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(x^2 - x)*(2*x - 1) + 1/8*log(-2*x + 2*sqrt(x^2 - x) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**2 - x), x)

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Giac [A]
time = 2.07, size = 37, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, \sqrt {x^{2} - x} {\left (2 \, x - 1\right )} + \frac {1}{8} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(x^2 - x)*(2*x - 1) + 1/8*log(abs(-2*x + 2*sqrt(x^2 - x) + 1))

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Mupad [B]
time = 0.20, size = 29, normalized size = 0.74 \begin {gather*} \sqrt {x^2-x}\,\left (\frac {x}{2}-\frac {1}{4}\right )-\frac {\ln \left (x+\sqrt {x\,\left (x-1\right )}-\frac {1}{2}\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - x)^(1/2),x)

[Out]

(x^2 - x)^(1/2)*(x/2 - 1/4) - log(x + (x*(x - 1))^(1/2) - 1/2)/8

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