3.4.60 \(\int \frac {x}{\sqrt {-4 x+x^2}} \, dx\) [360]

Optimal. Leaf size=28 \[ \sqrt {-4 x+x^2}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {-4 x+x^2}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {654, 634, 212} \begin {gather*} \sqrt {x^2-4 x}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[-4*x + x^2],x]

[Out]

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

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 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]

Rule 654

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 42, normalized size = 1.50 \begin {gather*} \frac {(-4+x) x+4 \sqrt {-4+x} \sqrt {x} \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-4+x}{x}}}\right )}{\sqrt {(-4+x) x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-4*x + x^2],x]

[Out]

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

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Maple [A]
time = 0.06, size = 26, normalized size = 0.93

method	result	size
default	\(\sqrt {x^{2}-4 x}+2 \ln \left (-2+x +\sqrt {x^{2}-4 x}\right )\)	\(26\)
trager	\(\sqrt {x^{2}-4 x}-2 \ln \left (2-x +\sqrt {x^{2}-4 x}\right )\)	\(28\)
risch	\(\frac {x \left (x -4\right )}{\sqrt {x \left (x -4\right )}}+2 \ln \left (-2+x +\sqrt {x^{2}-4 x}\right )\)	\(29\)
meijerg	\(\frac {4 i \sqrt {-\mathrm {signum}\left (x -4\right )}\, \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {-\frac {x}{4}+1}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {x}}{2}\right )\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x -4\right )}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2-4*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.40, size = 29, normalized size = 1.04 \begin {gather*} \sqrt {x^{2} - 4 \, x} + 2 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4 \, x} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-4*x)^(1/2),x, algorithm="maxima")

[Out]

sqrt(x^2 - 4*x) + 2*log(2*x + 2*sqrt(x^2 - 4*x) - 4)

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Fricas [A]
time = 0.58, size = 27, normalized size = 0.96 \begin {gather*} \sqrt {x^{2} - 4 \, x} - 2 \, \log \left (-x + \sqrt {x^{2} - 4 \, x} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-4*x)^(1/2),x, algorithm="fricas")

[Out]

sqrt(x^2 - 4*x) - 2*log(-x + sqrt(x^2 - 4*x) + 2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x \left (x - 4\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2-4*x)**(1/2),x)

[Out]

Integral(x/sqrt(x*(x - 4)), x)

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Giac [A]
time = 1.43, size = 28, normalized size = 1.00 \begin {gather*} \sqrt {x^{2} - 4 \, x} - 2 \, \log \left ({\left | -x + \sqrt {x^{2} - 4 \, x} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-4*x)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.15, size = 23, normalized size = 0.82 \begin {gather*} 2\,\ln \left (x+\sqrt {x\,\left (x-4\right )}-2\right )+\sqrt {x^2-4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2 - 4*x)^(1/2),x)

[Out]

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

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