\(\int \frac {x}{\sqrt {-4 x+x^2}} \, dx\) [70]

Optimal result

Integrand size = 13, antiderivative size = 28 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {-4 x+x^2}+4 \text {arctanh}\left (\frac {x}{\sqrt {-4 x+x^2}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {654, 634, 212} \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=4 \text {arctanh}\left (\frac {x}{\sqrt {x^2-4 x}}\right )+\sqrt {x^2-4 x} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\frac {(-4+x) x-4 \sqrt {-4+x} \sqrt {x} \log \left (\sqrt {-4+x}-\sqrt {x}\right )}{\sqrt {(-4+x) x}} \]

[In]

Integrate[x/Sqrt[-4*x + x^2],x]

[Out]

((-4 + x)*x - 4*Sqrt[-4 + x]*Sqrt[x]*Log[Sqrt[-4 + x] - Sqrt[x]])/Sqrt[(-4 + x)*x]

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

method	result	size
default	\(\sqrt {x^{2}-4 x}+2 \ln \left (-2+x +\sqrt {x^{2}-4 x}\right )\)	\(26\)
trager	\(\sqrt {x^{2}-4 x}+2 \ln \left (-2+x +\sqrt {x^{2}-4 x}\right )\)	\(26\)
risch	\(\frac {\left (x -4\right ) x}{\sqrt {\left (x -4\right ) x}}+2 \ln \left (-2+x +\sqrt {x^{2}-4 x}\right )\)	\(29\)
pseudoelliptic	\(-2 \ln \left (\frac {\sqrt {\left (x -4\right ) x}-x}{x}\right )+2 \ln \left (\frac {x +\sqrt {\left (x -4\right ) x}}{x}\right )+\sqrt {\left (x -4\right ) x}\)	\(43\)
meijerg	\(\frac {4 i \sqrt {-\operatorname {signum}\left (x -4\right )}\, \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {-\frac {x}{4}+1}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {x}}{2}\right )\right )}{\sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x -4\right )}}\)	\(50\)

[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))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x} - 2 \, \log \left (-x + \sqrt {x^{2} - 4 \, x} + 2\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 x} + 2 \log {\left (2 x + 2 \sqrt {x^{2} - 4 x} - 4 \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x} + 2 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4 \, x} - 4\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=\sqrt {x^{2} - 4 \, x} - 2 \, \log \left ({\left | -x + \sqrt {x^{2} - 4 \, x} + 2 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\sqrt {-4 x+x^2}} \, dx=2\,\ln \left (x+\sqrt {x\,\left (x-4\right )}-2\right )+\sqrt {x^2-4\,x} \]

[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)