### 3.360 $$\int \frac{x}{\sqrt{-4 x+x^2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0077534, antiderivative size = 28, normalized size of antiderivative = 1., 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 = {640, 620, 206} $\sqrt{x^2-4 x}+4 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-4 x}}\right )$

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 640

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]

Rule 620

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 206

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

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 \operatorname{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]  time = 0.0285069, size = 40, normalized size = 1.43 $\frac{(x-4) x-4 \sqrt{-(x-4) x} \sin ^{-1}\left (\sqrt{1-\frac{x}{4}}\right )}{\sqrt{(x-4) x}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[x/Sqrt[-4*x + x^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 26, normalized size = 0.9 \begin{align*} \sqrt{{x}^{2}-4\,x}+2\,\ln \left ( x-2+\sqrt{{x}^{2}-4\,x} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.929766, size = 39, normalized size = 1.39 \begin{align*} \sqrt{x^{2} - 4 \, x} + 2 \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 4 \, x} - 4\right ) \end{align*}

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 = 1.96705, size = 69, normalized size = 2.46 \begin{align*} \sqrt{x^{2} - 4 \, x} - 2 \, \log \left (-x + \sqrt{x^{2} - 4 \, x} + 2\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x \left (x - 4\right )}}\, dx \end{align*}

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 = 1.06386, size = 38, normalized size = 1.36 \begin{align*} \sqrt{x^{2} - 4 \, x} - 2 \, \log \left ({\left | -x + \sqrt{x^{2} - 4 \, x} + 2 \right |}\right ) \end{align*}

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))