### 3.311 $$\int \frac{\sqrt{2 x-x^2}}{2-2 x} \, dx$$

Optimal. Leaf size=36 $\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{1}{2} \sqrt{2 x-x^2}$

[Out]

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

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Rubi [A]  time = 0.0168601, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {685, 688, 207} $\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{1}{2} \sqrt{2 x-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[2*x - x^2]/(2 - 2*x),x]

[Out]

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

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(d*p*(b^2 - 4*a*c))/(b*e*(m + 2*p + 1)), Int[(d + e*x)^m*(a +
b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] &&
NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p]))
&& RationalQ[m] && IntegerQ[2*p]

Rule 688

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rule 207

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

Rubi steps

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

Mathematica [A]  time = 0.038693, size = 46, normalized size = 1.28 $\frac{(x-2) x-2 \sqrt{x-2} \sqrt{x} \tan ^{-1}\left (\sqrt{\frac{x-2}{x}}\right )}{2 \sqrt{-(x-2) x}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[2*x - x^2]/(2 - 2*x),x]

[Out]

((-2 + x)*x - 2*Sqrt[-2 + x]*Sqrt[x]*ArcTan[Sqrt[(-2 + x)/x]])/(2*Sqrt[-((-2 + x)*x)])

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Maple [A]  time = 0.047, size = 29, normalized size = 0.8 \begin{align*} -{\frac{1}{2}\sqrt{- \left ( -1+x \right ) ^{2}+1}}+{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.6917, size = 61, normalized size = 1.69 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac{2}{{\left | x - 1 \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.21014, size = 130, normalized size = 3.61 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\sqrt{- x^{2} + 2 x}}{x - 1}\, dx}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sqrt(-x**2 + 2*x)/(x - 1), x)/2

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Giac [A]  time = 1.37813, size = 54, normalized size = 1.5 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \log \left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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