3.883 \(\int \frac{x+\sqrt{2 x-x^2}}{2-2 x} \, dx\)

Optimal. Leaf size=51 \[ -\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{x}{2}-\frac{1}{2} \log (1-x) \]

[Out]

-x/2 - Sqrt[2*x - x^2]/2 + ArcTanh[Sqrt[2*x - x^2]]/2 - Log[1 - x]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0967327, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6742, 43, 685, 688, 207} \[ -\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{x}{2}-\frac{1}{2} \log (1-x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/2 - Sqrt[2*x - x^2]/2 + ArcTanh[Sqrt[2*x - x^2]]/2 - Log[1 - x]/2

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A]  time = 0.0462925, size = 39, normalized size = 0.76 \[ \frac{1}{2} \left (-x-\sqrt{-(x-2) x}-\log (1-x)+\tanh ^{-1}\left (\sqrt{-(x-2) x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*} -{\frac{x}{2}}-{\frac{\ln \left ( x-1 \right ) }{2}}-{\frac{1}{2}\sqrt{- \left ( x-1 \right ) ^{2}+1}}+{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( x-1 \right ) ^{2}+1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.57894, size = 73, normalized size = 1.43 \begin{align*} -\frac{1}{2} \, x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \log \left (x - 1\right ) + \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+(-x^2+2*x)^(1/2))/(2-2*x),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.45869, size = 163, normalized size = 3.2 \begin{align*} -\frac{1}{2} \, x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \log \left (x - 1\right ) + \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+(-x^2+2*x)^(1/2))/(2-2*x),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12349, size = 68, normalized size = 1.33 \begin{align*} -\frac{1}{2} \, x - \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 ) - \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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