### 3.313 $$\int \frac{1}{(2-2 x) (2 x-x^2)^{3/2}} \, dx$$

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]

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

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Rubi [A]  time = 0.0169694, 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 = {687, 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[1/((2 - 2*x)*(2*x - x^2)^(3/2)),x]

[Out]

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

Rule 687

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*c*e*(m + 2*p + 3))/(e*(p + 1)*(b^2 - 4*a
*c)), 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] && LtQ[p, -1] &&  !GtQ[m, 1] && 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{1}{(2-2 x) \left (2 x-x^2\right )^{3/2}} \, 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 = 0.0203524, size = 42, normalized size = 1.17 $-\frac{2 \sqrt{x-2} \sqrt{x} \tan ^{-1}\left (\sqrt{\frac{x-2}{x}}\right )+1}{2 \sqrt{-(x-2) x}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((2 - 2*x)*(2*x - x^2)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.048, size = 29, normalized size = 0.8 \begin{align*} -{\frac{1}{2}{\frac{1}{\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(1/(2-2*x)/(-x^2+2*x)^(3/2),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 = 1.12103, 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(1/(2-2*x)/(-x^2+2*x)^(3/2),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 = 2.01464, size = 169, normalized size = 4.69 \begin{align*} \frac{{\left (x^{2} - 2 \, x\right )} \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) -{\left (x^{2} - 2 \, x\right )} \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) + \sqrt{-x^{2} + 2 \, x}}{2 \,{\left (x^{2} - 2 \, x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2-2*x)/(-x^2+2*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{1}{- x^{3} \sqrt{- x^{2} + 2 x} + 3 x^{2} \sqrt{- x^{2} + 2 x} - 2 x \sqrt{- x^{2} + 2 x}}\, dx}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2-2*x)/(-x**2+2*x)**(3/2),x)

[Out]

-Integral(1/(-x**3*sqrt(-x**2 + 2*x) + 3*x**2*sqrt(-x**2 + 2*x) - 2*x*sqrt(-x**2 + 2*x)), x)/2

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Giac [A]  time = 1.35855, size = 66, normalized size = 1.83 \begin{align*} \frac{\sqrt{-x^{2} + 2 \, x}}{2 \,{\left (x^{2} - 2 \, x\right )}} - \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(1/(2-2*x)/(-x^2+2*x)^(3/2),x, algorithm="giac")

[Out]

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