3.139 \(\int \frac{1}{\sqrt{x} \sqrt{x (3-3 x+x^2)}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (2-x) \sqrt{x}}{2 \sqrt{x^3-3 x^2+3 x}}\right )}{\sqrt{3}} \]

[Out]

-(ArcTanh[(Sqrt[3]*(2 - x)*Sqrt[x])/(2*Sqrt[3*x - 3*x^2 + x^3])]/Sqrt[3])

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Rubi [A]  time = 0.0474514, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1997, 1913, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (2-x) \sqrt{x}}{2 \sqrt{x^3-3 x^2+3 x}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[x*(3 - 3*x + x^2)]),x]

[Out]

-(ArcTanh[(Sqrt[3]*(2 - x)*Sqrt[x])/(2*Sqrt[3*x - 3*x^2 + x^3])]/Sqrt[3])

Rule 1997

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] &&
GeneralizedTrinomialQ[u, x] &&  !GeneralizedTrinomialMatchQ[u, x]

Rule 1913

Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-2/(n - q), Sub
st[Int[1/(4*a - x^2), x], x, (x^(m + 1)*(2*a + b*x^(n - q)))/Sqrt[a*x^q + b*x^n + c*x^r]], x] /; FreeQ[{a, b,
c, m, n, q, r}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && NeQ[b^2 - 4*a*c, 0] && EqQ[m, q/2 - 1]

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{1}{\sqrt{x} \sqrt{x \left (3-3 x+x^2\right )}} \, dx &=\int \frac{1}{\sqrt{x} \sqrt{3 x-3 x^2+x^3}} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{(6-3 x) \sqrt{x}}{\sqrt{3 x-3 x^2+x^3}}\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (2-x) \sqrt{x}}{2 \sqrt{3 x-3 x^2+x^3}}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0197291, size = 62, normalized size = 1.44 \[ \frac{\sqrt{x} \sqrt{x^2-3 x+3} \tanh ^{-1}\left (\frac{\sqrt{3} (x-2)}{2 \sqrt{x^2-3 x+3}}\right )}{\sqrt{3} \sqrt{x \left (x^2-3 x+3\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*Sqrt[3 - 3*x + x^2]*ArcTanh[(Sqrt[3]*(-2 + x))/(2*Sqrt[3 - 3*x + x^2])])/(Sqrt[3]*Sqrt[x*(3 - 3*x + x
^2)])

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Maple [A]  time = 0.011, size = 50, normalized size = 1.2 \begin{align*}{\frac{\sqrt{3}}{3}\sqrt{x}\sqrt{{x}^{2}-3\,x+3}{\it Artanh} \left ({\frac{ \left ( -2+x \right ) \sqrt{3}}{2}{\frac{1}{\sqrt{{x}^{2}-3\,x+3}}}} \right ){\frac{1}{\sqrt{x \left ({x}^{2}-3\,x+3 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(x*(x^2-3*x+3))^(1/2),x)

[Out]

1/3*x^(1/2)/(x*(x^2-3*x+3))^(1/2)*(x^2-3*x+3)^(1/2)*3^(1/2)*arctanh(1/2*(-2+x)*3^(1/2)/(x^2-3*x+3)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (x^{2} - 3 \, x + 3\right )} x} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.39325, size = 132, normalized size = 3.07 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{7 \, x^{3} + 4 \, \sqrt{3} \sqrt{x^{3} - 3 \, x^{2} + 3 \, x}{\left (x - 2\right )} \sqrt{x} - 24 \, x^{2} + 24 \, x}{x^{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*log((7*x^3 + 4*sqrt(3)*sqrt(x^3 - 3*x^2 + 3*x)*(x - 2)*sqrt(x) - 24*x^2 + 24*x)/x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.14413, size = 69, normalized size = 1.6 \begin{align*} -\frac{1}{3} \, \sqrt{3} \log \left ({\left | -x + \sqrt{3} + \sqrt{x^{2} - 3 \, x + 3} \right |}\right ) + \frac{1}{3} \, \sqrt{3} \log \left ({\left | -x - \sqrt{3} + \sqrt{x^{2} - 3 \, x + 3} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*log(abs(-x + sqrt(3) + sqrt(x^2 - 3*x + 3))) + 1/3*sqrt(3)*log(abs(-x - sqrt(3) + sqrt(x^2 - 3*x
+ 3)))