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

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

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Rubi [A]  time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1997, 1913, 206} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {3} (2-x) \sqrt {x}}{2 \sqrt {x^3-3 x^2+3 x}}\right )}{\sqrt {3}} \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 62, normalized size = 1.44 \begin {gather*} \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 )}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.31, size = 45, normalized size = 1.05 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x}}{x^{3/2}-\sqrt {x^3-3 x^2+3 x}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.16, size = 49, normalized size = 1.14 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.33, size = 47, normalized size = 1.09 \begin {gather*} \frac {1}{3} \, \sqrt {3} \log \left (x + \sqrt {3} - \sqrt {x^{2} - 3 \, x + 3}\right ) - \frac {1}{3} \, \sqrt {3} \log \left (-x + \sqrt {3} + \sqrt {x^{2} - 3 \, x + 3}\right ) \end {gather*}

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

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maple [A]  time = 0.01, size = 50, normalized size = 1.16 \begin {gather*} \frac {\sqrt {x^{2}-3 x +3}\, \sqrt {3}\, \sqrt {x}\, \arctanh \left (\frac {\left (x -2\right ) \sqrt {3}}{2 \sqrt {x^{2}-3 x +3}}\right )}{3 \sqrt {\left (x^{2}-3 x +3\right ) x}} \end {gather*}

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {{\left (x^{2} - 3 \, x + 3\right )} x} \sqrt {x}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x}\,\sqrt {x\,\left (x^2-3\,x+3\right )}} \,d x \end {gather*}

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]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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