\(\int \frac {1}{\sqrt {x} \sqrt {x (3-3 x+x^2)}} \, dx\) [139]

Optimal result

Integrand size = 20, antiderivative size = 43 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (3-3 x+x^2\right )}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {3} (2-x) \sqrt {x}}{2 \sqrt {3 x-3 x^2+x^3}}\right )}{\sqrt {3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2022, 1927, 212} \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (3-3 x+x^2\right )}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {3} (2-x) \sqrt {x}}{2 \sqrt {x^3-3 x^2+3 x}}\right )}{\sqrt {3}} \]

[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 212

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

Rule 1927

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 2022

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*} \text {integral}& = \int \frac {1}{\sqrt {x} \sqrt {3 x-3 x^2+x^3}} \, dx \\ & = -\left (2 \text {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] (verified)

Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (3-3 x+x^2\right )}} \, dx=\frac {2 \sqrt {x} \sqrt {3-3 x+x^2} \text {arctanh}\left (\frac {x-\sqrt {3-3 x+x^2}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {x \left (3-3 x+x^2\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {\sqrt {x}\, \sqrt {x^{2}-3 x +3}\, \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (x -2\right ) \sqrt {3}}{2 \sqrt {x^{2}-3 x +3}}\right )}{3 \sqrt {x \left (x^{2}-3 x +3\right )}}\)	\(50\)

[In]

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (3-3 x+x^2\right )}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {7 \, x^{3} + 4 \, \sqrt {3} \sqrt {x^{3} - 3 \, x^{2} + 3 \, x} {\left (x - 2\right )} \sqrt {x} - 24 \, x^{2} + 24 \, x}{x^{3}}\right ) \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (3-3 x+x^2\right )}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {x \left (3-3 x+x^2\right )}} \, dx=\int { \frac {1}{\sqrt {{\left (x^{2} - 3 \, x + 3\right )} x} \sqrt {x}} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (3-3 x+x^2\right )}} \, dx=\frac {1}{3} \, \sqrt {3} \log \left (x + \sqrt {3} - \sqrt {x^{2} - 3 \, x + 3}\right ) - \frac {1}{3} \, \sqrt {3} \log \left (-x + \sqrt {3} + \sqrt {x^{2} - 3 \, x + 3}\right ) \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (3-3 x+x^2\right )}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {x\,\left (x^2-3\,x+3\right )}} \,d x \]

[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)