\(\int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx\) [2431]

Optimal result

Integrand size = 18, antiderivative size = 31 \[ \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} (1+x)}{\sqrt {2+4 x-3 x^2}}\right )}{\sqrt {2}} \]

[Out]

-1/2*arctanh((1+x)*2^(1/2)/(-3*x^2+4*x+2)^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {738, 212} \[ \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} (x+1)}{\sqrt {-3 x^2+4 x+2}}\right )}{\sqrt {2}} \]

[In]

Int[1/(x*Sqrt[2 + 4*x - 3*x^2]),x]

[Out]

-(ArcTanh[(Sqrt[2]*(1 + x))/Sqrt[2 + 4*x - 3*x^2]]/Sqrt[2])

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 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+4 x}{\sqrt {2+4 x-3 x^2}}\right )\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} (1+x)}{\sqrt {2+4 x-3 x^2}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx=\frac {-\log (x)+\log \left (-2-2 x+\sqrt {4+8 x-6 x^2}\right )}{\sqrt {2}} \]

[In]

Integrate[1/(x*Sqrt[2 + 4*x - 3*x^2]),x]

[Out]

(-Log[x] + Log[-2 - 2*x + Sqrt[4 + 8*x - 6*x^2]])/Sqrt[2]

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4+4 x \right ) \sqrt {2}}{4 \sqrt {-3 x^{2}+4 x +2}}\right )}{2}\)	\(29\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-\sqrt {-3 x^{2}+4 x +2}}{x}\right )}{2}\)	\(44\)

[In]

int(1/x/(-3*x^2+4*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*2^(1/2)*arctanh(1/4*(4+4*x)*2^(1/2)/(-3*x^2+4*x+2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (x + 1\right )} + x^{2} - 8 \, x - 4}{x^{2}}\right ) \]

[In]

integrate(1/x/(-3*x^2+4*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log(-(2*sqrt(2)*sqrt(-3*x^2 + 4*x + 2)*(x + 1) + x^2 - 8*x - 4)/x^2)

Sympy [F]

\[ \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx=\int \frac {1}{x \sqrt {- 3 x^{2} + 4 x + 2}}\, dx \]

[In]

integrate(1/x/(-3*x**2+4*x+2)**(1/2),x)

[Out]

Integral(1/(x*sqrt(-3*x**2 + 4*x + 2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {-3 \, x^{2} + 4 \, x + 2}}{{\left | x \right |}} + \frac {4}{{\left | x \right |}} + 4\right ) \]

[In]

integrate(1/x/(-3*x^2+4*x+2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*log(2*sqrt(2)*sqrt(-3*x^2 + 4*x + 2)/abs(x) + 4/abs(x) + 4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (25) = 50\).

Time = 0.33 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.16 \[ \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx=-\frac {1}{6} \, \sqrt {6} \sqrt {3} \log \left (\frac {{\left | -14 \, \sqrt {10} - 14 \, \sqrt {6} + \frac {28 \, {\left (\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} - \sqrt {10}\right )}}{3 \, x - 2} \right |}}{{\left | -14 \, \sqrt {10} + 14 \, \sqrt {6} + \frac {28 \, {\left (\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} - \sqrt {10}\right )}}{3 \, x - 2} \right |}}\right ) \]

[In]

integrate(1/x/(-3*x^2+4*x+2)^(1/2),x, algorithm="giac")

[Out]

-1/6*sqrt(6)*sqrt(3)*log(abs(-14*sqrt(10) - 14*sqrt(6) + 28*(sqrt(3)*sqrt(-3*x^2 + 4*x + 2) - sqrt(10))/(3*x -
2))/abs(-14*sqrt(10) + 14*sqrt(6) + 28*(sqrt(3)*sqrt(-3*x^2 + 4*x + 2) - sqrt(10))/(3*x - 2)))

Mupad [B] (verification not implemented)

Time = 10.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx=-\frac {\sqrt {2}\,\ln \left (\frac {2\,x+\sqrt {-6\,x^2+8\,x+4}+2}{x}\right )}{2} \]

[In]

int(1/(x*(4*x - 3*x^2 + 2)^(1/2)),x)

[Out]

-(2^(1/2)*log((2*x + (8*x - 6*x^2 + 4)^(1/2) + 2)/x))/2