\(\int \frac {1}{x \sqrt {2+4 x+3 x^2}} \, dx\) [2430]

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 = 38, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x \sqrt {2+4 x+3 x^2}} \, dx=\sqrt {2} \text {arctanh}\left (\sqrt {\frac {3}{2}} x-\frac {\sqrt {2+4 x+3 x^2}}{\sqrt {2}}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.24 (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 = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x \sqrt {2+4 x+3 x^2}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {3 \, x^{2} + 4 \, x + 2} {\left (x + 1\right )} - 5 \, x^{2} - 8 \, x - 4}{x^{2}}\right ) \]

[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) - 5*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 = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x \sqrt {2+4 x+3 x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \operatorname {arsinh}\left (\frac {\sqrt {2} x}{{\left | x \right |}} + \frac {\sqrt {2}}{{\left | x \right |}}\right ) \]

[In]

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

[Out]

-1/2*sqrt(2)*arcsinh(sqrt(2)*x/abs(x) + sqrt(2)/abs(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (25) = 50\).

Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.94 \[ \int \frac {1}{x \sqrt {2+4 x+3 x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (-\sqrt {3} x + \sqrt {2} + \sqrt {3 \, x^{2} + 4 \, x + 2}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {3} x - \sqrt {2} + \sqrt {3 \, x^{2} + 4 \, x + 2} \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.08 (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