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

Optimal. Leaf size=31 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} (x+1)}{\sqrt {3 x^2+4 x+2}}\right )}{\sqrt {2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {724, 206} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} (x+1)}{\sqrt {3 x^2+4 x+2}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[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 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 724

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*} \int \frac {1}{x \sqrt {2+4 x+3 x^2}} \, dx &=-\left (2 \operatorname {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*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.90 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x+1}{\sqrt {\frac {3 x^2}{2}+2 x+1}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.10, size = 38, normalized size = 1.23 \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\sqrt {\frac {3}{2}} x-\frac {\sqrt {3 x^2+4 x+2}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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]]

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fricas [A]  time = 0.40, size = 40, normalized size = 1.29 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.22, size = 60, normalized size = 1.94 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [A]  time = 0.05, size = 29, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {2}\, \arctanh \left (\frac {\left (4 x +4\right ) \sqrt {2}}{4 \sqrt {3 x^{2}+4 x +2}}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(3*x^2+4*x+2)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.96, size = 24, normalized size = 0.77 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \operatorname {arsinh}\left (\frac {\sqrt {2} x}{{\left | x \right |}} + \frac {\sqrt {2}}{{\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 1.16, size = 27, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {2}\,\ln \left (\frac {2\,x+\sqrt {6\,x^2+8\,x+4}+2}{x}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {3 x^{2} + 4 x + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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