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

Optimal. Leaf size=23 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+3}}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 63, 207} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+3}}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[3 + x^2]),x]

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {3+x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {3+x}} \, dx,x,x^2\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{-3+x^2} \, dx,x,\sqrt {3+x^2}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {3+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+3}}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[3 + x^2]),x]

[Out]

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

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fricas [A]  time = 0.40, size = 24, normalized size = 1.04 \[ \frac {1}{3} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - \sqrt {x^{2} + 3}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.91, size = 37, normalized size = 1.61 \[ -\frac {1}{6} \, \sqrt {3} \log \left (\sqrt {3} + \sqrt {x^{2} + 3}\right ) + \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} + \sqrt {x^{2} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*log(sqrt(3) + sqrt(x^2 + 3)) + 1/6*sqrt(3)*log(-sqrt(3) + sqrt(x^2 + 3))

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maple [A]  time = 0.01, size = 18, normalized size = 0.78 \[ -\frac {\sqrt {3}\, \arctanh \left (\frac {\sqrt {3}}{\sqrt {x^{2}+3}}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.96, size = 14, normalized size = 0.61 \[ -\frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {\sqrt {3}}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*arcsinh(sqrt(3)/abs(x))

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mupad [B]  time = 0.06, size = 18, normalized size = 0.78 \[ -\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sqrt {x^2+3}}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^2 + 3)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.06, size = 15, normalized size = 0.65 \[ - \frac {\sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {3}}{x} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*asinh(sqrt(3)/x)/3

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