3.2.38 \(\int \frac {1}{x \sqrt {3+x^2}} \, dx\) [138]

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

[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 = {272, 65, 213} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+3}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[3 + x^2]),x]

[Out]

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

Rule 65

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 213

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

Rule 272

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} \text {Subst}\left (\int \frac {1}{x \sqrt {3+x}} \, dx,x,x^2\right )\\ &=\text {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 = 0.03, size = 23, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {3+x^2}}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

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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Mathics [A]
time = 2.17, size = 13, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {3} \text {ArcSinh}\left [\frac {\sqrt {3}}{x}\right ]}{3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-Sqrt[3] ArcSinh[Sqrt[3] / x] / 3

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Maple [A]
time = 0.08, size = 18, normalized size = 0.78

method	result	size
default	\(-\frac {\sqrt {3}\, \arctanh \left (\frac {\sqrt {3}}{\sqrt {x^{2}+3}}\right )}{3}\)	\(18\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\sqrt {x^{2}+3}-\RootOf \left (\textit {\_Z}^{2}-3\right )}{x}\right )}{3}\)	\(30\)
meijerg	\(\frac {\sqrt {3}\, \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )-\ln \left (3\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {\frac {x^{2}}{3}+1}}{2}\right )\right )}{6 \sqrt {\pi }}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, size = 14, normalized size = 0.61 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {\sqrt {3}}{{\left | x \right |}}\right ) \end {gather*}

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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Fricas [A]
time = 0.33, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - \sqrt {x^{2} + 3}}{x}\right ) \end {gather*}

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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Sympy [A]
time = 0.45, size = 15, normalized size = 0.65 \begin {gather*} - \frac {\sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {3}}{x} \right )}}{3} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
time = 0.00, size = 51, normalized size = 2.22 \begin {gather*} \frac {1}{6} \sqrt {3} \ln \left (\sqrt {x^{2}+3}-\sqrt {3}\right )-\frac {1}{6} \sqrt {3} \ln \left (\sqrt {x^{2}+3}+\sqrt {3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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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