3.14.87 \(\int \frac {1}{x \sqrt {2+x^6}} \, dx\) [1387]

Optimal. Leaf size=25 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{3 \sqrt {2}} \]

[Out]

-1/6*arctanh(1/2*(x^6+2)^(1/2)*2^(1/2))*2^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 25, 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^6+2}}{\sqrt {2}}\right )}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[2 + x^6]),x]

[Out]

-1/3*ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/Sqrt[2]

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 {2+x^6}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {2+x^6}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{3 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[2 + x^6]),x]

[Out]

-1/3*ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/Sqrt[2]

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Maple [A]
time = 0.35, size = 26, normalized size = 1.04

method	result	size
default	\(\frac {\sqrt {2}\, \ln \left (\frac {\sqrt {x^{6}+2}-\sqrt {2}}{\sqrt {x^{6}}}\right )}{6}\)	\(26\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\sqrt {x^{6}+2}+\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}}\right )}{6}\)	\(28\)
meijerg	\(\frac {\sqrt {2}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{6}}{2}}}{2}\right )+\left (-3 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^6+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/6*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))/(x^6)^(1/2))

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Maxima [A]
time = 0.52, size = 34, normalized size = 1.36 \begin {gather*} \frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

1/12*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2)))

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Fricas [A]
time = 0.35, size = 27, normalized size = 1.08 \begin {gather*} \frac {1}{12} \, \sqrt {2} \log \left (\frac {x^{6} - 2 \, \sqrt {2} \sqrt {x^{6} + 2} + 4}{x^{6}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(2)*log((x^6 - 2*sqrt(2)*sqrt(x^6 + 2) + 4)/x^6)

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Sympy [A]
time = 0.41, size = 17, normalized size = 0.68 \begin {gather*} - \frac {\sqrt {2} \operatorname {asinh}{\left (\frac {\sqrt {2}}{x^{3}} \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**6+2)**(1/2),x)

[Out]

-sqrt(2)*asinh(sqrt(2)/x**3)/6

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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 = 0.99, size = 37, normalized size = 1.48 \begin {gather*} -\frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {x^{6} + 2}\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} + \sqrt {x^{6} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

-1/12*sqrt(2)*log(sqrt(2) + sqrt(x^6 + 2)) + 1/12*sqrt(2)*log(-sqrt(2) + sqrt(x^6 + 2))

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Mupad [B]
time = 0.07, size = 18, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^6 + 2)^(1/2)),x)

[Out]

-(2^(1/2)*atanh((2^(1/2)*(x^6 + 2)^(1/2))/2))/6

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