3.1387 \(\int \frac{1}{x \sqrt{2+x^6}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0118845, antiderivative size = 25, normalized size of antiderivative = 1., 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} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[2 + x^6]),x]

[Out]

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

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]]

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])

Rubi steps

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

Mathematica [A]  time = 0.0049945, size = 25, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[2 + x^6]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 26, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{6}\ln \left ({ \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^6+2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.46987, size = 46, normalized size = 1.84 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) \end{align*}

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)))

________________________________________________________________________________________

Fricas [A]  time = 1.5193, size = 80, normalized size = 3.2 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\frac{x^{6} - 2 \, \sqrt{2} \sqrt{x^{6} + 2} + 4}{x^{6}}\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.934494, size = 17, normalized size = 0.68 \begin{align*} - \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{6} \end{align*}

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

________________________________________________________________________________________

Giac [B]  time = 1.18465, size = 50, normalized size = 2. \begin{align*} -\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{align*}

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))