3.1389 \(\int \frac{1}{x^{13} \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=58 \[ \frac{\sqrt{x^6+2}}{32 x^6}-\frac{\sqrt{x^6+2}}{24 x^{12}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{32 \sqrt{2}} \]

[Out]

-Sqrt[2 + x^6]/(24*x^12) + Sqrt[2 + x^6]/(32*x^6) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(32*Sqrt[2])

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Rubi [A]  time = 0.023114, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ \frac{\sqrt{x^6+2}}{32 x^6}-\frac{\sqrt{x^6+2}}{24 x^{12}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^13*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(24*x^12) + Sqrt[2 + x^6]/(32*x^6) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(32*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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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^{13} \sqrt{2+x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{2+x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{2+x^6}}{24 x^{12}}-\frac{1}{16} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{2+x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{2+x^6}}{24 x^{12}}+\frac{\sqrt{2+x^6}}{32 x^6}+\frac{1}{64} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{2+x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{2+x^6}}{24 x^{12}}+\frac{\sqrt{2+x^6}}{32 x^6}+\frac{1}{32} \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{2+x^6}\right )\\ &=-\frac{\sqrt{2+x^6}}{24 x^{12}}+\frac{\sqrt{2+x^6}}{32 x^6}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2+x^6}}{\sqrt{2}}\right )}{32 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0052494, size = 30, normalized size = 0.52 \[ -\frac{1}{24} \sqrt{x^6+2} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{x^6}{2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^13*Sqrt[2 + x^6]),x]

[Out]

-(Sqrt[2 + x^6]*Hypergeometric2F1[1/2, 3, 3/2, 1 + x^6/2])/24

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Maple [A]  time = 0.019, size = 51, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{12}+2\,{x}^{6}-8}{96\,{x}^{12}}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{\sqrt{2}}{64}\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^13/(x^6+2)^(1/2),x)

[Out]

1/96*(3*x^12+2*x^6-8)/x^12/(x^6+2)^(1/2)+1/64*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))/(x^6)^(1/2))

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Maxima [A]  time = 1.50159, size = 100, normalized size = 1.72 \begin{align*} \frac{1}{128} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{3 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - 10 \, \sqrt{x^{6} + 2}}{96 \,{\left (4 \, x^{6} -{\left (x^{6} + 2\right )}^{2} + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

1/128*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) - 1/96*(3*(x^6 + 2)^(3/2) - 10*sqrt(x^
6 + 2))/(4*x^6 - (x^6 + 2)^2 + 4)

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Fricas [A]  time = 1.47128, size = 140, normalized size = 2.41 \begin{align*} \frac{3 \, \sqrt{2} x^{12} \log \left (\frac{x^{6} - 2 \, \sqrt{2} \sqrt{x^{6} + 2} + 4}{x^{6}}\right ) + 4 \,{\left (3 \, x^{6} - 4\right )} \sqrt{x^{6} + 2}}{384 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/384*(3*sqrt(2)*x^12*log((x^6 - 2*sqrt(2)*sqrt(x^6 + 2) + 4)/x^6) + 4*(3*x^6 - 4)*sqrt(x^6 + 2))/x^12

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Sympy [A]  time = 4.31716, size = 66, normalized size = 1.14 \begin{align*} - \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{64} + \frac{1}{32 x^{3} \sqrt{1 + \frac{2}{x^{6}}}} + \frac{1}{48 x^{9} \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{12 x^{15} \sqrt{1 + \frac{2}{x^{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**13/(x**6+2)**(1/2),x)

[Out]

-sqrt(2)*asinh(sqrt(2)/x**3)/64 + 1/(32*x**3*sqrt(1 + 2/x**6)) + 1/(48*x**9*sqrt(1 + 2/x**6)) - 1/(12*x**15*sq
rt(1 + 2/x**6))

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Giac [A]  time = 1.21357, size = 80, normalized size = 1.38 \begin{align*} \frac{1}{128} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) + \frac{3 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - 10 \, \sqrt{x^{6} + 2}}{96 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

1/128*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) + 1/96*(3*(x^6 + 2)^(3/2) - 10*sqrt(x^
6 + 2))/x^12