3.1388 \(\int \frac{1}{x^7 \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=42 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{12 \sqrt{2}}-\frac{\sqrt{x^6+2}}{12 x^6} \]

[Out]

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

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Rubi [A]  time = 0.017391, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{12 \sqrt{2}}-\frac{\sqrt{x^6+2}}{12 x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(12*x^6) + ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(12*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^7 \sqrt{2+x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{2+x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{2+x^6}}{12 x^6}-\frac{1}{24} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{2+x}} \, dx,x,x^6\right )\\ &=-\frac{\sqrt{2+x^6}}{12 x^6}-\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{2+x^6}\right )\\ &=-\frac{\sqrt{2+x^6}}{12 x^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2+x^6}}{\sqrt{2}}\right )}{12 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0075629, size = 42, normalized size = 1. \[ \frac{\sqrt{2} x^6 \tanh ^{-1}\left (\sqrt{\frac{x^6}{2}+1}\right )-2 \sqrt{x^6+2}}{24 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[2 + x^6]),x]

[Out]

(-2*Sqrt[2 + x^6] + Sqrt[2]*x^6*ArcTanh[Sqrt[1 + x^6/2]])/(24*x^6)

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Maple [A]  time = 0.02, size = 39, normalized size = 0.9 \begin{align*} -{\frac{1}{12\,{x}^{6}}\sqrt{{x}^{6}+2}}-{\frac{\sqrt{2}}{24}\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^7/(x^6+2)^(1/2),x)

[Out]

-1/12*(x^6+2)^(1/2)/x^6-1/24*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))/(x^6)^(1/2))

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Maxima [A]  time = 1.50427, size = 63, normalized size = 1.5 \begin{align*} -\frac{1}{48} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{\sqrt{x^{6} + 2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

-1/48*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) - 1/12*sqrt(x^6 + 2)/x^6

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Fricas [A]  time = 1.42565, size = 117, normalized size = 2.79 \begin{align*} \frac{\sqrt{2} x^{6} \log \left (\frac{x^{6} + 2 \, \sqrt{2} \sqrt{x^{6} + 2} + 4}{x^{6}}\right ) - 4 \, \sqrt{x^{6} + 2}}{48 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/48*(sqrt(2)*x^6*log((x^6 + 2*sqrt(2)*sqrt(x^6 + 2) + 4)/x^6) - 4*sqrt(x^6 + 2))/x^6

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Sympy [A]  time = 1.98439, size = 31, normalized size = 0.74 \begin{align*} \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{24} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{12 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**6+2)**(1/2),x)

[Out]

sqrt(2)*asinh(sqrt(2)/x**3)/24 - sqrt(1 + 2/x**6)/(12*x**3)

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Giac [A]  time = 1.17579, size = 63, normalized size = 1.5 \begin{align*} -\frac{1}{48} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{\sqrt{x^{6} + 2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

-1/48*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) - 1/12*sqrt(x^6 + 2)/x^6