3.1413 \(\int \frac{1}{x^7 (2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ -\frac{1}{12 x^6 \sqrt{x^6+2}}-\frac{1}{8 \sqrt{x^6+2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

[Out]

-1/(8*Sqrt[2 + x^6]) - 1/(12*x^6*Sqrt[2 + x^6]) + ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(8*Sqrt[2])

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Rubi [A]  time = 0.0228854, antiderivative size = 58, normalized size of antiderivative = 1.05, 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} \[ -\frac{\sqrt{x^6+2}}{8 x^6}+\frac{1}{6 x^6 \sqrt{x^6+2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(2 + x^6)^(3/2)),x]

[Out]

1/(6*x^6*Sqrt[2 + x^6]) - Sqrt[2 + x^6]/(8*x^6) + ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(8*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 \left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^2 (2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac{1}{6 x^6 \sqrt{2+x^6}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6 x^6 \sqrt{2+x^6}}-\frac{\sqrt{2+x^6}}{8 x^6}-\frac{1}{16} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6 x^6 \sqrt{2+x^6}}-\frac{\sqrt{2+x^6}}{8 x^6}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{2+x^6}\right )\\ &=\frac{1}{6 x^6 \sqrt{2+x^6}}-\frac{\sqrt{2+x^6}}{8 x^6}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2+x^6}}{\sqrt{2}}\right )}{8 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0056604, size = 30, normalized size = 0.55 \[ -\frac{\, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{x^6}{2}+1\right )}{12 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(2 + x^6)^(3/2)),x]

[Out]

-Hypergeometric2F1[-1/2, 2, 1/2, 1 + x^6/2]/(12*Sqrt[2 + x^6])

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Maple [A]  time = 0.017, size = 46, normalized size = 0.8 \begin{align*} -{\frac{3\,{x}^{6}+2}{24\,{x}^{6}}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{\sqrt{2}}{16}\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)^(3/2),x)

[Out]

-1/24*(3*x^6+2)/x^6/(x^6+2)^(1/2)-1/16*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))/(x^6)^(1/2))

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Maxima [A]  time = 1.51678, size = 85, normalized size = 1.55 \begin{align*} -\frac{1}{32} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{3 \, x^{6} + 2}{24 \,{\left ({\left (x^{6} + 2\right )}^{\frac{3}{2}} - 2 \, \sqrt{x^{6} + 2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

-1/32*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) - 1/24*(3*x^6 + 2)/((x^6 + 2)^(3/2) -
2*sqrt(x^6 + 2))

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Fricas [A]  time = 1.45942, size = 166, normalized size = 3.02 \begin{align*} \frac{3 \, \sqrt{2}{\left (x^{12} + 2 \, x^{6}\right )} \log \left (\frac{x^{6} + 2 \, \sqrt{2} \sqrt{x^{6} + 2} + 4}{x^{6}}\right ) - 4 \,{\left (3 \, x^{6} + 2\right )} \sqrt{x^{6} + 2}}{96 \,{\left (x^{12} + 2 \, x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

1/96*(3*sqrt(2)*(x^12 + 2*x^6)*log((x^6 + 2*sqrt(2)*sqrt(x^6 + 2) + 4)/x^6) - 4*(3*x^6 + 2)*sqrt(x^6 + 2))/(x^
12 + 2*x^6)

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Sympy [A]  time = 2.72542, size = 49, normalized size = 0.89 \begin{align*} \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{16} - \frac{1}{8 x^{3} \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{12 x^{9} \sqrt{1 + \frac{2}{x^{6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*asinh(sqrt(2)/x**3)/16 - 1/(8*x**3*sqrt(1 + 2/x**6)) - 1/(12*x**9*sqrt(1 + 2/x**6))

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Giac [A]  time = 1.15875, size = 85, normalized size = 1.55 \begin{align*} -\frac{1}{32} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{3 \, x^{6} + 2}{24 \,{\left ({\left (x^{6} + 2\right )}^{\frac{3}{2}} - 2 \, \sqrt{x^{6} + 2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

-1/32*sqrt(2)*log(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) - 1/24*(3*x^6 + 2)/((x^6 + 2)^(3/2) -
2*sqrt(x^6 + 2))