3.1412 \(\int \frac{1}{x (2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{6 \sqrt{x^6+2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.01586, antiderivative size = 39, 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} \[ \frac{1}{6 \sqrt{x^6+2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(2 + x^6)^(3/2)),x]

[Out]

1/(6*Sqrt[2 + x^6]) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(6*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 \left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x (2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac{1}{6 \sqrt{2+x^6}}+\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6 \sqrt{2+x^6}}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\sqrt{2+x^6}\right )\\ &=\frac{1}{6 \sqrt{2+x^6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2+x^6}}{\sqrt{2}}\right )}{6 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0050294, size = 30, normalized size = 0.77 \[ \frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{x^6}{2}+1\right )}{6 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(2 + x^6)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 36, normalized size = 0.9 \begin{align*}{\frac{1}{6}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{\sqrt{2}}{12}\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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.4877, size = 59, normalized size = 1.51 \begin{align*} \frac{1}{24} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) + \frac{1}{6 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44085, size = 134, normalized size = 3.44 \begin{align*} \frac{\sqrt{2}{\left (x^{6} + 2\right )} \log \left (\frac{x^{6} - 2 \, \sqrt{2} \sqrt{x^{6} + 2} + 4}{x^{6}}\right ) + 4 \, \sqrt{x^{6} + 2}}{24 \,{\left (x^{6} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(sqrt(2)*(x^6 + 2)*log((x^6 - 2*sqrt(2)*sqrt(x^6 + 2) + 4)/x^6) + 4*sqrt(x^6 + 2))/(x^6 + 2)

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Sympy [B]  time = 1.3509, size = 194, normalized size = 4.97 \begin{align*} \frac{x^{6} \log{\left (x^{6} \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} - \frac{2 x^{6} \log{\left (\sqrt{\frac{x^{6}}{2} + 1} + 1 \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} - \frac{x^{6} \log{\left (2 \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} + \frac{2 \sqrt{2} \sqrt{x^{6} + 2}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} + \frac{2 \log{\left (x^{6} \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} - \frac{4 \log{\left (\sqrt{\frac{x^{6}}{2} + 1} + 1 \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} - \frac{2 \log{\left (2 \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**6*log(x**6)/(12*sqrt(2)*x**6 + 24*sqrt(2)) - 2*x**6*log(sqrt(x**6/2 + 1) + 1)/(12*sqrt(2)*x**6 + 24*sqrt(2)
) - x**6*log(2)/(12*sqrt(2)*x**6 + 24*sqrt(2)) + 2*sqrt(2)*sqrt(x**6 + 2)/(12*sqrt(2)*x**6 + 24*sqrt(2)) + 2*l
og(x**6)/(12*sqrt(2)*x**6 + 24*sqrt(2)) - 4*log(sqrt(x**6/2 + 1) + 1)/(12*sqrt(2)*x**6 + 24*sqrt(2)) - 2*log(2
)/(12*sqrt(2)*x**6 + 24*sqrt(2))

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Giac [A]  time = 1.15335, size = 59, normalized size = 1.51 \begin{align*} \frac{1}{24} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) + \frac{1}{6 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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