3.15.12 \(\int \frac {1}{x (2+x^6)^{3/2}} \, dx\) [1412]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {272, 53, 65, 213} \begin {gather*} \frac {1}{6 \sqrt {x^6+2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{6 \sqrt {2}} \end {gather*}

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 53

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 65

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 272

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]]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{6 \sqrt {2+x^6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{6 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[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])

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Maple [A]
time = 0.34, size = 36, normalized size = 0.92

method	result	size
risch	\(\frac {1}{6 \sqrt {x^{6}+2}}+\frac {\sqrt {2}\, \ln \left (\frac {\sqrt {x^{6}+2}-\sqrt {2}}{\sqrt {x^{6}}}\right )}{12}\)	\(36\)
trager	\(\frac {1}{6 \sqrt {x^{6}+2}}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\sqrt {x^{6}+2}+\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}}\right )}{12}\)	\(38\)
meijerg	\(\frac {\sqrt {2}\, \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {1+\frac {x^{6}}{2}}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{6}}{2}}}{2}\right )+\frac {\left (2-3 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{2}\right )}{12 \sqrt {\pi }}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^6+2)^(3/2),x,method=_RETURNVERBOSE)

[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 = 0.50, size = 44, normalized size = 1.13 \begin {gather*} \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 {gather*}

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 = 0.37, size = 50, normalized size = 1.28 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (34) = 68\).
time = 0.66, size = 194, normalized size = 4.97 \begin {gather*} \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 {gather*}

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 = 1.38, size = 44, normalized size = 1.13 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 1.22, size = 28, normalized size = 0.72 \begin {gather*} \frac {1}{6\,\sqrt {x^6+2}}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{12} \end {gather*}

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

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