\(\int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx\) [1389]

Optimal result

Integrand size = 13, antiderivative size = 58 \[ \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx=-\frac {\sqrt {2+x^6}}{24 x^{12}}+\frac {\sqrt {2+x^6}}{32 x^6}-\frac {\text {arctanh}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{32 \sqrt {2}} \]

[Out]

-1/64*arctanh(1/2*(x^6+2)^(1/2)*2^(1/2))*2^(1/2)-1/24*(x^6+2)^(1/2)/x^12+1/32*(x^6+2)^(1/2)/x^6

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 44, 65, 213} \[ \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {x^6+2}}{\sqrt {2}}\right )}{32 \sqrt {2}}+\frac {\sqrt {x^6+2}}{32 x^6}-\frac {\sqrt {x^6+2}}{24 x^{12}} \]

[In]

Int[1/(x^13*Sqrt[2 + x^6]),x]

[Out]

-1/24*Sqrt[2 + x^6]/x^12 + Sqrt[2 + x^6]/(32*x^6) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(32*Sqrt[2])

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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*} \text {integral}& = \frac {1}{6} \text {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} \text {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} \text {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} \text {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 [A] (verified)

Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx=\frac {\sqrt {2+x^6} \left (-4+3 x^6\right )}{96 x^{12}}-\frac {\text {arctanh}\left (\frac {\sqrt {2+x^6}}{\sqrt {2}}\right )}{32 \sqrt {2}} \]

[In]

Integrate[1/(x^13*Sqrt[2 + x^6]),x]

[Out]

(Sqrt[2 + x^6]*(-4 + 3*x^6))/(96*x^12) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(32*Sqrt[2])

Maple [A] (verified)

Time = 4.63 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}}{\sqrt {x^{6}+2}}\right ) x^{12}+6 \sqrt {x^{6}+2}\, x^{6}-8 \sqrt {x^{6}+2}}{192 x^{12}}\)	\(48\)
trager	\(\frac {\left (3 x^{6}-4\right ) \sqrt {x^{6}+2}}{96 x^{12}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-\sqrt {x^{6}+2}}{x^{3}}\right )}{64}\)	\(51\)
risch	\(\frac {3 x^{12}+2 x^{6}-8}{96 x^{12} \sqrt {x^{6}+2}}+\frac {\sqrt {2}\, \left (\left (-3 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{6}}{2}}}{2}\right )\right )}{128 \sqrt {\pi }}\)	\(67\)
meijerg	\(\frac {\sqrt {2}\, \left (-\frac {2 \sqrt {\pi }}{x^{12}}+\frac {\sqrt {\pi }}{x^{6}}+\frac {3 \left (\frac {7}{6}-3 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-\frac {7}{4} x^{12}-4 x^{6}+8\right )}{4 x^{12}}-\frac {\sqrt {\pi }\, \left (-6 x^{6}+8\right ) \sqrt {1+\frac {x^{6}}{2}}}{4 x^{12}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{6}}{2}}}{2}\right )}{4}\right )}{48 \sqrt {\pi }}\)	\(103\)

[In]

int(1/x^13/(x^6+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/192*(-3*2^(1/2)*arctanh(2^(1/2)/(x^6+2)^(1/2))*x^12+6*(x^6+2)^(1/2)*x^6-8*(x^6+2)^(1/2))/x^12

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx=\frac {3 \, \sqrt {2} x^{12} \log \left (\frac {x^{6} - 2 \, \sqrt {2} \sqrt {x^{6} + 2} + 4}{x^{6}}\right ) + 4 \, {\left (3 \, x^{6} - 4\right )} \sqrt {x^{6} + 2}}{384 \, x^{12}} \]

[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

Sympy [A] (verification not implemented)

Time = 2.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx=- \frac {\sqrt {2} \operatorname {asinh}{\left (\frac {\sqrt {2}}{x^{3}} \right )}}{64} + \frac {1}{32 x^{3} \sqrt {1 + \frac {2}{x^{6}}}} + \frac {1}{48 x^{9} \sqrt {1 + \frac {2}{x^{6}}}} - \frac {1}{12 x^{15} \sqrt {1 + \frac {2}{x^{6}}}} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx=\frac {1}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) - \frac {3 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - 10 \, \sqrt {x^{6} + 2}}{96 \, {\left (4 \, x^{6} - {\left (x^{6} + 2\right )}^{2} + 4\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx=\frac {1}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {x^{6} + 2}}{\sqrt {2} + \sqrt {x^{6} + 2}}\right ) + \frac {3 \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - 10 \, \sqrt {x^{6} + 2}}{96 \, x^{12}} \]

[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

Mupad [B] (verification not implemented)

Time = 5.69 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^{13} \sqrt {2+x^6}} \, dx=\frac {\frac {5\,\sqrt {x^6+2}}{48}-\frac {{\left (x^6+2\right )}^{3/2}}{32}}{4\,x^6-{\left (x^6+2\right )}^2+4}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^6+2}}{2}\right )}{64} \]

[In]

int(1/(x^13*(x^6 + 2)^(1/2)),x)

[Out]

((5*(x^6 + 2)^(1/2))/48 - (x^6 + 2)^(3/2)/32)/(4*x^6 - (x^6 + 2)^2 + 4) - (2^(1/2)*atanh((2^(1/2)*(x^6 + 2)^(1
/2))/2))/64