\(\int \frac {\sqrt {1+x^6}}{x^7} \, dx\) [386]

Optimal result

Integrand size = 13, antiderivative size = 31 \[ \int \frac {\sqrt {1+x^6}}{x^7} \, dx=-\frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{6} \text {arctanh}\left (\sqrt {1+x^6}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 43, 65, 213} \[ \int \frac {\sqrt {1+x^6}}{x^7} \, dx=-\frac {1}{6} \text {arctanh}\left (\sqrt {x^6+1}\right )-\frac {\sqrt {x^6+1}}{6 x^6} \]

[In]

Int[Sqrt[1 + x^6]/x^7,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[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 {\sqrt {1+x}}{x^2} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right ) \\ & = -\frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{6} \text {arctanh}\left (\sqrt {1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^6}}{x^7} \, dx=-\frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{6} \text {arctanh}\left (\sqrt {1+x^6}\right ) \]

[In]

Integrate[Sqrt[1 + x^6]/x^7,x]

[Out]

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

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {-\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{6}+1}}\right ) x^{6}-\sqrt {x^{6}+1}}{6 x^{6}}\)	\(29\)
trager	\(-\frac {\sqrt {x^{6}+1}}{6 x^{6}}-\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{6}\)	\(30\)
risch	\(-\frac {\sqrt {x^{6}+1}}{6 x^{6}}+\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{12 \sqrt {\pi }}\)	\(50\)
meijerg	\(-\frac {-\frac {\sqrt {\pi }\, \left (4 x^{6}+8\right )}{4 x^{6}}+\frac {2 \sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{6}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )-\left (-2 \ln \left (2\right )-1+6 \ln \left (x \right )\right ) \sqrt {\pi }+\frac {2 \sqrt {\pi }}{x^{6}}}{12 \sqrt {\pi }}\)	\(77\)

[In]

int((x^6+1)^(1/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {1+x^6}}{x^7} \, dx=-\frac {x^{6} \log \left (\sqrt {x^{6} + 1} + 1\right ) - x^{6} \log \left (\sqrt {x^{6} + 1} - 1\right ) + 2 \, \sqrt {x^{6} + 1}}{12 \, x^{6}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1+x^6}}{x^7} \, dx=- \frac {\operatorname {asinh}{\left (\frac {1}{x^{3}} \right )}}{6} - \frac {\sqrt {1 + \frac {1}{x^{6}}}}{6 x^{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {1+x^6}}{x^7} \, dx=-\frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {1+x^6}}{x^7} \, dx=-\frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1+x^6}}{x^7} \, dx=-\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{6}-\frac {\sqrt {x^6+1}}{6\,x^6} \]

[In]

int((x^6 + 1)^(1/2)/x^7,x)

[Out]

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