\(\int \frac {\sqrt {1+x^6}}{x^{13}} \, dx\) [492]

Optimal result

Integrand size = 13, antiderivative size = 38 \[ \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx=\frac {\left (-2-x^6\right ) \sqrt {1+x^6}}{24 x^{12}}+\frac {1}{24} \text {arctanh}\left (\sqrt {1+x^6}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 43, 44, 65, 213} \[ \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx=\frac {1}{24} \text {arctanh}\left (\sqrt {x^6+1}\right )-\frac {\sqrt {x^6+1}}{24 x^6}-\frac {\sqrt {x^6+1}}{12 x^{12}} \]

[In]

Int[Sqrt[1 + x^6]/x^13,x]

[Out]

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

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 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 {\sqrt {1+x}}{x^3} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt {1+x^6}}{12 x^{12}}+\frac {1}{24} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {\sqrt {1+x^6}}{24 x^6}-\frac {1}{48} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {\sqrt {1+x^6}}{24 x^6}-\frac {1}{24} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right ) \\ & = -\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {\sqrt {1+x^6}}{24 x^6}+\frac {1}{24} \text {arctanh}\left (\sqrt {1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx=\frac {1}{24} \left (-\frac {\sqrt {1+x^6} \left (2+x^6\right )}{x^{12}}+\text {arctanh}\left (\sqrt {1+x^6}\right )\right ) \]

[In]

Integrate[Sqrt[1 + x^6]/x^13,x]

[Out]

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

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92

method	result	size
trager	\(-\frac {\left (x^{6}+2\right ) \sqrt {x^{6}+1}}{24 x^{12}}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{24}\)	\(35\)
pseudoelliptic	\(\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{6}+1}}\right ) x^{12}-\sqrt {x^{6}+1}\, x^{6}-2 \sqrt {x^{6}+1}}{24 x^{12}}\)	\(40\)
risch	\(-\frac {x^{12}+3 x^{6}+2}{24 x^{12} \sqrt {x^{6}+1}}-\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{48 \sqrt {\pi }}\)	\(60\)
meijerg	\(-\frac {-\frac {\sqrt {\pi }\, \left (x^{12}+8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (4 x^{6}+8\right ) \sqrt {x^{6}+1}}{8 x^{12}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{12}}+\frac {\sqrt {\pi }}{x^{6}}}{12 \sqrt {\pi }}\)	\(93\)

[In]

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

[Out]

-1/24*(x^6+2)/x^12*(x^6+1)^(1/2)+1/24*ln(((x^6+1)^(1/2)+1)/x^3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx=\frac {x^{12} \log \left (\sqrt {x^{6} + 1} + 1\right ) - x^{12} \log \left (\sqrt {x^{6} + 1} - 1\right ) - 2 \, {\left (x^{6} + 2\right )} \sqrt {x^{6} + 1}}{48 \, x^{12}} \]

[In]

integrate((x^6+1)^(1/2)/x^13,x, algorithm="fricas")

[Out]

1/48*(x^12*log(sqrt(x^6 + 1) + 1) - x^12*log(sqrt(x^6 + 1) - 1) - 2*(x^6 + 2)*sqrt(x^6 + 1))/x^12

Sympy [A] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx=\frac {\operatorname {asinh}{\left (\frac {1}{x^{3}} \right )}}{24} - \frac {1}{24 x^{3} \sqrt {1 + \frac {1}{x^{6}}}} - \frac {1}{8 x^{9} \sqrt {1 + \frac {1}{x^{6}}}} - \frac {1}{12 x^{15} \sqrt {1 + \frac {1}{x^{6}}}} \]

[In]

integrate((x**6+1)**(1/2)/x**13,x)

[Out]

asinh(x**(-3))/24 - 1/(24*x**3*sqrt(1 + x**(-6))) - 1/(8*x**9*sqrt(1 + x**(-6))) - 1/(12*x**15*sqrt(1 + x**(-6
)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (28) = 56\).

Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx=\frac {{\left (x^{6} + 1\right )}^{\frac {3}{2}} + \sqrt {x^{6} + 1}}{24 \, {\left (2 \, x^{6} - {\left (x^{6} + 1\right )}^{2} + 1\right )}} + \frac {1}{48} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{48} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[In]

integrate((x^6+1)^(1/2)/x^13,x, algorithm="maxima")

[Out]

1/24*((x^6 + 1)^(3/2) + sqrt(x^6 + 1))/(2*x^6 - (x^6 + 1)^2 + 1) + 1/48*log(sqrt(x^6 + 1) + 1) - 1/48*log(sqrt
(x^6 + 1) - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx=-\frac {\sqrt {x^{6} + 1} + \frac {1}{\sqrt {x^{6} + 1}}}{24 \, {\left ({\left (\sqrt {x^{6} + 1} + \frac {1}{\sqrt {x^{6} + 1}}\right )}^{2} - 4\right )}} + \frac {1}{96} \, \log \left (\sqrt {x^{6} + 1} + \frac {1}{\sqrt {x^{6} + 1}} + 2\right ) - \frac {1}{96} \, \log \left (\sqrt {x^{6} + 1} + \frac {1}{\sqrt {x^{6} + 1}} - 2\right ) \]

[In]

integrate((x^6+1)^(1/2)/x^13,x, algorithm="giac")

[Out]

-1/24*(sqrt(x^6 + 1) + 1/sqrt(x^6 + 1))/((sqrt(x^6 + 1) + 1/sqrt(x^6 + 1))^2 - 4) + 1/96*log(sqrt(x^6 + 1) + 1
/sqrt(x^6 + 1) + 2) - 1/96*log(sqrt(x^6 + 1) + 1/sqrt(x^6 + 1) - 2)

Mupad [B] (verification not implemented)

Time = 5.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx=\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{24}+\frac {\frac {\sqrt {x^6+1}}{24}+\frac {{\left (x^6+1\right )}^{3/2}}{24}}{2\,x^6-{\left (x^6+1\right )}^2+1} \]

[In]

int((x^6 + 1)^(1/2)/x^13,x)

[Out]

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