\(\int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx\) [453]

Optimal result

Integrand size = 13, antiderivative size = 36 \[ \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{24 x^{12}}+\frac {1}{24} \arctan \left (\sqrt {-1+x^6}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31, 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, 209} \[ \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx=\frac {1}{24} \arctan \left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{24 x^6}-\frac {\sqrt {x^6-1}}{12 x^{12}} \]

[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) + ArcTan[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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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}{\sqrt {-1+x} x^2} \, 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}{\sqrt {-1+x} 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} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx=\frac {1}{24} \left (\frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^{12}}+\arctan \left (\sqrt {-1+x^6}\right )\right ) \]

[In]

Integrate[Sqrt[-1 + x^6]/x^13,x]

[Out]

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

Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {-\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{12}+\sqrt {x^{6}-1}\, x^{6}-2 \sqrt {x^{6}-1}}{24 x^{12}}\)	\(40\)
trager	\(\frac {\left (x^{6}-2\right ) \sqrt {x^{6}-1}}{24 x^{12}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\)	\(49\)
risch	\(\frac {x^{12}-3 x^{6}+2}{24 x^{12} \sqrt {x^{6}-1}}+\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{48 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(84\)
meijerg	\(-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (x^{12}-8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right ) \sqrt {-x^{6}+1}}{8 x^{12}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{12}}-\frac {\sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\)	\(120\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx=\frac {x^{12} \arctan \left (\sqrt {x^{6} - 1}\right ) + \sqrt {x^{6} - 1} {\left (x^{6} - 2\right )}}{24 \, x^{12}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.93 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.44 \[ \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{24} - \frac {i}{24 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{8 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i}{12 x^{15} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{24} + \frac {1}{24 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{8 x^{9} \sqrt {1 - \frac {1}{x^{6}}}} + \frac {1}{12 x^{15} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((I*acosh(x**(-3))/24 - I/(24*x**3*sqrt(-1 + x**(-6))) + I/(8*x**9*sqrt(-1 + x**(-6))) - I/(12*x**15*
sqrt(-1 + x**(-6))), 1/Abs(x**6) > 1), (-asin(x**(-3))/24 + 1/(24*x**3*sqrt(1 - 1/x**6)) - 1/(8*x**9*sqrt(1 -
1/x**6)) + 1/(12*x**15*sqrt(1 - 1/x**6)), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}} - \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {1}{24} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[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) + 1/24*arctan(sqrt(x^6 - 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx=\frac {1}{96} \, \pi + \frac {\sqrt {x^{6} - 1} - \frac {1}{\sqrt {x^{6} - 1}}}{24 \, {\left ({\left (\sqrt {x^{6} - 1} - \frac {1}{\sqrt {x^{6} - 1}}\right )}^{2} + 4\right )}} + \frac {1}{48} \, \arctan \left (\frac {x^{6} - 2}{2 \, \sqrt {x^{6} - 1}}\right ) \]

[In]

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

[Out]

1/96*pi + 1/24*(sqrt(x^6 - 1) - 1/sqrt(x^6 - 1))/((sqrt(x^6 - 1) - 1/sqrt(x^6 - 1))^2 + 4) + 1/48*arctan(1/2*(
x^6 - 2)/sqrt(x^6 - 1))

Mupad [B] (verification not implemented)

Time = 5.53 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{24}-\frac {\frac {\sqrt {x^6-1}}{24}-\frac {{\left (x^6-1\right )}^{3/2}}{24}}{x^{12}} \]

[In]

int((x^6 - 1)^(1/2)/x^13,x)

[Out]

atan((x^6 - 1)^(1/2))/24 - ((x^6 - 1)^(1/2)/24 - (x^6 - 1)^(3/2)/24)/x^12