\(\int \frac {\sqrt {-1+x^6}}{x^7} \, dx\) [384]

Optimal result

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

[Out]

-1/6*(x^6-1)^(1/2)/x^6+1/6*arctan((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, 209} \[ \int \frac {\sqrt {-1+x^6}}{x^7} \, dx=\frac {1}{6} \arctan \left (\sqrt {x^6-1}\right )-\frac {\sqrt {x^6-1}}{6 x^6} \]

[In]

Int[Sqrt[-1 + x^6]/x^7,x]

[Out]

-1/6*Sqrt[-1 + x^6]/x^6 + ArcTan[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 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^2} \, dx,x,x^6\right ) \\ & = -\frac {\sqrt {-1+x^6}}{6 x^6}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} 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} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[Sqrt[-1 + x^6]/x^7,x]

[Out]

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

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {-\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{6}-\sqrt {x^{6}-1}}{6 x^{6}}\)	\(29\)
trager	\(-\frac {\sqrt {x^{6}-1}}{6 x^{6}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{6}-1}}{x^{3}}\right )}{6}\)	\(44\)
risch	\(-\frac {\sqrt {x^{6}-1}}{6 x^{6}}+\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(74\)
meijerg	\(\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right )}{4 x^{6}}-\frac {2 \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{6}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-\left (-2 \ln \left (2\right )-1+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-\frac {2 \sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\)	\(103\)

[In]

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

[Out]

1/6*(-arctan(1/(x^6-1)^(1/2))*x^6-(x^6-1)^(1/2))/x^6

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-1+x^6}}{x^7} \, dx=\frac {x^{6} \arctan \left (\sqrt {x^{6} - 1}\right ) - \sqrt {x^{6} - 1}}{6 \, x^{6}} \]

[In]

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

[Out]

1/6*(x^6*arctan(sqrt(x^6 - 1)) - sqrt(x^6 - 1))/x^6

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.85 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {-1+x^6}}{x^7} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{6} + \frac {i}{6 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i}{6 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{6} - \frac {\sqrt {1 - \frac {1}{x^{6}}}}{6 x^{3}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {-1+x^6}}{x^7} \, dx=-\frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} + \frac {1}{6} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {-1+x^6}}{x^7} \, dx=-\frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} + \frac {1}{6} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {-1+x^6}}{x^7} \, dx=\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{6}-\frac {\sqrt {x^6-1}}{6\,x^6} \]

[In]

int((x^6 - 1)^(1/2)/x^7,x)

[Out]

atan((x^6 - 1)^(1/2))/6 - (x^6 - 1)^(1/2)/(6*x^6)