\(\int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx\) [102]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx=\frac {\left (-1+x^6\right )^{3/2}}{9 x^9} \]

[Out]

1/9*(x^6-1)^(3/2)/x^9

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx=\frac {\left (x^6-1\right )^{3/2}}{9 x^9} \]

[In]

Int[Sqrt[-1 + x^6]/x^10,x]

[Out]

(-1 + x^6)^(3/2)/(9*x^9)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^6\right )^{3/2}}{9 x^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx=\frac {\left (-1+x^6\right )^{3/2}}{9 x^9} \]

[In]

Integrate[Sqrt[-1 + x^6]/x^10,x]

[Out]

(-1 + x^6)^(3/2)/(9*x^9)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
trager	\(\frac {\left (x^{6}-1\right )^{\frac {3}{2}}}{9 x^{9}}\)	\(13\)
pseudoelliptic	\(\frac {\left (x^{6}-1\right )^{\frac {3}{2}}}{9 x^{9}}\)	\(13\)
risch	\(\frac {x^{12}-2 x^{6}+1}{9 x^{9} \sqrt {x^{6}-1}}\)	\(23\)
gosper	\(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \sqrt {x^{6}-1}}{9 x^{9}}\)	\(33\)
meijerg	\(-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-x^{6}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, x^{9}}\)	\(33\)

[In]

int((x^6-1)^(1/2)/x^10,x,method=_RETURNVERBOSE)

[Out]

1/9*(x^6-1)^(3/2)/x^9

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx=\frac {x^{9} + {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \]

[In]

integrate((x^6-1)^(1/2)/x^10,x, algorithm="fricas")

[Out]

1/9*(x^9 + (x^6 - 1)^(3/2))/x^9

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.56 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.81 \[ \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx=\begin {cases} \frac {i \sqrt {-1 + \frac {1}{x^{6}}}}{9} - \frac {i \sqrt {-1 + \frac {1}{x^{6}}}}{9 x^{6}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {\sqrt {1 - \frac {1}{x^{6}}}}{9} - \frac {\sqrt {1 - \frac {1}{x^{6}}}}{9 x^{6}} & \text {otherwise} \end {cases} \]

[In]

integrate((x**6-1)**(1/2)/x**10,x)

[Out]

Piecewise((I*sqrt(-1 + x**(-6))/9 - I*sqrt(-1 + x**(-6))/(9*x**6), 1/Abs(x**6) > 1), (sqrt(1 - 1/x**6)/9 - sqr
t(1 - 1/x**6)/(9*x**6), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \]

[In]

integrate((x^6-1)^(1/2)/x^10,x, algorithm="maxima")

[Out]

1/9*(x^6 - 1)^(3/2)/x^9

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx=\frac {{\left (-\frac {1}{x^{6}} + 1\right )}^{\frac {3}{2}}}{9 \, \mathrm {sgn}\left (x\right )} - \frac {1}{9} \, \mathrm {sgn}\left (x\right ) \]

[In]

integrate((x^6-1)^(1/2)/x^10,x, algorithm="giac")

[Out]

1/9*(-1/x^6 + 1)^(3/2)/sgn(x) - 1/9*sgn(x)

Mupad [B] (verification not implemented)

Time = 5.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx=\frac {{\left (x^6-1\right )}^{3/2}}{9\,x^9} \]

[In]

int((x^6 - 1)^(1/2)/x^10,x)

[Out]

(x^6 - 1)^(3/2)/(9*x^9)