\(\int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx\) [287]

Optimal result

Integrand size = 13, antiderivative size = 26 \[ \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx=\frac {\sqrt {-1+x^6} \left (-3+x^6+2 x^{12}\right )}{45 x^{15}} \]

[Out]

1/45*(x^6-1)^(1/2)*(2*x^12+x^6-3)/x^15

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 270} \[ \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx=\frac {\left (x^6-1\right )^{3/2}}{15 x^{15}}+\frac {2 \left (x^6-1\right )^{3/2}}{45 x^9} \]

[In]

Int[Sqrt[-1 + x^6]/x^16,x]

[Out]

(-1 + x^6)^(3/2)/(15*x^15) + (2*(-1 + x^6)^(3/2))/(45*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]

Rule 277

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^6\right )^{3/2}}{15 x^{15}}+\frac {2}{5} \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx \\ & = \frac {\left (-1+x^6\right )^{3/2}}{15 x^{15}}+\frac {2 \left (-1+x^6\right )^{3/2}}{45 x^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx=\frac {\sqrt {-1+x^6} \left (-3+x^6+2 x^{12}\right )}{45 x^{15}} \]

[In]

Integrate[Sqrt[-1 + x^6]/x^16,x]

[Out]

(Sqrt[-1 + x^6]*(-3 + x^6 + 2*x^12))/(45*x^15)

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(\frac {\left (2 x^{6}+3\right ) \left (x^{6}-1\right )^{\frac {3}{2}}}{45 x^{15}}\)	\(20\)
trager	\(\frac {\sqrt {x^{6}-1}\, \left (2 x^{12}+x^{6}-3\right )}{45 x^{15}}\)	\(23\)
risch	\(\frac {2 x^{18}-x^{12}-4 x^{6}+3}{45 x^{15} \sqrt {x^{6}-1}}\)	\(30\)
gosper	\(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (2 x^{6}+3\right ) \sqrt {x^{6}-1}}{45 x^{15}}\)	\(40\)
meijerg	\(-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {2}{3} x^{12}-\frac {1}{3} x^{6}+1\right ) \sqrt {-x^{6}+1}}{15 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, x^{15}}\)	\(45\)

[In]

int((x^6-1)^(1/2)/x^16,x,method=_RETURNVERBOSE)

[Out]

1/45*(2*x^6+3)*(x^6-1)^(3/2)/x^15

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx=\frac {2 \, x^{15} + {\left (2 \, x^{12} + x^{6} - 3\right )} \sqrt {x^{6} - 1}}{45 \, x^{15}} \]

[In]

integrate((x^6-1)^(1/2)/x^16,x, algorithm="fricas")

[Out]

1/45*(2*x^15 + (2*x^12 + x^6 - 3)*sqrt(x^6 - 1))/x^15

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.54 \[ \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx=\begin {cases} \frac {2 \sqrt {x^{6} - 1}}{45 x^{3}} + \frac {\sqrt {x^{6} - 1}}{45 x^{9}} - \frac {\sqrt {x^{6} - 1}}{15 x^{15}} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {2 i \sqrt {1 - x^{6}}}{45 x^{3}} + \frac {i \sqrt {1 - x^{6}}}{45 x^{9}} - \frac {i \sqrt {1 - x^{6}}}{15 x^{15}} & \text {otherwise} \end {cases} \]

[In]

integrate((x**6-1)**(1/2)/x**16,x)

[Out]

Piecewise((2*sqrt(x**6 - 1)/(45*x**3) + sqrt(x**6 - 1)/(45*x**9) - sqrt(x**6 - 1)/(15*x**15), Abs(x**6) > 1),
(2*I*sqrt(1 - x**6)/(45*x**3) + I*sqrt(1 - x**6)/(45*x**9) - I*sqrt(1 - x**6)/(15*x**15), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} - \frac {{\left (x^{6} - 1\right )}^{\frac {5}{2}}}{15 \, x^{15}} \]

[In]

integrate((x^6-1)^(1/2)/x^16,x, algorithm="maxima")

[Out]

1/9*(x^6 - 1)^(3/2)/x^9 - 1/15*(x^6 - 1)^(5/2)/x^15

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx=-\frac {3 \, {\left (\frac {1}{x^{6}} - 1\right )}^{2} \sqrt {-\frac {1}{x^{6}} + 1} - 5 \, {\left (-\frac {1}{x^{6}} + 1\right )}^{\frac {3}{2}}}{45 \, \mathrm {sgn}\left (x\right )} - \frac {2}{45} \, \mathrm {sgn}\left (x\right ) \]

[In]

integrate((x^6-1)^(1/2)/x^16,x, algorithm="giac")

[Out]

-1/45*(3*(1/x^6 - 1)^2*sqrt(-1/x^6 + 1) - 5*(-1/x^6 + 1)^(3/2))/sgn(x) - 2/45*sgn(x)

Mupad [B] (verification not implemented)

Time = 5.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {-1+x^6}}{x^{16}} \, dx=\frac {5\,{\left (x^6-1\right )}^{3/2}+2\,{\left (x^6-1\right )}^{5/2}}{45\,x^{15}} \]

[In]

int((x^6 - 1)^(1/2)/x^16,x)

[Out]

(5*(x^6 - 1)^(3/2) + 2*(x^6 - 1)^(5/2))/(45*x^15)