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

Optimal. Leaf size=16 \[ \frac {\left (x^6-1\right )^{3/2}}{9 x^9} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \begin {gather*} \frac {\left (x^6-1\right )^{3/2}}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^6]/x^10,x]

[Out]

(-1 + x^6)^(3/2)/(9*x^9)

Rule 264

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*} \int \frac {\sqrt {-1+x^6}}{x^{10}} \, dx &=\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (x^6-1\right )^{3/2}}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x^6]/x^10,x]

[Out]

(-1 + x^6)^(3/2)/(9*x^9)

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IntegrateAlgebraic [A]  time = 0.11, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^6\right )^{3/2}}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-1 + x^6]/x^10,x]

[Out]

(-1 + x^6)^(3/2)/(9*x^9)

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fricas [A]  time = 0.47, size = 16, normalized size = 1.00 \begin {gather*} \frac {x^{9} + {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(x^9 + (x^6 - 1)^(3/2))/x^9

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giac [A]  time = 0.62, size = 20, normalized size = 1.25 \begin {gather*} \frac {{\left (-\frac {1}{x^{6}} + 1\right )}^{\frac {3}{2}}}{9 \, \mathrm {sgn}\relax (x)} - \frac {1}{9} \, \mathrm {sgn}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.07, size = 13, normalized size = 0.81

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(x^6 - 1)^(3/2)/x^9

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mupad [B]  time = 0.28, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^6-1\right )}^{3/2}}{9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 - 1)^(1/2)/x^10,x)

[Out]

(x^6 - 1)^(3/2)/(9*x^9)

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sympy [B]  time = 0.87, size = 61, normalized size = 3.81 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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