3.1.100 \(\int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx\)

Optimal. Leaf size=16 \[ \frac {\sqrt {x^6-1}}{3 x^3} \]

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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 {\sqrt {x^6-1}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(3*x^3)

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 {1}{x^4 \sqrt {-1+x^6}} \, dx &=\frac {\sqrt {-1+x^6}}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6-1}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(3*x^3)

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IntegrateAlgebraic [A]  time = 0.14, size = 16, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^4*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(3*x^3)

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fricas [A]  time = 0.46, size = 16, normalized size = 1.00 \begin {gather*} \frac {x^{3} + \sqrt {x^{6} - 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/3*(x^3 + sqrt(x^6 - 1))/x^3

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giac [A]  time = 0.36, size = 20, normalized size = 1.25 \begin {gather*} \frac {\sqrt {-\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\relax (x)} - \frac {1}{3} \, \mathrm {sgn}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6-1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 13, normalized size = 0.81

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.45, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6-1)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(x^6 - 1)/x^3

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mupad [B]  time = 0.30, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x^6-1}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^6 - 1)^(1/2)),x)

[Out]

(x^6 - 1)^(1/2)/(3*x^3)

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sympy [A]  time = 0.58, size = 29, normalized size = 1.81 \begin {gather*} \begin {cases} \frac {i \sqrt {-1 + \frac {1}{x^{6}}}}{3} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {\sqrt {1 - \frac {1}{x^{6}}}}{3} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(x**6-1)**(1/2),x)

[Out]

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

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