3.2.84 \(\int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx\)

Optimal. Leaf size=20 \[ \frac {3 \left (x^6-x^2\right )^{2/3}}{8 x^4} \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2014} \begin {gather*} \frac {3 \left (x^6-x^2\right )^{2/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(-x^2 + x^6)^(1/3)),x]

[Out]

(3*(-x^2 + x^6)^(2/3))/(8*x^4)

Rule 2014

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

Rubi steps

\begin {align*} \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx &=\frac {3 \left (-x^2+x^6\right )^{2/3}}{8 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {3 \left (x^2 \left (x^4-1\right )\right )^{2/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(-x^2 + x^6)^(1/3)),x]

[Out]

(3*(x^2*(-1 + x^4))^(2/3))/(8*x^4)

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IntegrateAlgebraic [A]  time = 0.32, size = 20, normalized size = 1.00 \begin {gather*} \frac {3 \left (-x^2+x^6\right )^{2/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^3*(-x^2 + x^6)^(1/3)),x]

[Out]

(3*(-x^2 + x^6)^(2/3))/(8*x^4)

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fricas [A]  time = 0.47, size = 16, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}}}{8 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^6-x^2)^(1/3),x, algorithm="fricas")

[Out]

3/8*(x^6 - x^2)^(2/3)/x^4

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giac [A]  time = 0.43, size = 11, normalized size = 0.55 \begin {gather*} \frac {3}{8} \, {\left (-\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^6-x^2)^(1/3),x, algorithm="giac")

[Out]

3/8*(-1/x^4 + 1)^(2/3)

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maple [A]  time = 0.07, size = 17, normalized size = 0.85

method	result	size
trager	\(\frac {3 \left (x^{6}-x^{2}\right )^{\frac {2}{3}}}{8 x^{4}}\)	\(17\)
risch	\(\frac {\frac {3 x^{4}}{8}-\frac {3}{8}}{x^{2} \left (x^{2} \left (x^{4}-1\right )\right )^{\frac {1}{3}}}\)	\(22\)
gosper	\(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{8 x^{2} \left (x^{6}-x^{2}\right )^{\frac {1}{3}}}\)	\(28\)
meijerg	\(-\frac {3 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{3}} \left (-x^{4}+1\right )^{\frac {2}{3}}}{8 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{3}} x^{\frac {8}{3}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(x^6-x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/8*(x^6-x^2)^(2/3)/x^4

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maxima [A]  time = 0.41, size = 30, normalized size = 1.50 \begin {gather*} \frac {3 \, {\left (x^{6} - x^{2}\right )}}{8 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2}\right )}^{\frac {7}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^6-x^2)^(1/3),x, algorithm="maxima")

[Out]

3/8*(x^6 - x^2)/((x^2 + 1)^(1/3)*(x^2 - 1)^(1/3)*(x^2)^(7/3))

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mupad [B]  time = 0.24, size = 16, normalized size = 0.80 \begin {gather*} \frac {3\,{\left (x^6-x^2\right )}^{2/3}}{8\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(x^6 - x^2)^(1/3)),x)

[Out]

(3*(x^6 - x^2)^(2/3))/(8*x^4)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(x**6-x**2)**(1/3),x)

[Out]

Integral(1/(x**3*(x**2*(x - 1)*(x + 1)*(x**2 + 1))**(1/3)), x)

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