3.4.2 \(\int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx\)

Optimal. Leaf size=27 \[ \frac {2 \left (4 x^2+3\right ) \left (x^4-x^2\right )^{3/4}}{21 x^5} \]

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2016, 2014} \begin {gather*} \frac {2 \left (x^4-x^2\right )^{3/4}}{7 x^5}+\frac {8 \left (x^4-x^2\right )^{3/4}}{21 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(-x^2 + x^4)^(1/4)),x]

[Out]

(2*(-x^2 + x^4)^(3/4))/(7*x^5) + (8*(-x^2 + x^4)^(3/4))/(21*x^3)

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])

Rule 2016

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*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx &=\frac {2 \left (-x^2+x^4\right )^{3/4}}{7 x^5}+\frac {4}{7} \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx\\ &=\frac {2 \left (-x^2+x^4\right )^{3/4}}{7 x^5}+\frac {8 \left (-x^2+x^4\right )^{3/4}}{21 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2 \left (x^2-1\right )\right )^{3/4} \left (4 x^2+3\right )}{21 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(-x^2 + x^4)^(1/4)),x]

[Out]

(2*(x^2*(-1 + x^2))^(3/4)*(3 + 4*x^2))/(21*x^5)

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IntegrateAlgebraic [A]  time = 0.12, size = 27, normalized size = 1.00 \begin {gather*} \frac {2 \left (3+4 x^2\right ) \left (-x^2+x^4\right )^{3/4}}{21 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^4*(-x^2 + x^4)^(1/4)),x]

[Out]

(2*(3 + 4*x^2)*(-x^2 + x^4)^(3/4))/(21*x^5)

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fricas [A]  time = 0.46, size = 23, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (4 \, x^{2} + 3\right )}}{21 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(x^4 - x^2)^(3/4)*(4*x^2 + 3)/x^5

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giac [A]  time = 0.42, size = 23, normalized size = 0.85 \begin {gather*} \frac {2}{7} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {7}{4}} - \frac {2}{3} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(-1/x^2 + 1)^(7/4) - 2/3*(-1/x^2 + 1)^(3/4)

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maple [A]  time = 0.08, size = 24, normalized size = 0.89

method	result	size
trager	\(\frac {2 \left (4 x^{2}+3\right ) \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{21 x^{5}}\)	\(24\)
risch	\(\frac {-\frac {2}{21} x^{2}-\frac {2}{7}+\frac {8}{21} x^{4}}{x^{3} \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}\)	\(29\)
gosper	\(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (4 x^{2}+3\right )}{21 x^{3} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\)	\(30\)
meijerg	\(-\frac {2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (1+\frac {4 x^{2}}{3}\right ) \left (-x^{2}+1\right )^{\frac {3}{4}}}{7 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {7}{2}}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(4*x^2+3)*(x^4-x^2)^(3/4)/x^5

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maxima [A]  time = 0.65, size = 29, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (4 \, x^{5} - x^{3} - 3 \, x\right )}}{21 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(4*x^5 - x^3 - 3*x)/((x + 1)^(1/4)*(x - 1)^(1/4)*x^(9/2))

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mupad [B]  time = 0.24, size = 35, normalized size = 1.30 \begin {gather*} \frac {8\,x^2\,{\left (x^4-x^2\right )}^{3/4}+6\,{\left (x^4-x^2\right )}^{3/4}}{21\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^4 - x^2)^(1/4)),x)

[Out]

(8*x^2*(x^4 - x^2)^(3/4) + 6*(x^4 - x^2)^(3/4))/(21*x^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**4*(x**2*(x - 1)*(x + 1))**(1/4)), x)

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