3.882 \(\int \frac{1}{x^7 \sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{\sqrt{1-x^4}}{3 x^2}-\frac{\sqrt{1-x^4}}{6 x^6} \]

[Out]

-Sqrt[1 - x^4]/(6*x^6) - Sqrt[1 - x^4]/(3*x^2)

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Rubi [A]  time = 0.0074301, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{\sqrt{1-x^4}}{3 x^2}-\frac{\sqrt{1-x^4}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(6*x^6) - Sqrt[1 - x^4]/(3*x^2)

Rule 271

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]

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^7 \sqrt{1-x^4}} \, dx &=-\frac{\sqrt{1-x^4}}{6 x^6}+\frac{2}{3} \int \frac{1}{x^3 \sqrt{1-x^4}} \, dx\\ &=-\frac{\sqrt{1-x^4}}{6 x^6}-\frac{\sqrt{1-x^4}}{3 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0042643, size = 25, normalized size = 0.68 \[ -\frac{\sqrt{1-x^4} \left (2 x^4+1\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[1 - x^4]),x]

[Out]

-(Sqrt[1 - x^4]*(1 + 2*x^4))/(6*x^6)

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Maple [A]  time = 0.004, size = 33, normalized size = 0.9 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 2\,{x}^{4}+1 \right ) }{6\,{x}^{6}}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(-x^4+1)^(1/2),x)

[Out]

1/6*(-1+x)*(1+x)*(x^2+1)*(2*x^4+1)/x^6/(-x^4+1)^(1/2)

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Maxima [A]  time = 1.02099, size = 39, normalized size = 1.05 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} - \frac{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^4 + 1)/x^2 - 1/6*(-x^4 + 1)^(3/2)/x^6

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Fricas [A]  time = 1.55222, size = 50, normalized size = 1.35 \begin{align*} -\frac{{\left (2 \, x^{4} + 1\right )} \sqrt{-x^{4} + 1}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(2*x^4 + 1)*sqrt(-x^4 + 1)/x^6

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Sympy [A]  time = 1.20448, size = 63, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{i \sqrt{x^{4} - 1}}{3 x^{2}} - \frac{i \sqrt{x^{4} - 1}}{6 x^{6}} & \text{for}\: \left |{x^{4}}\right | > 1 \\- \frac{\sqrt{1 - x^{4}}}{3 x^{2}} - \frac{\sqrt{1 - x^{4}}}{6 x^{6}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-I*sqrt(x**4 - 1)/(3*x**2) - I*sqrt(x**4 - 1)/(6*x**6), Abs(x**4) > 1), (-sqrt(1 - x**4)/(3*x**2) -
sqrt(1 - x**4)/(6*x**6), True))

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Giac [A]  time = 1.15847, size = 26, normalized size = 0.7 \begin{align*} -\frac{1}{6} \,{\left (\frac{1}{x^{4}} - 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

-1/6*(1/x^4 - 1)^(3/2) - 1/2*sqrt(1/x^4 - 1)