3.903 \(\int \frac{1}{x^7 (1-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{4 x^2}{3 \sqrt{1-x^4}}-\frac{2}{3 \sqrt{1-x^4} x^2}-\frac{1}{6 \sqrt{1-x^4} x^6} \]

[Out]

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

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Rubi [A]  time = 0.0122783, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{4 x^2}{3 \sqrt{1-x^4}}-\frac{2}{3 \sqrt{1-x^4} x^2}-\frac{1}{6 \sqrt{1-x^4} x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(1 - x^4)^(3/2)),x]

[Out]

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

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

Mathematica [A]  time = 0.0054927, size = 30, normalized size = 0.55 \[ -\frac{-8 x^8+4 x^4+1}{6 x^6 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(1 - x^4)^(3/2)),x]

[Out]

-(1 + 4*x^4 - 8*x^8)/(6*x^6*Sqrt[1 - x^4])

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Maple [A]  time = 0.005, size = 38, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 8\,{x}^{8}-4\,{x}^{4}-1 \right ) }{6\,{x}^{6}} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-1+x)*(1+x)*(x^2+1)*(8*x^8-4*x^4-1)/x^6/(-x^4+1)^(3/2)

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Maxima [A]  time = 0.997549, size = 58, normalized size = 1.05 \begin{align*} \frac{x^{2}}{2 \, \sqrt{-x^{4} + 1}} - \frac{\sqrt{-x^{4} + 1}}{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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46177, size = 73, normalized size = 1.33 \begin{align*} -\frac{{\left (8 \, x^{8} - 4 \, x^{4} - 1\right )} \sqrt{-x^{4} + 1}}{6 \,{\left (x^{10} - x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/6*(8*x^8 - 4*x^4 - 1)*sqrt(-x^4 + 1)/(x^10 - x^6)

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Sympy [A]  time = 1.37735, size = 151, normalized size = 2.75 \begin{align*} \begin{cases} - \frac{8 x^{8} \sqrt{-1 + \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac{4 x^{4} \sqrt{-1 + \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac{\sqrt{-1 + \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} & \text{for}\: \frac{1}{\left |{x^{4}}\right |} > 1 \\- \frac{8 i x^{8} \sqrt{1 - \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac{4 i x^{4} \sqrt{1 - \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac{i \sqrt{1 - \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-8*x**8*sqrt(-1 + x**(-4))/(6*x**8 - 6*x**4) + 4*x**4*sqrt(-1 + x**(-4))/(6*x**8 - 6*x**4) + sqrt(-
1 + x**(-4))/(6*x**8 - 6*x**4), 1/Abs(x**4) > 1), (-8*I*x**8*sqrt(1 - 1/x**4)/(6*x**8 - 6*x**4) + 4*I*x**4*sqr
t(1 - 1/x**4)/(6*x**8 - 6*x**4) + I*sqrt(1 - 1/x**4)/(6*x**8 - 6*x**4), True))

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Giac [A]  time = 1.19588, size = 54, normalized size = 0.98 \begin{align*} -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} - \frac{1}{6} \,{\left (\frac{1}{x^{4}} - 1\right )}^{\frac{3}{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)^(3/2),x, algorithm="giac")

[Out]

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