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

Optimal. Leaf size=45 \[ \frac{5}{6} \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{1}{2 x^3 \sqrt{1-x^4}} \]

[Out]

1/(2*x^3*Sqrt[1 - x^4]) - (5*Sqrt[1 - x^4])/(6*x^3) + (5*EllipticF[ArcSin[x], -1])/6

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Rubi [A]  time = 0.0097015, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {290, 325, 221} \[ -\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{1}{2 x^3 \sqrt{1-x^4}}+\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(1 - x^4)^(3/2)),x]

[Out]

1/(2*x^3*Sqrt[1 - x^4]) - (5*Sqrt[1 - x^4])/(6*x^3) + (5*EllipticF[ArcSin[x], -1])/6

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

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] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{2 x^3 \sqrt{1-x^4}}+\frac{5}{2} \int \frac{1}{x^4 \sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^3 \sqrt{1-x^4}}-\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{5}{6} \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=\frac{1}{2 x^3 \sqrt{1-x^4}}-\frac{5 \sqrt{1-x^4}}{6 x^3}+\frac{5}{6} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0026915, size = 20, normalized size = 0.44 \[ -\frac{\, _2F_1\left (-\frac{3}{4},\frac{3}{2};\frac{1}{4};x^4\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(1 - x^4)^(3/2)),x]

[Out]

-Hypergeometric2F1[-3/4, 3/2, 1/4, x^4]/(3*x^3)

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Maple [A]  time = 0.011, size = 59, normalized size = 1.3 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{6}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/(-x^4+1)^(1/2)-1/3*(-x^4+1)^(1/2)/x^3+5/6*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*EllipticF(x,I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{4} + 1}}{x^{12} - 2 \, x^{8} + x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-x^4 + 1)/(x^12 - 2*x^8 + x^4), x)

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Sympy [A]  time = 1.04721, size = 34, normalized size = 0.76 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-3/4)*hyper((-3/4, 3/2), (1/4,), x**4*exp_polar(2*I*pi))/(4*x**3*gamma(1/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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