3.497 \(\int \frac{1}{x^4 \sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=35 \[ \frac{\sqrt{-x^3-1}}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]

[Out]

Sqrt[-1 - x^3]/(3*x^3) - ArcTan[Sqrt[-1 - x^3]]/3

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Rubi [A]  time = 0.0147958, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 204} \[ \frac{\sqrt{-x^3-1}}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[-1 - x^3]),x]

[Out]

Sqrt[-1 - x^3]/(3*x^3) - ArcTan[Sqrt[-1 - x^3]]/3

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0225215, size = 40, normalized size = 1.14 \[ \frac{1}{3} \sqrt{-x^3-1} \left (\frac{1}{x^3}-\frac{\tanh ^{-1}\left (\sqrt{x^3+1}\right )}{\sqrt{x^3+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-1 - x^3]),x]

[Out]

(Sqrt[-1 - x^3]*(x^(-3) - ArcTanh[Sqrt[1 + x^3]]/Sqrt[1 + x^3]))/3

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Maple [A]  time = 0.021, size = 28, normalized size = 0.8 \begin{align*} -{\frac{1}{3}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }+{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{3}-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*arctan((-x^3-1)^(1/2))+1/3*(-x^3-1)^(1/2)/x^3

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Maxima [A]  time = 1.53544, size = 36, normalized size = 1.03 \begin{align*} \frac{\sqrt{-x^{3} - 1}}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(-x^3 - 1)/x^3 - 1/3*arctan(sqrt(-x^3 - 1))

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Fricas [A]  time = 1.62839, size = 76, normalized size = 2.17 \begin{align*} -\frac{x^{3} \arctan \left (\sqrt{-x^{3} - 1}\right ) - \sqrt{-x^{3} - 1}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(x^3*arctan(sqrt(-x^3 - 1)) - sqrt(-x^3 - 1))/x^3

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Sympy [C]  time = 2.45636, size = 29, normalized size = 0.83 \begin{align*} - \frac{i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{i \sqrt{1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*asinh(x**(-3/2))/3 + I*sqrt(1 + x**(-3))/(3*x**(3/2))

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Giac [A]  time = 1.09588, size = 36, normalized size = 1.03 \begin{align*} \frac{\sqrt{-x^{3} - 1}}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(-x^3 - 1)/x^3 - 1/3*arctan(sqrt(-x^3 - 1))