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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 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])

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]]

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 = 0.02, 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

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 31, normalized size = 0.89 \[ -\frac {x^{3} \arctan \left (\sqrt {-x^{3} - 1}\right ) - \sqrt {-x^{3} - 1}}{3 \, x^{3}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.24, size = 27, normalized size = 0.77 \[ \frac {\sqrt {-x^{3} - 1}}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 28, normalized size = 0.80 \[ -\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{3}+\frac {\sqrt {-x^{3}-1}}{3 x^{3}} \]

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

________________________________________________________________________________________

maxima [A]  time = 2.98, size = 27, normalized size = 0.77 \[ \frac {\sqrt {-x^{3} - 1}}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \]

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))

________________________________________________________________________________________

mupad [B]  time = 0.98, size = 194, normalized size = 5.54 \[ \frac {\sqrt {-x^3-1}}{3\,x^3}+\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(- x^3 - 1)^(1/2)),x)

[Out]

(- x^3 - 1)^(1/2)/(3*x^3) + (((3^(1/2)*1i)/2 + 3/2)*(x^3 + 1)^(1/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/
2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/
2)*ellipticPi((3^(1/2)*1i)/2 + 3/2, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^
(1/2)*1i)/2 - 3/2)))/((- x^3 - 1)^(1/2)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/
2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))

________________________________________________________________________________________

sympy [C]  time = 2.15, size = 29, normalized size = 0.83 \[ - \frac {i \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {i \sqrt {1 + \frac {1}{x^{3}}}}{3 x^{\frac {3}{2}}} \]

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))

________________________________________________________________________________________