3.5.97 \(\int \frac {1}{x^4 \sqrt {-1-x^3}} \, dx\) [497]

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

[Out]

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

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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 = {272, 44, 65, 210} \begin {gather*} \frac {\sqrt {-x^3-1}}{3 x^3}-\frac {1}{3} \text {ArcTan}\left (\sqrt {-x^3-1}\right ) \end {gather*}

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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 210

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

Rule 272

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} \text {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} \text {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} \text {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*}

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Mathematica [A]
time = 0.03, size = 35, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1-x^3}}{3 x^3}-\frac {1}{3} \tan ^{-1}\left (\sqrt {-1-x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-1 - x^3]),x]

[Out]

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

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Maple [A]
time = 0.33, size = 28, normalized size = 0.80

method	result	size
default	\(-\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{3}+\frac {\sqrt {-x^{3}-1}}{3 x^{3}}\)	\(28\)
elliptic	\(-\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{3}+\frac {\sqrt {-x^{3}-1}}{3 x^{3}}\)	\(28\)
risch	\(-\frac {x^{3}+1}{3 x^{3} \sqrt {-x^{3}-1}}-\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{3}\)	\(33\)
trager	\(\frac {\sqrt {-x^{3}-1}}{3 x^{3}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {x^{3} \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {-x^{3}-1}}{x^{3}}\right )}{6}\)	\(60\)
meijerg	\(-\frac {i \left (\frac {\sqrt {\pi }\, \left (4 x^{3}+8\right )}{8 x^{3}}-\frac {\sqrt {\pi }\, \sqrt {x^{3}+1}}{x^{3}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{x^{3}}\right )}{3 \sqrt {\pi }}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.49, size = 27, normalized size = 0.77 \begin {gather*} \frac {\sqrt {-x^{3} - 1}}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \end {gather*}

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 = 0.36, size = 31, normalized size = 0.89 \begin {gather*} -\frac {x^{3} \arctan \left (\sqrt {-x^{3} - 1}\right ) - \sqrt {-x^{3} - 1}}{3 \, x^{3}} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.91, size = 29, normalized size = 0.83 \begin {gather*} - \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 {gather*}

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 = 2.21, size = 27, normalized size = 0.77 \begin {gather*} \frac {\sqrt {-x^{3} - 1}}{3 \, x^{3}} - \frac {1}{3} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.98, size = 194, normalized size = 5.54 \begin {gather*} \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 )}} \end {gather*}

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))

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