3.5.79 \(\int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx\) [479]

Optimal. Leaf size=31 \[ \frac {\sqrt {-1+x^3}}{3 x^3}+\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^3}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 44, 65, 209} \begin {gather*} \frac {1}{3} \text {ArcTan}\left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1}}{3 x^3} \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 209

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 28, normalized size = 0.90 \begin {gather*} \frac {1}{3} \left (\frac {\sqrt {-1+x^3}}{x^3}+\tan ^{-1}\left (\sqrt {-1+x^3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-1 + x^3]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.39, size = 24, normalized size = 0.77

method	result	size
default	\(\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}+\frac {\sqrt {x^{3}-1}}{3 x^{3}}\)	\(24\)
risch	\(\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}+\frac {\sqrt {x^{3}-1}}{3 x^{3}}\)	\(24\)
elliptic	\(\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}+\frac {\sqrt {x^{3}-1}}{3 x^{3}}\)	\(24\)
trager	\(\frac {\sqrt {x^{3}-1}}{3 x^{3}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {x^{3} \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{3}-1}}{x^{3}}\right )}{6}\)	\(56\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right )}{8 x^{3}}+\frac {\sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{3}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{3}}\right )}{3 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{3}-1\right )}}\)	\(100\)

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

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 23, normalized size = 0.74 \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))

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 25, normalized size = 0.81 \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

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.91, size = 80, normalized size = 2.58 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} - \frac {i}{3 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{3 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {\sqrt {1 - \frac {1}{x^{3}}}}{3 x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*acosh(x**(-3/2))/3 - I/(3*x**(3/2)*sqrt(-1 + x**(-3))) + I/(3*x**(9/2)*sqrt(-1 + x**(-3))), 1/Abs
(x**3) > 1), (-asin(x**(-3/2))/3 + sqrt(1 - 1/x**3)/(3*x**(3/2)), True))

________________________________________________________________________________________

Giac [A]
time = 2.22, size = 23, normalized size = 0.74 \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))

________________________________________________________________________________________

Mupad [B]
time = 1.00, size = 177, normalized size = 5.71 \begin {gather*} \frac {\sqrt {x^3-1}}{3\,x^3}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\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+\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}}}\,\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+\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^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(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)*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)))/(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) +
x^3)^(1/2)

________________________________________________________________________________________