Integrand size = 13, antiderivative size = 92 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^4} \, dx=-\frac {\sqrt [3]{-1+x^3}}{3 x^3}-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {1}{18} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:
-1/3*(x^3-1)^(1/3)/x^3+1/9*arctan(-1/3*3^(1/2)+2/3*(x^3-1)^(1/3)*3^(1/2))* 3^(1/2)+1/9*ln(1+(x^3-1)^(1/3))-1/18*ln(1-(x^3-1)^(1/3)+(x^3-1)^(2/3))
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^4} \, dx=\frac {1}{18} \left (-\frac {6 \sqrt [3]{-1+x^3}}{x^3}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [3]{-1+x^3}\right )-\log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[(-1 + x^3)^(1/3)/x^4,x]
Output:
((-6*(-1 + x^3)^(1/3))/x^3 - 2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqr t[3]] + 2*Log[1 + (-1 + x^3)^(1/3)] - Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3 )^(2/3)])/18
Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 51, 70, 16, 1083, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt [3]{x^3-1}}{x^4} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{x^3-1}}{x^6}dx^3\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \int \frac {1}{x^3 \left (x^3-1\right )^{2/3}}dx^3-\frac {\sqrt [3]{x^3-1}}{x^3}\right )\) |
\(\Big \downarrow \) 70 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )\right )-\frac {\sqrt [3]{x^3-1}}{x^3}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\sqrt [3]{x^3-1}}{x^3}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (-3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3-1}-1\right )-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\sqrt [3]{x^3-1}}{x^3}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^3-1}-1}{\sqrt {3}}\right )-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\sqrt [3]{x^3-1}}{x^3}\right )\) |
Input:
Int[(-1 + x^3)^(1/3)/x^4,x]
Output:
(-((-1 + x^3)^(1/3)/x^3) + (Sqrt[3]*ArcTan[(-1 + 2*(-1 + x^3)^(1/3))/Sqrt[ 3]] - Log[x^3]/2 + (3*Log[1 + (-1 + x^3)^(1/3)])/2)/3)/3
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) , x] + (Simp[3/(2*b*q) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 /3)], x] + Simp[3/(2*b*q^2) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )-\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(75\) |
risch | \(-\frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{3 x^{3}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{9 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(79\) |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{3}+\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right ) x^{3}-2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right ) x^{3}+6 \left (x^{3}-1\right )^{\frac {1}{3}}}{18 \left (-1+\left (x^{3}-1\right )^{\frac {1}{3}}-\left (x^{3}-1\right )^{\frac {2}{3}}\right ) \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}\) | \(107\) |
trager | \(-\frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{3 x^{3}}+\frac {\ln \left (-\frac {13504 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x^{3}-21632 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x^{3}+44016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+8628 x^{3}+19749 \left (x^{3}-1\right )^{\frac {2}{3}}+157992 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-108032 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+100472 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-7190}{x^{3}}\right )}{9}+\frac {8 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) \ln \left (\frac {105536 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2} x^{3}+103848 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) x^{3}+44016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+10066 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-113976 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-844288 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )^{2}-19749 \left (x^{3}-1\right )^{\frac {1}{3}}-263528 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )-18694}{x^{3}}\right )}{9}\) | \(294\) |
Input:
int((x^3-1)^(1/3)/x^4,x,method=_RETURNVERBOSE)
Output:
1/9/GAMMA(2/3)*signum(x^3-1)^(1/3)/(-signum(x^3-1))^(1/3)*(-1/3*GAMMA(2/3) *x^3*hypergeom([1,1,5/3],[2,3],x^3)-(1/6*Pi*3^(1/2)-3/2*ln(3)-1+3*ln(x)+I* Pi)*GAMMA(2/3)-3*GAMMA(2/3)/x^3)
Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^4} \, dx=\frac {2 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - x^{3} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{3} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 6 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, x^{3}} \] Input:
integrate((x^3-1)^(1/3)/x^4,x, algorithm="fricas")
Output:
1/18*(2*sqrt(3)*x^3*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - x^ 3*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 2*x^3*log((x^3 - 1)^(1/3) + 1) - 6*(x^3 - 1)^(1/3))/x^3
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^4} \, dx=- \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \] Input:
integrate((x**3-1)**(1/3)/x**4,x)
Output:
-gamma(2/3)*hyper((-1/3, 2/3), (5/3,), exp_polar(2*I*pi)/x**3)/(3*x**2*gam ma(5/3))
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^4} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{18} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:
integrate((x^3-1)^(1/3)/x^4,x, algorithm="maxima")
Output:
1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) - 1/3*(x^3 - 1)^(1 /3)/x^3 - 1/18*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 1/9*log((x^3 - 1)^(1/3) + 1)
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^4} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{18} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \] Input:
integrate((x^3-1)^(1/3)/x^4,x, algorithm="giac")
Output:
1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) - 1/3*(x^3 - 1)^(1 /3)/x^3 - 1/18*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 1/9*log(abs((x ^3 - 1)^(1/3) + 1))
Time = 8.95 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^4} \, dx=\frac {\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{9}+\frac {1}{9}\right )}{9}+\ln \left ({\left (x^3-1\right )}^{1/3}-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\frac {{\left (x^3-1\right )}^{1/3}}{3\,x^3}-\ln \left (\frac {1}{2}-{\left (x^3-1\right )}^{1/3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right ) \] Input:
int((x^3 - 1)^(1/3)/x^4,x)
Output:
log((x^3 - 1)^(1/3)/9 + 1/9)/9 + log((3^(1/2)*1i)/2 + (x^3 - 1)^(1/3) - 1/ 2)*((3^(1/2)*1i)/18 - 1/18) - (x^3 - 1)^(1/3)/(3*x^3) - log((3^(1/2)*1i)/2 - (x^3 - 1)^(1/3) + 1/2)*((3^(1/2)*1i)/18 + 1/18)
\[ \int \frac {\sqrt [3]{-1+x^3}}{x^4} \, dx=\frac {-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{x^{4}-x}d x \right ) x^{3}}{3 x^{3}} \] Input:
int((x^3-1)^(1/3)/x^4,x)
Output:
( - (x**3 - 1)**(1/3) + int((x**3 - 1)**(1/3)/(x**4 - x),x)*x**3)/(3*x**3)