Integrand size = 18, antiderivative size = 109 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-81-99 x^3+30 x^6+50 x^9\right )}{972 x^{12}}-\frac {25 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {25}{729} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {25 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{1458} \]
1/972*(x^3-1)^(1/3)*(50*x^9+30*x^6-99*x^3-81)/x^12+25/729*arctan(-1/3*3^(1 /2)+2/3*(x^3-1)^(1/3)*3^(1/2))*3^(1/2)+25/729*ln(1+(x^3-1)^(1/3))-25/1458* ln(1-(x^3-1)^(1/3)+(x^3-1)^(2/3))
Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {\frac {3 \sqrt [3]{-1+x^3} \left (-81-99 x^3+30 x^6+50 x^9\right )}{x^{12}}-100 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+100 \log \left (1+\sqrt [3]{-1+x^3}\right )-50 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{2916} \]
((3*(-1 + x^3)^(1/3)*(-81 - 99*x^3 + 30*x^6 + 50*x^9))/x^12 - 100*Sqrt[3]* ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]] + 100*Log[1 + (-1 + x^3)^(1/3)] - 50*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/2916
Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {948, 87, 51, 52, 52, 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.
| \(\displaystyle \int \frac {\sqrt [3]{x^3-1} \left (x^3+1\right )}{x^{13}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {\sqrt [3]{x^3-1} \left (x^3+1\right )}{x^{15}}dx^3\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{3} \left (\frac {5}{3} \int \frac {\sqrt [3]{x^3-1}}{x^{12}}dx^3+\frac {\left (x^3-1\right )^{4/3}}{4 x^{12}}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{3} \left (\frac {5}{3} \left (\frac {1}{9} \int \frac {1}{x^9 \left (x^3-1\right )^{2/3}}dx^3-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^{12}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{3} \left (\frac {5}{3} \left (\frac {1}{9} \left (\frac {5}{6} \int \frac {1}{x^6 \left (x^3-1\right )^{2/3}}dx^3+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^{12}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{3} \left (\frac {5}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \int \frac {1}{x^3 \left (x^3-1\right )^{2/3}}dx^3+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^{12}}\right )\) |
| \(\Big \downarrow \) 70 |
| \(\displaystyle \frac {1}{3} \left (\frac {5}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^{12}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (\frac {5}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^{12}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{3} \left (\frac {5}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (-3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3-1}-1\right )-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^{12}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {5}{3} \left (\frac {1}{9} \left (\frac {5}{6} \left (\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^3-1}-1}{\sqrt {3}}\right )-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )+\frac {\sqrt [3]{x^3-1}}{x^3}\right )+\frac {\sqrt [3]{x^3-1}}{2 x^6}\right )-\frac {\sqrt [3]{x^3-1}}{3 x^9}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^{12}}\right )\) |
((-1 + x^3)^(4/3)/(4*x^12) + (5*(-1/3*(-1 + x^3)^(1/3)/x^9 + ((-1 + x^3)^( 1/3)/(2*x^6) + (5*((-1 + x^3)^(1/3)/x^3 + (2*(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)) /6)/9))/3)/3
3.16.87.3.1 Defintions of rubi rules used
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[((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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && 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 = 1.65 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(\frac {50 x^{12}-20 x^{9}-129 x^{6}+18 x^{3}+81}{972 x^{12} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {25 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{729 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(101\) |
| pseudoelliptic | \(\frac {\left (150 x^{9}+90 x^{6}-297 x^{3}-243\right ) \left (x^{3}-1\right )^{\frac {1}{3}}-50 x^{12} \left (-2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )+\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )-2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )\right )}{2916 {\left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}^{4} {\left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{4}}\) | \(119\) |
| meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {22 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {14}{3}\right ], \left [2, 6\right ], x^{3}\right )}{243}+\frac {10 \left (\frac {47}{120}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{81}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x^{12}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{9}}-\frac {\Gamma \left (\frac {2}{3}\right )}{6 x^{6}}-\frac {5 \Gamma \left (\frac {2}{3}\right )}{27 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(185\) |
| trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (50 x^{9}+30 x^{6}-99 x^{3}-81\right )}{972 x^{12}}-\frac {25 \ln \left (-\frac {24125636608 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}+40366080 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}-22536192 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-193005092864 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}+22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-69656576 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right )}{729}-\frac {102400 \ln \left (-\frac {24125636608 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}+40366080 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}-22536192 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-193005092864 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}+22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-69656576 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right ) \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )}{729}+\frac {102400 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \ln \left (-\frac {24125636608 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}-28585984 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}+22536192 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}-193005092864 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}-22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-24584192 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}+8245}{x^{3}}\right )}{729}\) | \(458\) |
1/972*(50*x^12-20*x^9-129*x^6+18*x^3+81)/x^12/(x^3-1)^(2/3)+25/729/GAMMA(2 /3)/signum(x^3-1)^(2/3)*(-signum(x^3-1))^(2/3)*(2/3*GAMMA(2/3)*x^3*hyperge om([1,1,5/3],[2,2],x^3)+(1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*GAMMA(2/3) )
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {100 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 50 \, x^{12} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 100 \, x^{12} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (50 \, x^{9} + 30 \, x^{6} - 99 \, x^{3} - 81\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{2916 \, x^{12}} \]
1/2916*(100*sqrt(3)*x^12*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - 50*x^12*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 100*x^12*log((x^3 - 1)^(1/3) + 1) + 3*(50*x^9 + 30*x^6 - 99*x^3 - 81)*(x^3 - 1)^(1/3))/x^12
Timed out. \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (87) = 174\).
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {25}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {20 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + 72 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 93 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 40 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{972 \, {\left ({\left (x^{3} - 1\right )}^{4} + 4 \, {\left (x^{3} - 1\right )}^{3} + 4 \, x^{3} + 6 \, {\left (x^{3} - 1\right )}^{2} - 3\right )}} + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} - \frac {25}{1458} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {25}{729} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
25/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/972*(20*(x^ 3 - 1)^(10/3) + 72*(x^3 - 1)^(7/3) + 93*(x^3 - 1)^(4/3) - 40*(x^3 - 1)^(1/ 3))/((x^3 - 1)^4 + 4*(x^3 - 1)^3 + 4*x^3 + 6*(x^3 - 1)^2 - 3) + 1/162*(5*( x^3 - 1)^(7/3) + 13*(x^3 - 1)^(4/3) - 10*(x^3 - 1)^(1/3))/((x^3 - 1)^3 + 3 *x^3 + 3*(x^3 - 1)^2 - 2) - 25/1458*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 25/729*log((x^3 - 1)^(1/3) + 1)
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {25}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {50 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + 180 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 111 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 100 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{972 \, x^{12}} - \frac {25}{1458} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {25}{729} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
25/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/972*(50*(x^ 3 - 1)^(10/3) + 180*(x^3 - 1)^(7/3) + 111*(x^3 - 1)^(4/3) - 100*(x^3 - 1)^ (1/3))/x^12 - 25/1458*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 25/729* log(abs((x^3 - 1)^(1/3) + 1))
Time = 6.23 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.43 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}+\frac {10\,\ln \left (\frac {100\,{\left (x^3-1\right )}^{1/3}}{59049}+\frac {100}{59049}\right )}{729}+\frac {\frac {31\,{\left (x^3-1\right )}^{4/3}}{324}-\frac {10\,{\left (x^3-1\right )}^{1/3}}{243}+\frac {2\,{\left (x^3-1\right )}^{7/3}}{27}+\frac {5\,{\left (x^3-1\right )}^{10/3}}{243}}{6\,{\left (x^3-1\right )}^2+4\,{\left (x^3-1\right )}^3+{\left (x^3-1\right )}^4+4\,x^3-3}+\frac {\frac {13\,{\left (x^3-1\right )}^{4/3}}{162}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {5\,{\left (x^3-1\right )}^{7/3}}{162}}{3\,{\left (x^3-1\right )}^2+{\left (x^3-1\right )}^3+3\,x^3-2}-\ln \left (\frac {5}{54}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )+\ln \left (\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}-\frac {5}{54}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (-\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )-\ln \left (\frac {5}{81}-\frac {10\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {\sqrt {3}\,5{}\mathrm {i}}{81}\right )\,\left (\frac {5}{729}+\frac {\sqrt {3}\,5{}\mathrm {i}}{729}\right )+\ln \left (\frac {10\,{\left (x^3-1\right )}^{1/3}}{81}-\frac {5}{81}+\frac {\sqrt {3}\,5{}\mathrm {i}}{81}\right )\,\left (-\frac {5}{729}+\frac {\sqrt {3}\,5{}\mathrm {i}}{729}\right ) \]
(5*log((25*(x^3 - 1)^(1/3))/6561 + 25/6561))/243 + (10*log((100*(x^3 - 1)^ (1/3))/59049 + 100/59049))/729 + ((31*(x^3 - 1)^(4/3))/324 - (10*(x^3 - 1) ^(1/3))/243 + (2*(x^3 - 1)^(7/3))/27 + (5*(x^3 - 1)^(10/3))/243)/(6*(x^3 - 1)^2 + 4*(x^3 - 1)^3 + (x^3 - 1)^4 + 4*x^3 - 3) + ((13*(x^3 - 1)^(4/3))/1 62 - (5*(x^3 - 1)^(1/3))/81 + (5*(x^3 - 1)^(7/3))/162)/(3*(x^3 - 1)^2 + (x ^3 - 1)^3 + 3*x^3 - 2) - log((3^(1/2)*5i)/54 - (5*(x^3 - 1)^(1/3))/27 + 5/ 54)*((3^(1/2)*5i)/486 + 5/486) + log((3^(1/2)*5i)/54 + (5*(x^3 - 1)^(1/3)) /27 - 5/54)*((3^(1/2)*5i)/486 - 5/486) - log((3^(1/2)*5i)/81 - (10*(x^3 - 1)^(1/3))/81 + 5/81)*((3^(1/2)*5i)/729 + 5/729) + log((3^(1/2)*5i)/81 + (1 0*(x^3 - 1)^(1/3))/81 - 5/81)*((3^(1/2)*5i)/729 - 5/729)