Integrand size = 20, antiderivative size = 108 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+2 x^6\right )}{x^6} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-3-3 x^3+10 x^6\right )}{15 x^5}+\frac {4 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {4}{9} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {2}{9} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
1/15*(x^3+1)^(2/3)*(10*x^6-3*x^3-3)/x^5+4/9*arctan(3^(1/2)*x/(x+2*(x^3+1)^ (1/3)))*3^(1/2)-4/9*ln(-x+(x^3+1)^(1/3))+2/9*ln(x^2+x*(x^3+1)^(1/3)+(x^3+1 )^(2/3))
Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+2 x^6\right )}{x^6} \, dx=\frac {1}{45} \left (\frac {3 \left (1+x^3\right )^{2/3} \left (-3-3 x^3+10 x^6\right )}{x^5}+20 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )-20 \log \left (-x+\sqrt [3]{1+x^3}\right )+10 \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]
((3*(1 + x^3)^(2/3)*(-3 - 3*x^3 + 10*x^6))/x^5 + 20*Sqrt[3]*ArcTan[(Sqrt[3 ]*x)/(x + 2*(1 + x^3)^(1/3))] - 20*Log[-x + (1 + x^3)^(1/3)] + 10*Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/45
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1811, 953, 809, 769}
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.
\(\displaystyle \int \frac {\left (x^3+1\right )^{2/3} \left (2 x^6+1\right )}{x^6} \, dx\) |
\(\Big \downarrow \) 1811 |
\(\displaystyle \frac {1}{3} \int \frac {\left (x^3+1\right )^{2/3} \left (4 x^3+3\right )}{x^6}dx+\frac {2 \left (x^3+1\right )^{5/3}}{3 x^2}\) |
\(\Big \downarrow \) 953 |
\(\displaystyle \frac {1}{3} \left (4 \int \frac {\left (x^3+1\right )^{2/3}}{x^3}dx-\frac {3 \left (x^3+1\right )^{5/3}}{5 x^5}\right )+\frac {2 \left (x^3+1\right )^{5/3}}{3 x^2}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle \frac {1}{3} \left (4 \left (\int \frac {1}{\sqrt [3]{x^3+1}}dx-\frac {\left (x^3+1\right )^{2/3}}{2 x^2}\right )-\frac {3 \left (x^3+1\right )^{5/3}}{5 x^5}\right )+\frac {2 \left (x^3+1\right )^{5/3}}{3 x^2}\) |
\(\Big \downarrow \) 769 |
\(\displaystyle \frac {1}{3} \left (4 \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{2/3}}{2 x^2}\right )-\frac {3 \left (x^3+1\right )^{5/3}}{5 x^5}\right )+\frac {2 \left (x^3+1\right )^{5/3}}{3 x^2}\) |
(2*(1 + x^3)^(5/3))/(3*x^2) + ((-3*(1 + x^3)^(5/3))/(5*x^5) + 4*(-1/2*(1 + x^3)^(2/3)/x^2 + ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Lo g[-x + (1 + x^3)^(1/3)]/2))/3
3.16.80.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[d/e^n Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && G tQ[m + n, -1]))
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ (n_))^(q_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 2*n*p - n + 1)*((d + e*x^n)^( q + 1)/(e*f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1))), x] + Simp[1/(e*(m + 2* n*p + n*q + 1)) Int[(f*x)^m*(d + e*x^n)^q*ExpandToSum[e*(m + 2*n*p + n*q + 1)*((a + c*x^(2*n))^p - c^p*x^(2*n*p)) - d*c^p*(m + 2*n*p - n + 1)*x^(2*n *p - n), x], x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && I GtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] && !IntegerQ[q] && NeQ[m + 2* n*p + n*q + 1, 0]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.55 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.26
method | result | size |
meijerg | \(2 x \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )-\frac {\left (x^{3}+1\right )^{\frac {5}{3}}}{5 x^{5}}\) | \(28\) |
risch | \(\frac {10 x^{9}+7 x^{6}-6 x^{3}-3}{15 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {4 x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )}{3}\) | \(45\) |
pseudoelliptic | \(\frac {-10 \ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+20 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (-30 x^{6}+9 x^{3}+9\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{45 x^{5} \left (\left (x^{3}+1\right )^{\frac {2}{3}}+x \left (x +\left (x^{3}+1\right )^{\frac {1}{3}}\right )\right ) \left (x -\left (x^{3}+1\right )^{\frac {1}{3}}\right )}\) | \(140\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (10 x^{6}-3 x^{3}-3\right )}{15 x^{5}}-\frac {4 \ln \left (317 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +2358 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-2120 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-1803 x \left (x^{3}+1\right )^{\frac {2}{3}}-555 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+2675 x^{3}-317 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1070\right )}{9}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-535 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +1803 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-1823 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2358 x \left (x^{3}+1\right )^{\frac {2}{3}}+555 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+1268 x^{3}+535 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-1922 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+951\right )}{9}\) | \(287\) |
Time = 0.54 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+2 x^6\right )}{x^6} \, dx=\frac {20 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) - 10 \, x^{5} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (10 \, x^{6} - 3 \, x^{3} - 3\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{45 \, x^{5}} \]
1/45*(20*sqrt(3)*x^5*arctan(-(25382*sqrt(3)*(x^3 + 1)^(1/3)*x^2 - 13720*sq rt(3)*(x^3 + 1)^(2/3)*x + sqrt(3)*(5831*x^3 + 7200))/(58653*x^3 + 8000)) - 10*x^5*log(3*(x^3 + 1)^(1/3)*x^2 - 3*(x^3 + 1)^(2/3)*x + 1) + 3*(10*x^6 - 3*x^3 - 3)*(x^3 + 1)^(2/3))/x^5
Result contains complex when optimal does not.
Time = 1.46 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+2 x^6\right )}{x^6} \, dx=\frac {2 x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} \]
2*x*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), x**3*exp_polar(I*pi))/(3*gamma(4 /3)) + (1 + x**(-3))**(2/3)*gamma(-5/3)/(3*gamma(-2/3)) + (1 + x**(-3))**( 2/3)*gamma(-5/3)/(3*x**3*gamma(-2/3))
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+2 x^6\right )}{x^6} \, dx=-\frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {2 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} + 1}{x^{3}} - 1\right )}} - \frac {{\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {2}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {4}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
-4/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3)/x + 1)) + 2/3*(x^3 + 1) ^(2/3)/(x^2*((x^3 + 1)/x^3 - 1)) - 1/5*(x^3 + 1)^(5/3)/x^5 + 2/9*log((x^3 + 1)^(1/3)/x + (x^3 + 1)^(2/3)/x^2 + 1) - 4/9*log((x^3 + 1)^(1/3)/x - 1)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+2 x^6\right )}{x^6} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{6}} \,d x } \]
Time = 5.65 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.35 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1+2 x^6\right )}{x^6} \, dx=2\,x\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ -x^3\right )-\frac {{\left (x^3+1\right )}^{2/3}+x^3\,{\left (x^3+1\right )}^{2/3}}{5\,x^5} \]