Integrand size = 18, antiderivative size = 108 \[ \int \frac {-1+x^6}{x^6 \sqrt [3]{x+x^3}} \, dx=\frac {3 \left (x+x^3\right )^{2/3} \left (5-6 x^2+9 x^4\right )}{80 x^6}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )-\frac {1}{2} \log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {1}{4} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \]
3/80*(x^3+x)^(2/3)*(9*x^4-6*x^2+5)/x^6+1/2*arctan(3^(1/2)*x/(x+2*(x^3+x)^( 1/3)))*3^(1/2)-1/2*ln(-x+(x^3+x)^(1/3))+1/4*ln(x^2+x*(x^3+x)^(1/3)+(x^3+x) ^(2/3))
Time = 1.72 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.54 \[ \int \frac {-1+x^6}{x^6 \sqrt [3]{x+x^3}} \, dx=\frac {15-3 x^2+9 x^4+27 x^6+40 \sqrt {3} x^{16/3} \sqrt [3]{1+x^2} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-40 x^{16/3} \sqrt [3]{1+x^2} \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )+20 x^{16/3} \sqrt [3]{1+x^2} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )}{80 x^5 \sqrt [3]{x+x^3}} \]
(15 - 3*x^2 + 9*x^4 + 27*x^6 + 40*Sqrt[3]*x^(16/3)*(1 + x^2)^(1/3)*ArcTan[ (Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(1 + x^2)^(1/3))] - 40*x^(16/3)*(1 + x^2)^( 1/3)*Log[-x^(2/3) + (1 + x^2)^(1/3)] + 20*x^(16/3)*(1 + x^2)^(1/3)*Log[x^( 4/3) + x^(2/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)])/(80*x^5*(x + x^3)^(1/3) )
Time = 0.30 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2449, 2009}
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.
\(\displaystyle \int \frac {x^6-1}{x^6 \sqrt [3]{x^3+x}} \, dx\) |
\(\Big \downarrow \) 2449 |
\(\displaystyle \int \left (\frac {1}{\sqrt [3]{x^3+x}}-\frac {1}{x^6 \sqrt [3]{x^3+x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3+x}}+\frac {3 \left (x^3+x\right )^{2/3}}{16 x^6}-\frac {9 \left (x^3+x\right )^{2/3}}{40 x^4}+\frac {27 \left (x^3+x\right )^{2/3}}{80 x^2}-\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \sqrt [3]{x^3+x}}\) |
(3*(x + x^3)^(2/3))/(16*x^6) - (9*(x + x^3)^(2/3))/(40*x^4) + (27*(x + x^3 )^(2/3))/(80*x^2) + (Sqrt[3]*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*x^(2/3 ))/(1 + x^2)^(1/3))/Sqrt[3]])/(2*(x + x^3)^(1/3)) - (3*x^(1/3)*(1 + x^2)^( 1/3)*Log[x^(2/3) - (1 + x^2)^(1/3)])/(4*(x + x^3)^(1/3))
3.16.77.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ [{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !Integer Q[p] && NeQ[n, j]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.98 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.39
method | result | size |
meijerg | \(\frac {3 \left (\frac {9}{5} x^{4}-\frac {6}{5} x^{2}+1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}}{16 x^{\frac {16}{3}}}+\frac {3 x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right )}{2}\) | \(42\) |
risch | \(\frac {-\frac {3}{80} x^{2}+\frac {3}{16}+\frac {9}{80} x^{4}+\frac {27}{80} x^{6}}{{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{5}}+\frac {3 x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right )}{2}\) | \(49\) |
pseudoelliptic | \(\frac {20 \ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right ) x^{6}-40 \sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) x^{6}-40 \ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right ) x^{6}+\left (27 x^{4}-18 x^{2}+15\right ) {\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}}{80 x^{6}}\) | \(116\) |
trager | \(\frac {3 \left (x^{3}+x \right )^{\frac {2}{3}} \left (9 x^{4}-6 x^{2}+5\right )}{80 x^{6}}-\frac {\ln \left (-144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+54 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x +114 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}-24 x \left (x^{3}+x \right )^{\frac {1}{3}}+144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+5 x^{2}+54 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+2\right )}{2}+3 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (-36 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-90 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x -48 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}+24 x \left (x^{3}+x \right )^{\frac {1}{3}}+36 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}-16 x^{2}+6 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-12\right )\) | \(301\) |
Time = 0.41 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04 \[ \int \frac {-1+x^6}{x^6 \sqrt [3]{x+x^3}} \, dx=\frac {40 \, \sqrt {3} x^{6} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) - 20 \, x^{6} \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) + 3 \, {\left (9 \, x^{4} - 6 \, x^{2} + 5\right )} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{80 \, x^{6}} \]
1/80*(40*sqrt(3)*x^6*arctan(-(196*sqrt(3)*(x^3 + x)^(1/3)*x - sqrt(3)*(539 *x^2 + 507) - 1274*sqrt(3)*(x^3 + x)^(2/3))/(2205*x^2 + 2197)) - 20*x^6*lo g(3*(x^3 + x)^(1/3)*x - 3*(x^3 + x)^(2/3) + 1) + 3*(9*x^4 - 6*x^2 + 5)*(x^ 3 + x)^(2/3))/x^6
\[ \int \frac {-1+x^6}{x^6 \sqrt [3]{x+x^3}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \sqrt [3]{x \left (x^{2} + 1\right )}}\, dx \]
\[ \int \frac {-1+x^6}{x^6 \sqrt [3]{x+x^3}} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{3} + x\right )}^{\frac {1}{3}} x^{6}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.76 \[ \int \frac {-1+x^6}{x^6 \sqrt [3]{x+x^3}} \, dx=\frac {3}{16} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {8}{3}} - \frac {3}{5} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{3}} - \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {3}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + \frac {1}{4} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
3/16*(1/x^2 + 1)^(8/3) - 3/5*(1/x^2 + 1)^(5/3) - 1/2*sqrt(3)*arctan(1/3*sq rt(3)*(2*(1/x^2 + 1)^(1/3) + 1)) + 3/4*(1/x^2 + 1)^(2/3) + 1/4*log((1/x^2 + 1)^(2/3) + (1/x^2 + 1)^(1/3) + 1) - 1/2*log(abs((1/x^2 + 1)^(1/3) - 1))
Time = 5.78 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int \frac {-1+x^6}{x^6 \sqrt [3]{x+x^3}} \, dx=\frac {3\,{\left (x^3+x\right )}^{2/3}\,\left (9\,x^4-6\,x^2+5\right )}{80\,x^6}+\frac {3\,x\,{\left (x^2+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ -x^2\right )}{2\,{\left (x^3+x\right )}^{1/3}} \]