Integrand size = 40, antiderivative size = 102 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (-4+15 x^3+4 x^6\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )+\log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Output:
1/10*(x^6-1)^(2/3)*(4*x^6+15*x^3-4)/x^5-arctan(3^(1/2)*x/(x+2*(x^6-1)^(1/3 )))*3^(1/2)+ln(-x+(x^6-1)^(1/3))-1/2*ln(x^2+x*(x^6-1)^(1/3)+(x^6-1)^(2/3))
Time = 1.83 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (-4+15 x^3+4 x^6\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )+\log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \] Input:
Integrate[((-1 + x^6)^(2/3)*(1 + x^6)*(-2 + x^3 + 2*x^6))/(x^6*(-1 - x^3 + x^6)),x]
Output:
((-1 + x^6)^(2/3)*(-4 + 15*x^3 + 4*x^6))/(10*x^5) - Sqrt[3]*ArcTan[(Sqrt[3 ]*x)/(x + 2*(-1 + x^6)^(1/3))] + Log[-x + (-1 + x^6)^(1/3)] - Log[x^2 + x* (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/2
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.23 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.81, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {7279, 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.
| \(\displaystyle \int \frac {\left (x^6-1\right )^{2/3} \left (x^6+1\right ) \left (2 x^6+x^3-2\right )}{x^6 \left (x^6-x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 \left (x^6-1\right )^{2/3}}{x^6}+2 \left (x^6-1\right )^{2/3}+\frac {3 \left (x^6-1\right )^{2/3} \left (2 x^3-1\right )}{x^6-x^3-1}-\frac {3 \left (x^6-1\right )^{2/3}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \left (1-\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \left (1+\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {2 \left (x^6-1\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-\frac {2 \left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 \left (1-x^6\right )^{2/3} x^5}+\frac {3 \left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 \left (1-x^6\right )^{2/3} x^2}\) |
Input:
Int[((-1 + x^6)^(2/3)*(1 + x^6)*(-2 + x^3 + 2*x^6))/(x^6*(-1 - x^3 + x^6)) ,x]
Output:
(-3*(1 - Sqrt[5])*x*(-1 + x^6)^(2/3)*AppellF1[1/6, -2/3, 1, 7/6, x^6, (2*x ^6)/(3 - Sqrt[5])])/((3 - Sqrt[5])*(1 - x^6)^(2/3)) - (3*(1 + Sqrt[5])*x*( -1 + x^6)^(2/3)*AppellF1[1/6, 1, -2/3, 7/6, (2*x^6)/(3 + Sqrt[5]), x^6])/( (3 + Sqrt[5])*(1 - x^6)^(2/3)) - (3*x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/ 3, 1, 5/3, x^6, (2*x^6)/(3 - Sqrt[5])])/(2*(3 - Sqrt[5])*(1 - x^6)^(2/3)) - (3*x^4*(-1 + x^6)^(2/3)*AppellF1[2/3, -2/3, 1, 5/3, x^6, (2*x^6)/(3 + Sq rt[5])])/(2*(3 + Sqrt[5])*(1 - x^6)^(2/3)) - (2*(-1 + x^6)^(2/3)*Hypergeom etric2F1[-5/6, -2/3, 1/6, x^6])/(5*x^5*(1 - x^6)^(2/3)) + (3*(-1 + x^6)^(2 /3)*Hypergeometric2F1[-2/3, -1/3, 2/3, x^6])/(2*x^2*(1 - x^6)^(2/3)) + (2* x*(-1 + x^6)^(2/3)*Hypergeometric2F1[-2/3, 1/6, 7/6, x^6])/(1 - x^6)^(2/3)
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 16.50 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04
| method | result | size |
| pseudoelliptic | \(\frac {-5 \ln \left (\frac {x^{2}+x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+10 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{6}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{5}+10 \ln \left (\frac {-x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (4 x^{6}+15 x^{3}-4\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{10 x^{5}}\) | \(106\) |
| trager | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}} \left (4 x^{6}+15 x^{3}-4\right )}{10 x^{5}}+\ln \left (-\frac {-1789441197416077361152076587008 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{6}+78347540255370505701524760864 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{6}+5614428583490269128672312324 x^{6}+14091849429651609219072603122688 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+363650051031240685770824996928 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -1018262760247845306182226669792 \left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}+704415254713687571563215730080 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-6818882387672964795952100759 x \left (x^{6}-1\right )^{\frac {2}{3}}-3788021364908757143446093718 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+6327371895679509652948161508 x^{3}+1789441197416077361152076587008 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-78347540255370505701524760864 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-5614428583490269128672312324}{x^{6}-x^{3}-1}\right )+96 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (\frac {-6570485565136040671726226079744 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{6}-626067714458317065861690969216 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{6}-1723230319777309741777851531 x^{6}+51742573825446320289844030377984 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+363650051031240685770824996928 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +654612709216604620411401672864 \left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}+78347540255370505701524760864 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+10606903752581721939398194477 x \left (x^{6}-1\right )^{\frac {2}{3}}-3788021364908757143446093718 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-194166796594626449777786088 x^{3}+6570485565136040671726226079744 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}+626067714458317065861690969216 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+1723230319777309741777851531}{x^{6}-x^{3}-1}\right )\) | \(416\) |
| risch | \(\frac {4 x^{12}+15 x^{9}-8 x^{6}-15 x^{3}+4}{10 x^{5} \left (x^{6}-1\right )^{\frac {1}{3}}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}-x^{3}-1}\right )-3 \ln \left (\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}-x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-\ln \left (\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}-x^{3}-1}\right )\) | \(490\) |
Input:
int((x^6-1)^(2/3)*(x^6+1)*(2*x^6+x^3-2)/x^6/(x^6-x^3-1),x,method=_RETURNVE RBOSE)
Output:
1/10*(-5*ln((x^2+x*(x^6-1)^(1/3)+(x^6-1)^(2/3))/x^2)*x^5+10*3^(1/2)*arctan (1/3*(x+2*(x^6-1)^(1/3))*3^(1/2)/x)*x^5+10*ln((-x+(x^6-1)^(1/3))/x)*x^5+(4 *x^6+15*x^3-4)*(x^6-1)^(2/3))/x^5
Time = 8.41 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.45 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {473996388635948633452428917614298985996886224511260115036680453514888144148250 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 19325031480489228255674265966448835967818926087643600184123099965366515892788 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (771225779807741020855977802972631216428368740202755221603971931588718036144 \, x^{6} + 245889484278411189833195613987401279765924206559249102388797804808538611984375 \, x^{3} - 771225779807741020855977802972631216428368740202755221603971931588718036144\right )}}{3 \, {\left (15407513785538665202033017569552164636906896740149986002803824712402669144 \, x^{6} - 227351086091515241263579358841494627179170556108548407412281480599473216796875 \, x^{3} - 15407513785538665202033017569552164636906896740149986002803824712402669144\right )}}\right ) - 5 \, x^{5} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} - x^{3} - 1}\right ) - {\left (4 \, x^{6} + 15 \, x^{3} - 4\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)*(2*x^6+x^3-2)/x^6/(x^6-x^3-1),x, algorithm ="fricas")
Output:
-1/10*(10*sqrt(3)*x^5*arctan(1/3*(4739963886359486334524289176142989859968 86224511260115036680453514888144148250*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 19325 031480489228255674265966448835967818926087643600184123099965366515892788*s qrt(3)*(x^6 - 1)^(2/3)*x + sqrt(3)*(77122577980774102085597780297263121642 8368740202755221603971931588718036144*x^6 + 245889484278411189833195613987 401279765924206559249102388797804808538611984375*x^3 - 7712257798077410208 55977802972631216428368740202755221603971931588718036144))/(15407513785538 665202033017569552164636906896740149986002803824712402669144*x^6 - 2273510 86091515241263579358841494627179170556108548407412281480599473216796875*x^ 3 - 1540751378553866520203301756955216463690689674014998600280382471240266 9144)) - 5*x^5*log((x^6 - x^3 + 3*(x^6 - 1)^(1/3)*x^2 - 3*(x^6 - 1)^(2/3)* x - 1)/(x^6 - x^3 - 1)) - (4*x^6 + 15*x^3 - 4)*(x^6 - 1)^(2/3))/x^5
Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\text {Timed out} \] Input:
integrate((x**6-1)**(2/3)*(x**6+1)*(2*x**6+x**3-2)/x**6/(x**6-x**3-1),x)
Output:
Timed out
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} - 2\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{6}} \,d x } \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)*(2*x^6+x^3-2)/x^6/(x^6-x^3-1),x, algorithm ="maxima")
Output:
integrate((2*x^6 + x^3 - 2)*(x^6 + 1)*(x^6 - 1)^(2/3)/((x^6 - x^3 - 1)*x^6 ), x)
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} - 2\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{6}} \,d x } \] Input:
integrate((x^6-1)^(2/3)*(x^6+1)*(2*x^6+x^3-2)/x^6/(x^6-x^3-1),x, algorithm ="giac")
Output:
integrate((2*x^6 + x^3 - 2)*(x^6 + 1)*(x^6 - 1)^(2/3)/((x^6 - x^3 - 1)*x^6 ), x)
Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\int -\frac {{\left (x^6-1\right )}^{2/3}\,\left (x^6+1\right )\,\left (2\,x^6+x^3-2\right )}{x^6\,\left (-x^6+x^3+1\right )} \,d x \] Input:
int(-((x^6 - 1)^(2/3)*(x^6 + 1)*(x^3 + 2*x^6 - 2))/(x^6*(x^3 - x^6 + 1)),x )
Output:
int(-((x^6 - 1)^(2/3)*(x^6 + 1)*(x^3 + 2*x^6 - 2))/(x^6*(x^3 - x^6 + 1)), x)
\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\frac {2 \left (x^{6}-1\right )^{\frac {2}{3}} x^{6}-2 \left (x^{6}-1\right )^{\frac {2}{3}}+15 \left (\int \frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{9}-x^{6}-x^{3}}d x \right ) x^{5}+15 \left (\int \frac {\left (x^{6}-1\right )^{\frac {2}{3}} x^{3}}{x^{6}-x^{3}-1}d x \right ) x^{5}}{5 x^{5}} \] Input:
int((x^6-1)^(2/3)*(x^6+1)*(2*x^6+x^3-2)/x^6/(x^6-x^3-1),x)
Output:
(2*(x**6 - 1)**(2/3)*x**6 - 2*(x**6 - 1)**(2/3) + 15*int((x**6 - 1)**(2/3) /(x**9 - x**6 - x**3),x)*x**5 + 15*int(((x**6 - 1)**(2/3)*x**3)/(x**6 - x* *3 - 1),x)*x**5)/(5*x**5)