Integrand size = 26, antiderivative size = 167 \[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {\left (-x^2+x^4\right )^{2/3}}{x \left (-1+x^2\right )}-\frac {2 \arctan \left (\frac {\sqrt [6]{3} x}{\sqrt [3]{-x^2+x^4}}\right )}{3 \sqrt [6]{3}}-\frac {\arctan \left (\frac {3^{5/6} x \sqrt [3]{-x^2+x^4}}{-3 x^2+3^{2/3} \left (-x^2+x^4\right )^{2/3}}\right )}{3 \sqrt [6]{3}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [3]{3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{3^{2/3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{3^{2/3}} \]
-(x^4-x^2)^(2/3)/x/(x^2-1)-2/9*arctan(3^(1/6)*x/(x^4-x^2)^(1/3))*3^(5/6)-1 /9*arctan(3^(5/6)*x*(x^4-x^2)^(1/3)/(-3*x^2+3^(2/3)*(x^4-x^2)^(2/3)))*3^(5 /6)-1/3*arctanh((1/3*3^(2/3)*x^2+1/3*(x^4-x^2)^(2/3)*3^(1/3))/x/(x^4-x^2)^ (1/3))*3^(1/3)
Time = 0.79 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.14 \[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {x^{2/3} \left (9 \sqrt [3]{x}+2\ 3^{5/6} \sqrt [3]{-1+x^2} \arctan \left (\frac {\sqrt [6]{3} \sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+3^{5/6} \sqrt [3]{-1+x^2} \arctan \left (\frac {3^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{-3 x^{2/3}+3^{2/3} \left (-1+x^2\right )^{2/3}}\right )+3 \sqrt [3]{3} \sqrt [3]{-1+x^2} \text {arctanh}\left (\frac {3 \sqrt [3]{3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 x^{2/3}+3^{2/3} \left (-1+x^2\right )^{2/3}}\right )\right )}{9 \sqrt [3]{x^2 \left (-1+x^2\right )}} \]
-1/9*(x^(2/3)*(9*x^(1/3) + 2*3^(5/6)*(-1 + x^2)^(1/3)*ArcTan[(3^(1/6)*x^(1 /3))/(-1 + x^2)^(1/3)] + 3^(5/6)*(-1 + x^2)^(1/3)*ArcTan[(3^(5/6)*x^(1/3)* (-1 + x^2)^(1/3))/(-3*x^(2/3) + 3^(2/3)*(-1 + x^2)^(2/3))] + 3*3^(1/3)*(-1 + x^2)^(1/3)*ArcTanh[(3*3^(1/3)*x^(1/3)*(-1 + x^2)^(1/3))/(3*x^(2/3) + 3^ (2/3)*(-1 + x^2)^(2/3))]))/(x^2*(-1 + x^2))^(1/3)
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.
| \(\displaystyle \int \frac {x^6+1}{\sqrt [3]{x^4-x^2} \left (x^6-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^2-1} \int -\frac {x^6+1}{x^{2/3} \sqrt [3]{x^2-1} \left (1-x^6\right )}dx}{\sqrt [3]{x^4-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^2-1} \int \frac {x^6+1}{x^{2/3} \sqrt [3]{x^2-1} \left (1-x^6\right )}dx}{\sqrt [3]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2-1} \int \frac {x^6+1}{\sqrt [3]{x^2-1} \left (1-x^6\right )}d\sqrt [3]{x}}{\sqrt [3]{x^4-x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2-1} \int \left (\frac {2}{\sqrt [3]{x^2-1} \left (1-x^6\right )}-\frac {1}{\sqrt [3]{x^2-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2-1} \left (-\frac {1}{9} \int \frac {1}{\left (-\sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left (1-\sqrt [3]{x}\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {1}{9} \int \frac {1}{\left (\sqrt [9]{-1} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left (\sqrt [9]{-1} \sqrt [3]{x}+1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {1}{9} \int \frac {1}{\left (-(-1)^{2/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left (1-(-1)^{2/9} \sqrt [3]{x}\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {1}{9} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left (\sqrt [3]{-1} \sqrt [3]{x}+1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {1}{9} \int \frac {1}{\left (-(-1)^{4/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left (1-(-1)^{4/9} \sqrt [3]{x}\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {1}{9} \int \frac {1}{\left ((-1)^{5/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left ((-1)^{5/9} \sqrt [3]{x}+1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {1}{9} \int \frac {1}{\left (-(-1)^{2/3} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left (1-(-1)^{2/3} \sqrt [3]{x}\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {1}{9} \int \frac {1}{\left ((-1)^{7/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left ((-1)^{7/9} \sqrt [3]{x}+1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {1}{9} \int \frac {1}{\left (-(-1)^{8/9} \sqrt [3]{x}-1\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}+\frac {1}{9} \int \frac {1}{\left (1-(-1)^{8/9} \sqrt [3]{x}\right ) \sqrt [3]{x^2-1}}d\sqrt [3]{x}-\frac {\sqrt [3]{x} \sqrt [3]{1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},x^2\right )}{\sqrt [3]{x^2-1}}\right )}{\sqrt [3]{x^4-x^2}}\) |
3.23.42.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 61.49 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.23
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (\arctan \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}-3 x \sqrt {3}}{3 x}\right )+2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}}{3 x}\right )+\arctan \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}+3 x \sqrt {3}}{3 x}\right )\right ) 3^{\frac {5}{6}}+\frac {3 \,3^{\frac {1}{3}} \left (\ln \left (\frac {-3^{\frac {2}{3}} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {3^{\frac {2}{3}} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{2}\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}}-9 x}{9 \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\) | \(205\) |
| risch | \(\text {Expression too large to display}\) | \(758\) |
| trager | \(\text {Expression too large to display}\) | \(2605\) |
1/9*(((arctan(1/3*(2*(x^4-x^2)^(1/3)*3^(5/6)-3*x*3^(1/2))/x)+2*arctan(1/3* (x^4-x^2)^(1/3)/x*3^(5/6))+arctan(1/3*(2*(x^4-x^2)^(1/3)*3^(5/6)+3*x*3^(1/ 2))/x))*3^(5/6)+3/2*3^(1/3)*(ln((-3^(2/3)*(x^4-x^2)^(1/3)*x+3^(1/3)*x^2+(x ^4-x^2)^(2/3))/x^2)-ln((3^(2/3)*(x^4-x^2)^(1/3)*x+3^(1/3)*x^2+(x^4-x^2)^(2 /3))/x^2)))*(x^4-x^2)^(1/3)-9*x)/(x^4-x^2)^(1/3)
Result contains complex when optimal does not.
Time = 4.02 (sec) , antiderivative size = 1561, normalized size of antiderivative = 9.35 \[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\text {Too large to display} \]
1/36*(3^(5/6)*(-1)^(1/6)*(x^3 + sqrt(-3)*(x^3 - x) - x)*log(-(3^(5/6)*(-1) ^(1/6)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 + sqrt(-3)*(419*x^5 + 288 0*x^4 - 2095*x^3 - 2880*x^2 + 419*x) + 419*x) + 6*3^(1/3)*(-1)^(2/3)*(240* x^5 - 419*x^4 - 1200*x^3 + 419*x^2 + sqrt(-3)*(240*x^5 - 419*x^4 - 1200*x^ 3 + 419*x^2 + 240*x) + 240*x) - 12*(x^4 - x^2)^(2/3)*(419*x^2 - sqrt(3)*(- 480*I*x^2 + 419*I*x + 480*I) + 1440*x - 419) + 6*(x^4 - x^2)^(1/3)*(3^(2/3 )*(-1)^(1/3)*(480*x^3 - 419*x^2 - sqrt(-3)*(480*x^3 - 419*x^2 - 480*x) - 4 80*x) - 3^(1/6)*(-1)^(5/6)*(419*x^3 + 1440*x^2 - sqrt(-3)*(419*x^3 + 1440* x^2 - 419*x) - 419*x)))/(x^5 + x^3 + x)) - 3^(5/6)*(-1)^(1/6)*(x^3 + sqrt( -3)*(x^3 - x) - x)*log((3^(5/6)*(-1)^(1/6)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 + sqrt(-3)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 + 419*x) + 419*x) - 6*3^(1/3)*(-1)^(2/3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 + s qrt(-3)*(240*x^5 - 419*x^4 - 1200*x^3 + 419*x^2 + 240*x) + 240*x) + 12*(x^ 4 - x^2)^(2/3)*(419*x^2 - sqrt(3)*(480*I*x^2 - 419*I*x - 480*I) + 1440*x - 419) - 6*(x^4 - x^2)^(1/3)*(3^(2/3)*(-1)^(1/3)*(480*x^3 - 419*x^2 - sqrt( -3)*(480*x^3 - 419*x^2 - 480*x) - 480*x) + 3^(1/6)*(-1)^(5/6)*(419*x^3 + 1 440*x^2 - sqrt(-3)*(419*x^3 + 1440*x^2 - 419*x) - 419*x)))/(x^5 + x^3 + x) ) + 3^(5/6)*(-1)^(1/6)*(x^3 - sqrt(-3)*(x^3 - x) - x)*log(-(3^(5/6)*(-1)^( 1/6)*(419*x^5 + 2880*x^4 - 2095*x^3 - 2880*x^2 - sqrt(-3)*(419*x^5 + 2880* x^4 - 2095*x^3 - 2880*x^2 + 419*x) + 419*x) + 6*3^(1/3)*(-1)^(2/3)*(240...
\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Integral((x**2 + 1)*(x**4 - x**2 + 1)/((x**2*(x - 1)*(x + 1))**(1/3)*(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)), x)
\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int { \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1+x^6}{\sqrt [3]{-x^2+x^4} \left (-1+x^6\right )} \, dx=\int \frac {x^6+1}{\left (x^6-1\right )\,{\left (x^4-x^2\right )}^{1/3}} \,d x \]