Integrand size = 22, antiderivative size = 214 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {3 \left (x^2+x^3\right )^{2/3} \left (3080-3300 x+3600 x^2+2495 x^3-2994 x^4+4491 x^5\right )}{52360 x^7}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^3}+\sqrt [3]{2} \left (x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Output:
Unintegrable
Time = 0.34 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {27720-1980 x+2700 x^2+54855 x^3-4491 x^4+13473 x^5+40419 x^6-26180\ 2^{2/3} \sqrt {3} x^{17/3} \sqrt [3]{1+x} \arctan \left (\frac {1+2 \sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}}{\sqrt {3}}\right )+26180\ 2^{2/3} x^{17/3} \sqrt [3]{1+x} \log \left (-1+\sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}\right )-13090\ 2^{2/3} x^{17/3} \sqrt [3]{1+x} \log \left (1+\sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}+2^{2/3} \left (\frac {x}{1+x}\right )^{2/3}\right )+52360 x^{17/3} \sqrt [3]{1+x} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right )+\log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]}{157080 x^5 \sqrt [3]{x^2 (1+x)}} \] Input:
Integrate[1/(x^6*(-1 + x^3)*(x^2 + x^3)^(1/3)),x]
Output:
(27720 - 1980*x + 2700*x^2 + 54855*x^3 - 4491*x^4 + 13473*x^5 + 40419*x^6 - 26180*2^(2/3)*Sqrt[3]*x^(17/3)*(1 + x)^(1/3)*ArcTan[(1 + 2*2^(1/3)*(x/(1 + x))^(1/3))/Sqrt[3]] + 26180*2^(2/3)*x^(17/3)*(1 + x)^(1/3)*Log[-1 + 2^( 1/3)*(x/(1 + x))^(1/3)] - 13090*2^(2/3)*x^(17/3)*(1 + x)^(1/3)*Log[1 + 2^( 1/3)*(x/(1 + x))^(1/3) + 2^(2/3)*(x/(1 + x))^(2/3)] + 52360*x^(17/3)*(1 + x)^(1/3)*RootSum[1 - #1^3 + #1^6 & , (-2*Log[(x/(1 + x))^(1/3) - #1] + Log [(x/(1 + x))^(1/3) - #1]*#1^3)/(-#1^2 + 2*#1^5) & ])/(157080*x^5*(x^2*(1 + x))^(1/3))
Result contains complex when optimal does not.
Time = 1.89 (sec) , antiderivative size = 991, normalized size of antiderivative = 4.63, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2467, 25, 2035, 7276, 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.
\(\displaystyle \int \frac {1}{x^6 \left (x^3-1\right ) \sqrt [3]{x^3+x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {x^{2/3} \sqrt [3]{x+1} \int -\frac {1}{x^{20/3} \sqrt [3]{x+1} \left (1-x^3\right )}dx}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {x^{2/3} \sqrt [3]{x+1} \int \frac {1}{x^{20/3} \sqrt [3]{x+1} \left (1-x^3\right )}dx}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \int \frac {1}{x^6 \sqrt [3]{x+1} \left (1-x^3\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \int \left (\frac {\sqrt [3]{x}+2}{9 \left (x^{2/3}+\sqrt [3]{x}+1\right ) \sqrt [3]{x+1}}-\frac {1}{9 \left (\sqrt [3]{x}-1\right ) \sqrt [3]{x+1}}+\frac {1}{x^3 \sqrt [3]{x+1}}+\frac {1}{x^6 \sqrt [3]{x+1}}+\frac {x+2}{3 \sqrt [3]{x+1} \left (x^2+x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x+1} \left (-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{x}+1\right )}{\sqrt [3]{x+1}}}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{x}+1\right )}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}+\frac {\arctan \left (\frac {2^{2/3} \sqrt [3]{x+1}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-\left (1-\sqrt [3]{x}\right )^2 \left (\sqrt [3]{x}+1\right )\right )}{36 \sqrt [3]{2}}-\frac {\log \left (\left (1-\sqrt [3]{x}\right )^2 \left (\sqrt [3]{x}+1\right )\right )}{108 \sqrt [3]{2}}+\frac {\log (1-x)}{54 \sqrt [3]{2}}+\frac {\log \left (2 x-i \sqrt {3}+1\right )}{18 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}+\frac {1}{18} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \log \left (2 x+i \sqrt {3}+1\right )-\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{x}+1\right )^2}{(x+1)^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{x}+1\right )}{\sqrt [3]{x+1}}+1\right )}{54 \sqrt [3]{2}}+\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{x}+1\right )}{\sqrt [3]{x+1}}+1\right )}{27 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{x+1}\right )}{18 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x}-\sqrt [3]{x+1}\right )}{9 \sqrt [3]{2}}-\frac {1}{6} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}-\sqrt [3]{x+1}\right )-\frac {\log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}}}+\frac {\log \left (\sqrt [3]{x}-2^{2/3} \sqrt [3]{x+1}+1\right )}{36 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{x}+2^{2/3} \sqrt [3]{x+1}-1\right )}{12 \sqrt [3]{2}}-\frac {4491 (x+1)^{2/3}}{52360 x^{2/3}}+\frac {1497 (x+1)^{2/3}}{26180 x^{5/3}}-\frac {499 (x+1)^{2/3}}{10472 x^{8/3}}-\frac {90 (x+1)^{2/3}}{1309 x^{11/3}}+\frac {15 (x+1)^{2/3}}{238 x^{14/3}}-\frac {(x+1)^{2/3}}{17 x^{17/3}}\right )}{\sqrt [3]{x^3+x^2}}\) |
Input:
Int[1/(x^6*(-1 + x^3)*(x^2 + x^3)^(1/3)),x]
Output:
(-3*x^(2/3)*(1 + x)^(1/3)*(-1/17*(1 + x)^(2/3)/x^(17/3) + (15*(1 + x)^(2/3 ))/(238*x^(14/3)) - (90*(1 + x)^(2/3))/(1309*x^(11/3)) - (499*(1 + x)^(2/3 ))/(10472*x^(8/3)) + (1497*(1 + x)^(2/3))/(26180*x^(5/3)) - (4491*(1 + x)^ (2/3))/(52360*x^(2/3)) - ArcTan[(1 - (2*2^(1/3)*(1 + x^(1/3)))/(1 + x)^(1/ 3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) + ArcTan[(1 + (2^(1/3)*(1 + x^(1/3)))/(1 + x)^(1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) + (2^(2/3)*ArcTan[(1 + (2*2^(1/3) *x^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) + ((-((I - Sqrt[3])/(I + Sq rt[3])))^(1/3)*ArcTan[(1 + (2*x^(1/3))/((-((I - Sqrt[3])/(I + Sqrt[3])))^( 1/3)*(1 + x)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]) + ArcTan[(1 + (2*(-((I - Sqrt[3 ])/(I + Sqrt[3])))^(1/3)*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]]/(3*Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)) + ArcTan[(1 + 2^(2/3)*(1 + x)^(1/3))/Sq rt[3]]/(9*2^(1/3)*Sqrt[3]) + Log[-((1 - x^(1/3))^2*(1 + x^(1/3)))]/(36*2^( 1/3)) - Log[(1 - x^(1/3))^2*(1 + x^(1/3))]/(108*2^(1/3)) + Log[1 - x]/(54* 2^(1/3)) + Log[1 - I*Sqrt[3] + 2*x]/(18*(-((I - Sqrt[3])/(I + Sqrt[3])))^( 1/3)) + ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*Log[1 + I*Sqrt[3] + 2*x])/ 18 - Log[1 + (2^(2/3)*(1 + x^(1/3))^2)/(1 + x)^(2/3) - (2^(1/3)*(1 + x^(1/ 3)))/(1 + x)^(1/3)]/(54*2^(1/3)) + Log[1 + (2^(1/3)*(1 + x^(1/3)))/(1 + x) ^(1/3)]/(27*2^(1/3)) + Log[2^(1/3) - (1 + x)^(1/3)]/(18*2^(1/3)) - Log[2^( 1/3)*x^(1/3) - (1 + x)^(1/3)]/(9*2^(1/3)) - ((-((I - Sqrt[3])/(I + Sqrt[3] )))^(1/3)*Log[x^(1/3)/(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3) - (1 + x)^...
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 = 38.68 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {-13090 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{7}+52360 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{7}+26180 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{7}+26180 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right ) x^{7}+40419 \left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}} \left (x^{4}-\frac {5}{3} x^{3}+\frac {20}{9} x^{2}-\frac {6380}{4491} x +\frac {3080}{4491}\right ) \left (1+x \right )}{157080 x^{7}}\) | \(189\) |
risch | \(\text {Expression too large to display}\) | \(1915\) |
trager | \(\text {Expression too large to display}\) | \(4412\) |
Input:
int(1/x^6/(x^3-1)/(x^3+x^2)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/157080*(-13090*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*(x^2*(1+x))^(1/3)*x+(x^2* (1+x))^(2/3))/x^2)*x^7+52360*sum(ln((-_R*x+(x^2*(1+x))^(1/3))/x)/_R,_R=Roo tOf(_Z^6-_Z^3+1))*x^7+26180*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*(x ^2*(1+x))^(1/3)+x)/x)*x^7+26180*2^(2/3)*ln((-2^(1/3)*x+(x^2*(1+x))^(1/3))/ x)*x^7+40419*(x^2*(1+x))^(2/3)*(x^4-5/3*x^3+20/9*x^2-6380/4491*x+3080/4491 )*(1+x))/x^7
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.10 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.55 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx =\text {Too large to display} \] Input:
integrate(1/x^6/(x^3-1)/(x^3+x^2)^(1/3),x, algorithm="fricas")
Output:
1/157080*(157080*2^(1/6)*sqrt(1/6)*x^7*arctan(2^(1/6)*sqrt(1/6)*(2^(1/3)*x + 2*(x^3 + x^2)^(1/3))/x) + 26180*2^(2/3)*x^7*log(-(2^(1/3)*x - (x^3 + x^ 2)^(1/3))/x) - 13090*2^(2/3)*x^7*log((2^(2/3)*x^2 + 2^(1/3)*(x^3 + x^2)^(1 /3)*x + (x^3 + x^2)^(2/3))/x^2) + 52360*x^7*(3/2*sqrt(-1/3) + 1/2)^(1/3)*l og(((3*sqrt(-1/3)*x - x)*(3/2*sqrt(-1/3) + 1/2)^(2/3) + 2*(x^3 + x^2)^(1/3 ))/x) + 52360*x^7*(-3/2*sqrt(-1/3) + 1/2)^(1/3)*log(-((3*sqrt(-1/3)*x + x) *(-3/2*sqrt(-1/3) + 1/2)^(2/3) - 2*(x^3 + x^2)^(1/3))/x) + 26180*(sqrt(-3) *x^7 - x^7)*(3/2*sqrt(-1/3) + 1/2)^(1/3)*log(-((3*sqrt(-1/3)*(sqrt(-3)*x + x) - sqrt(-3)*x - x)*(3/2*sqrt(-1/3) + 1/2)^(2/3) - 4*(x^3 + x^2)^(1/3))/ x) - 26180*(sqrt(-3)*x^7 + x^7)*(3/2*sqrt(-1/3) + 1/2)^(1/3)*log(((3*sqrt( -1/3)*(sqrt(-3)*x - x) - sqrt(-3)*x + x)*(3/2*sqrt(-1/3) + 1/2)^(2/3) + 4* (x^3 + x^2)^(1/3))/x) + 26180*(sqrt(-3)*x^7 - x^7)*(-3/2*sqrt(-1/3) + 1/2) ^(1/3)*log(((3*sqrt(-1/3)*(sqrt(-3)*x + x) + sqrt(-3)*x + x)*(-3/2*sqrt(-1 /3) + 1/2)^(2/3) + 4*(x^3 + x^2)^(1/3))/x) - 26180*(sqrt(-3)*x^7 + x^7)*(- 3/2*sqrt(-1/3) + 1/2)^(1/3)*log(-((3*sqrt(-1/3)*(sqrt(-3)*x - x) + sqrt(-3 )*x - x)*(-3/2*sqrt(-1/3) + 1/2)^(2/3) - 4*(x^3 + x^2)^(1/3))/x) + 9*(4491 *x^5 - 2994*x^4 + 2495*x^3 + 3600*x^2 - 3300*x + 3080)*(x^3 + x^2)^(2/3))/ x^7
Not integrable
Time = 0.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \] Input:
integrate(1/x**6/(x**3-1)/(x**3+x**2)**(1/3),x)
Output:
Integral(1/(x**6*(x**2*(x + 1))**(1/3)*(x - 1)*(x**2 + x + 1)), x)
Not integrable
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )} x^{6}} \,d x } \] Input:
integrate(1/x^6/(x^3-1)/(x^3+x^2)^(1/3),x, algorithm="maxima")
Output:
integrate(1/((x^3 + x^2)^(1/3)*(x^3 - 1)*x^6), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 45.20 (sec) , antiderivative size = 986, normalized size of antiderivative = 4.61 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^6/(x^3-1)/(x^3+x^2)^(1/3),x, algorithm="giac")
Output:
3/17*(1/x + 1)^(17/3) - 15/14*(1/x + 1)^(14/3) + 30/11*(1/x + 1)^(11/3) - 27/8*(1/x + 1)^(8/3) + 1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^( 1/3) + 2*(1/x + 1)^(1/3))) - 1/3*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos(4 /9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*p i)^4*sin(4/9*pi) + 10*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3 )*cos(4/9*pi)^2 - sqrt(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)*sin(4/9*pi))*arcta n(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) - 1/3*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^ 3*sin(2/9*pi)^2 + 5*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^4 - 5*cos(2/9*pi)^4*si n(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2 /9*pi)^2 - sqrt(3)*sin(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*( (-I*sqrt(3) - 1)*cos(2/9*pi) + 2*(1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*s in(2/9*pi))) + 1/3*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1 /9*pi)^2 + 5*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*p i) - 10*cos(1/9*pi)^2*sin(1/9*pi)^3 + sin(1/9*pi)^5 - sqrt(3)*cos(1/9*pi)^ 2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sq rt(3) - 1)*cos(1/9*pi) - 2*(1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9 *pi))) - 1/6*(5*sqrt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 10*sqrt(3)*cos(4/9*pi) ^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*pi)^ 3*sin(4/9*pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*s...
Not integrable
Time = 9.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\int \frac {1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^6-x^9\right )} \,d x \] Input:
int(1/(x^6*(x^2 + x^3)^(1/3)*(x^3 - 1)),x)
Output:
-int(1/((x^2 + x^3)^(1/3)*(x^6 - x^9)), x)
Not integrable
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^{\frac {29}{3}} \left (x +1\right )^{\frac {1}{3}}-x^{\frac {20}{3}} \left (x +1\right )^{\frac {1}{3}}}d x \] Input:
int(1/x^6/(x^3-1)/(x^3+x^2)^(1/3),x)
Output:
int(1/(x**(2/3)*(x + 1)**(1/3)*x**9 - x**(2/3)*(x + 1)**(1/3)*x**6),x)