Integrand size = 21, antiderivative size = 208 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=-\frac {3 \left (-x^2+x^3\right )^{2/3}}{4 (-1+x) x}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{4 \sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{4 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{8 \sqrt [3]{2}}+\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=-\frac {x^{2/3} \left (12 \sqrt [3]{x}+2\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2\ 2^{2/3} \sqrt [3]{-1+x} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )+2^{2/3} \sqrt [3]{-1+x} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )-4 \sqrt [3]{-1+x} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{16 \sqrt [3]{(-1+x) x^2}} \]
-1/16*(x^(2/3)*(12*x^(1/3) + 2*2^(2/3)*Sqrt[3]*(-1 + x)^(1/3)*ArcTan[(Sqrt [3]*x^(1/3))/(2^(2/3)*(-1 + x)^(1/3) + x^(1/3))] - 2*2^(2/3)*(-1 + x)^(1/3 )*Log[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] + 2^(2/3)*(-1 + x)^(1/3)*Log[2^( 1/3)*(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3)*x^(1/3) + 2*x^(2/3)] - 4*(-1 + x)^(1/3)*RootSum[2 - 2*#1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/ 3) - x^(1/3)*#1])/#1 & ]))/((-1 + x)*x^2)^(1/3)
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.59, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, 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.
\(\displaystyle \int \frac {1}{\sqrt [3]{x^3-x^2} \left (x^4-1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \int -\frac {1}{\sqrt [3]{x-1} x^{2/3} \left (1-x^4\right )}dx}{\sqrt [3]{x^3-x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} x^{2/3} \left (1-x^4\right )}dx}{\sqrt [3]{x^3-x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {3 \sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} \left (1-x^4\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {3 \sqrt [3]{x-1} x^{2/3} \int \left (\frac {1}{2 \left (1-x^2\right ) \sqrt [3]{x-1}}+\frac {1}{2 \left (x^2+1\right ) \sqrt [3]{x-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3-x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 \sqrt [3]{x-1} x^{2/3} \left (\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{1-i} \sqrt [3]{x}}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{4 \sqrt [3]{1-i} \sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{1+i} \sqrt [3]{x}}{\sqrt [3]{x-1}}}{\sqrt {3}}\right )}{4 \sqrt [3]{1+i} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {\sqrt [3]{x}}{4 \sqrt [3]{x-1}}-\frac {\log \left (-\sqrt [3]{x-1}+\sqrt [3]{1-i} \sqrt [3]{x}\right )}{8 \sqrt [3]{1-i}}-\frac {\log \left (-\sqrt [3]{x-1}+\sqrt [3]{1+i} \sqrt [3]{x}\right )}{8 \sqrt [3]{1+i}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{8 \sqrt [3]{2}}+\frac {\log (-x-1)}{24 \sqrt [3]{2}}+\frac {\log (-x+i)}{24 \sqrt [3]{1+i}}+\frac {\log (x+i)}{24 \sqrt [3]{1-i}}\right )}{\sqrt [3]{x^3-x^2}}\) |
(-3*(-1 + x)^(1/3)*x^(2/3)*(x^(1/3)/(4*(-1 + x)^(1/3)) + ArcTan[(1 + (2*(1 - I)^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]]/(4*(1 - I)^(1/3)*Sqrt[3]) + ArcTan[(1 + (2*(1 + I)^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]]/(4*(1 + I)^ (1/3)*Sqrt[3]) + ArcTan[(1 + (2*2^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]]/ (4*2^(1/3)*Sqrt[3]) - Log[-(-1 + x)^(1/3) + (1 - I)^(1/3)*x^(1/3)]/(8*(1 - I)^(1/3)) - Log[-(-1 + x)^(1/3) + (1 + I)^(1/3)*x^(1/3)]/(8*(1 + I)^(1/3) ) - Log[-(-1 + x)^(1/3) + 2^(1/3)*x^(1/3)]/(8*2^(1/3)) + Log[-1 - x]/(24*2 ^(1/3)) + Log[I - x]/(24*(1 + I)^(1/3)) + Log[I + x]/(24*(1 - I)^(1/3))))/ (-x^2 + x^3)^(1/3)
3.25.100.3.1 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 = 0.00 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(-\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+12 x}{16 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\) | \(189\) |
trager | \(\text {Expression too large to display}\) | \(18221\) |
risch | \(\text {Expression too large to display}\) | \(19795\) |
-1/16*(-2*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*((-1+x)*x^2)^(1/3)+x )/x)*((-1+x)*x^2)^(1/3)-2*2^(2/3)*ln((-2^(1/3)*x+((-1+x)*x^2)^(1/3))/x)*(( -1+x)*x^2)^(1/3)+2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*((-1+x)*x^2)^(1/3)*x+((-1 +x)*x^2)^(2/3))/x^2)*((-1+x)*x^2)^(1/3)-4*sum(ln((-_R*x+((-1+x)*x^2)^(1/3) )/x)/_R,_R=RootOf(_Z^6-2*_Z^3+2))*((-1+x)*x^2)^(1/3)+12*x)/((-1+x)*x^2)^(1 /3)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.40 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} {\left (x^{2} - x\right )} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} x + 2 \, \sqrt {2} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) - 2^{\frac {2}{3}} \left (-i + 1\right )^{\frac {1}{3}} {\left (x^{2} - \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} \log \left (\frac {2^{\frac {1}{3}} \left (-i + 1\right )^{\frac {2}{3}} {\left (\left (i + 1\right ) \, \sqrt {-3} x + \left (i + 1\right ) \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2^{\frac {2}{3}} \left (i + 1\right )^{\frac {1}{3}} {\left (x^{2} - \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} \log \left (\frac {2^{\frac {1}{3}} \left (i + 1\right )^{\frac {2}{3}} {\left (-\left (i - 1\right ) \, \sqrt {-3} x - \left (i - 1\right ) \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2^{\frac {2}{3}} \left (i + 1\right )^{\frac {1}{3}} {\left (x^{2} + \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} \log \left (\frac {2^{\frac {1}{3}} \left (i + 1\right )^{\frac {2}{3}} {\left (\left (i - 1\right ) \, \sqrt {-3} x - \left (i - 1\right ) \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2^{\frac {2}{3}} \left (-i + 1\right )^{\frac {1}{3}} {\left (x^{2} + \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} \log \left (\frac {2^{\frac {1}{3}} \left (-i + 1\right )^{\frac {2}{3}} {\left (-\left (i + 1\right ) \, \sqrt {-3} x + \left (i + 1\right ) \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 2 \cdot 2^{\frac {2}{3}} \left (i + 1\right )^{\frac {1}{3}} {\left (x^{2} - x\right )} \log \left (\frac {\left (i - 1\right ) \cdot 2^{\frac {1}{3}} \left (i + 1\right )^{\frac {2}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 2 \cdot 2^{\frac {2}{3}} \left (-i + 1\right )^{\frac {1}{3}} {\left (x^{2} - x\right )} \log \left (\frac {-\left (i + 1\right ) \cdot 2^{\frac {1}{3}} \left (-i + 1\right )^{\frac {2}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 2 \cdot 2^{\frac {2}{3}} {\left (x^{2} - x\right )} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2^{\frac {2}{3}} {\left (x^{2} - x\right )} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 12 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{16 \, {\left (x^{2} - x\right )}} \]
1/16*(2*sqrt(3)*2^(2/3)*(x^2 - x)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*x + 2*sqrt(2)*(x^3 - x^2)^(1/3))/x) - 2^(2/3)*(-I + 1)^(1/3)*(x^2 - sqrt(-3)*( x^2 - x) - x)*log((2^(1/3)*(-I + 1)^(2/3)*((I + 1)*sqrt(-3)*x + (I + 1)*x) + 4*(x^3 - x^2)^(1/3))/x) - 2^(2/3)*(I + 1)^(1/3)*(x^2 - sqrt(-3)*(x^2 - x) - x)*log((2^(1/3)*(I + 1)^(2/3)*(-(I - 1)*sqrt(-3)*x - (I - 1)*x) + 4*( x^3 - x^2)^(1/3))/x) - 2^(2/3)*(I + 1)^(1/3)*(x^2 + sqrt(-3)*(x^2 - x) - x )*log((2^(1/3)*(I + 1)^(2/3)*((I - 1)*sqrt(-3)*x - (I - 1)*x) + 4*(x^3 - x ^2)^(1/3))/x) - 2^(2/3)*(-I + 1)^(1/3)*(x^2 + sqrt(-3)*(x^2 - x) - x)*log( (2^(1/3)*(-I + 1)^(2/3)*(-(I + 1)*sqrt(-3)*x + (I + 1)*x) + 4*(x^3 - x^2)^ (1/3))/x) + 2*2^(2/3)*(I + 1)^(1/3)*(x^2 - x)*log(((I - 1)*2^(1/3)*(I + 1) ^(2/3)*x + 2*(x^3 - x^2)^(1/3))/x) + 2*2^(2/3)*(-I + 1)^(1/3)*(x^2 - x)*lo g((-(I + 1)*2^(1/3)*(-I + 1)^(2/3)*x + 2*(x^3 - x^2)^(1/3))/x) + 2*2^(2/3) *(x^2 - x)*log(-(2^(1/3)*x - (x^3 - x^2)^(1/3))/x) - 2^(2/3)*(x^2 - x)*log ((2^(2/3)*x^2 + 2^(1/3)*(x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) - 12 *(x^3 - x^2)^(2/3))/(x^2 - x)
Not integrable
Time = 0.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {1}{{\left (x^{4} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {1}{{\left (x^{4} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\int \frac {1}{\left (x^4-1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]