Integrand size = 26, antiderivative size = 243 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx=\frac {1}{6} (-1+3 x) \sqrt [3]{-x^2+x^3}-\frac {17 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )}{3 \sqrt {3}}+\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )-\frac {17}{9} \log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )+\frac {17}{18} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{2^{2/3}} \]
1/6*(-1+3*x)*(x^3-x^2)^(1/3)-17/9*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3-x^2)^ (1/3)))+3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^3-x^2)^(1/3)))*2^(1/3)-17/9 *ln(-x+(x^3-x^2)^(1/3))+ln(-2*x+2^(2/3)*(x^3-x^2)^(1/3))*2^(1/3)+17/18*ln( x^2+x*(x^3-x^2)^(1/3)+(x^3-x^2)^(2/3))-1/2*ln(2*x^2+2^(2/3)*x*(x^3-x^2)^(1 /3)+2^(1/3)*(x^3-x^2)^(2/3))*2^(1/3)
Time = 0.45 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx=\frac {(-1+x)^{2/3} x^{4/3} \left (-3 \sqrt [3]{-1+x} x^{2/3}+9 \sqrt [3]{-1+x} x^{5/3}-34 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+18 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+18 \sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )-34 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+17 \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )-9 \sqrt [3]{2} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )\right )}{18 \left ((-1+x) x^2\right )^{2/3}} \]
((-1 + x)^(2/3)*x^(4/3)*(-3*(-1 + x)^(1/3)*x^(2/3) + 9*(-1 + x)^(1/3)*x^(5 /3) - 34*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(2*(-1 + x)^(1/3) + x^(1/3))] + 18*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(2^(2/3)*(-1 + x)^(1/3) + x^(1 /3))] + 18*2^(1/3)*Log[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] - 34*Log[(-1 + x)^(1/3) - x^(1/3)] + 17*Log[(-1 + x)^(2/3) + (-1 + x)^(1/3)*x^(1/3) + x^( 2/3)] - 9*2^(1/3)*Log[2^(1/3)*(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3)*x^(1 /3) + 2*x^(2/3)]))/(18*((-1 + x)*x^2)^(2/3))
Time = 0.74 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.80, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2467, 25, 2003, 2035, 25, 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.
| \(\displaystyle \int \frac {\left (x^2+1\right ) \sqrt [3]{x^3-x^2}}{x^2-1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x^2} \int -\frac {\sqrt [3]{x-1} x^{2/3} \left (x^2+1\right )}{1-x^2}dx}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x^3-x^2} \int \frac {\sqrt [3]{x-1} x^{2/3} \left (x^2+1\right )}{1-x^2}dx}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle -\frac {\sqrt [3]{x^3-x^2} \int \frac {x^{2/3} \left (x^2+1\right )}{(-x-1) (x-1)^{2/3}}dx}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^3-x^2} \int -\frac {x^{4/3} \left (x^2+1\right )}{(x-1)^{2/3} (x+1)}d\sqrt [3]{x}}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \sqrt [3]{x^3-x^2} \int \frac {x^{4/3} \left (x^2+1\right )}{(x-1)^{2/3} (x+1)}d\sqrt [3]{x}}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x^3-x^2} \int \left (\frac {x^{7/3}}{(x-1)^{2/3}}-\frac {x^{4/3}}{(x-1)^{2/3}}+\frac {2 \sqrt [3]{x}}{(x-1)^{2/3}}-\frac {2 \sqrt [3]{x}}{(x-1)^{2/3} (x+1)}\right )d\sqrt [3]{x}}{\sqrt [3]{x-1} x^{2/3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^3-x^2} \left (\frac {17 \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\sqrt [3]{2} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \sqrt [3]{x-1} x^{5/3}+\frac {1}{18} \sqrt [3]{x-1} x^{2/3}+\frac {17}{18} \log \left (\sqrt [3]{x}-\sqrt [3]{x-1}\right )-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{2^{2/3}}+\frac {\log (x+1)}{3\ 2^{2/3}}\right )}{\sqrt [3]{x-1} x^{2/3}}\) |
(-3*(-x^2 + x^3)^(1/3)*(((-1 + x)^(1/3)*x^(2/3))/18 - ((-1 + x)^(1/3)*x^(5 /3))/6 + (17*ArcTan[(1 + (2*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - (2^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(1/3))/(-1 + x)^(1/3))/Sqrt[3]])/Sqrt [3] + (17*Log[-(-1 + x)^(1/3) + x^(1/3)])/18 - Log[-(-1 + x)^(1/3) + 2^(1/ 3)*x^(1/3)]/2^(2/3) + Log[1 + x]/(3*2^(2/3))))/((-1 + x)^(1/3)*x^(2/3))
3.27.86.3.1 Defintions of rubi rules used
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
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 = 8.77 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {x^{4} \left (-18 \,2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right )+34 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+18 \,2^{\frac {1}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )-9 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )+9 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +17 \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-3 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-34 \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )\right )}{18 {\left (\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}\right )}^{2} {\left (\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x \right )}^{2}}\) | \(248\) |
| trager | \(\text {Expression too large to display}\) | \(1772\) |
| risch | \(\text {Expression too large to display}\) | \(2018\) |
1/18*x^4*(-18*2^(1/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2^(2/3)*((-1+x)*x^2)^(1/ 3)+x)/x)+34*3^(1/2)*arctan(1/3*(2*((-1+x)*x^2)^(1/3)+x)*3^(1/2)/x)+18*2^(1 /3)*ln((-2^(1/3)*x+((-1+x)*x^2)^(1/3))/x)-9*2^(1/3)*ln((2^(2/3)*x^2+2^(1/3 )*((-1+x)*x^2)^(1/3)*x+((-1+x)*x^2)^(2/3))/x^2)+9*((-1+x)*x^2)^(1/3)*x+17* ln((((-1+x)*x^2)^(2/3)+((-1+x)*x^2)^(1/3)*x+x^2)/x^2)-3*((-1+x)*x^2)^(1/3) -34*ln((((-1+x)*x^2)^(1/3)-x)/x))/(((-1+x)*x^2)^(2/3)+((-1+x)*x^2)^(1/3)*x +x^2)^2/(((-1+x)*x^2)^(1/3)-x)^2
Time = 0.26 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.91 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx=-\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + \sqrt {3} x}{3 \, x}\right ) + \frac {17}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{6} \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} + 2^{\frac {1}{3}} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {17}{9} \, \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {17}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
-sqrt(3)*2^(1/3)*arctan(1/3*(sqrt(3)*2^(2/3)*(x^3 - x^2)^(1/3) + sqrt(3)*x )/x) + 17/9*sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - x^2)^(1/3))/x ) + 1/6*(x^3 - x^2)^(1/3)*(3*x - 1) + 2^(1/3)*log(-(2^(1/3)*x - (x^3 - x^2 )^(1/3))/x) - 1/2*2^(1/3)*log((2^(2/3)*x^2 + 2^(1/3)*(x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) - 17/9*log(-(x - (x^3 - x^2)^(1/3))/x) + 17/18*lo g((x^2 + (x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2)
\[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx=\int \frac {\sqrt [3]{x^{2} \left (x - 1\right )} \left (x^{2} + 1\right )}{\left (x - 1\right ) \left (x + 1\right )}\, dx \]
\[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{x^{2} - 1} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.72 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx=\frac {1}{6} \, {\left ({\left (-\frac {1}{x} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )} x^{2} - \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {17}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) + \frac {17}{18} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {17}{9} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
1/6*((-1/x + 1)^(4/3) + 2*(-1/x + 1)^(1/3))*x^2 - sqrt(3)*2^(1/3)*arctan(1 /6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x + 1)^(1/3))) + 17/9*sqrt(3)*arctan(1 /3*sqrt(3)*(2*(-1/x + 1)^(1/3) + 1)) - 1/2*2^(1/3)*log(2^(2/3) + 2^(1/3)*( -1/x + 1)^(1/3) + (-1/x + 1)^(2/3)) + 2^(1/3)*log(abs(-2^(1/3) + (-1/x + 1 )^(1/3))) + 17/18*log((-1/x + 1)^(2/3) + (-1/x + 1)^(1/3) + 1) - 17/9*log( abs((-1/x + 1)^(1/3) - 1))
Timed out. \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx=\int \frac {\left (x^2+1\right )\,{\left (x^3-x^2\right )}^{1/3}}{x^2-1} \,d x \]