Integrand size = 27, antiderivative size = 234 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=-\frac {3 \left (-1+5 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x+x^3}}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x+x^3}\right )}{2^{2/3}}+\frac {1}{4} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x+x^3}+\sqrt [3]{2} \left (-x+x^3\right )^{2/3}\right )}{2\ 2^{2/3}} \]
-3/4*(5*x^2-1)*(x^3-x)^(1/3)/x^3-1/2*arctan(3^(1/2)*x/(x+2*(x^3-x)^(1/3))) *3^(1/2)-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^3-x)^(1/3)))*2^(1/3)-1 /2*ln(-x+(x^3-x)^(1/3))-1/2*ln(-2*x+2^(2/3)*(x^3-x)^(1/3))*2^(1/3)+1/4*ln( x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))+1/4*ln(2*x^2+2^(2/3)*x*(x^3-x)^(1/3)+2^ (1/3)*(x^3-x)^(2/3))*2^(1/3)
Time = 1.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=\frac {\sqrt [3]{x \left (-1+x^2\right )} \left (3 \sqrt [3]{-1+x^2}-15 x^2 \sqrt [3]{-1+x^2}-2 \sqrt {3} x^{8/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )-2 \sqrt [3]{2} \sqrt {3} x^{8/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}}\right )-2 x^{8/3} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-2 \sqrt [3]{2} x^{8/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )+x^{8/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )+\sqrt [3]{2} x^{8/3} \log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{-1+x^2}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )\right )}{4 x^3 \sqrt [3]{-1+x^2}} \]
((x*(-1 + x^2))^(1/3)*(3*(-1 + x^2)^(1/3) - 15*x^2*(-1 + x^2)^(1/3) - 2*Sq rt[3]*x^(8/3)*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(-1 + x^2)^(1/3))] - 2 *2^(1/3)*Sqrt[3]*x^(8/3)*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2^(2/3)*(-1 + x^2)^(1/3))] - 2*x^(8/3)*Log[-x^(2/3) + (-1 + x^2)^(1/3)] - 2*2^(1/3)*x^( 8/3)*Log[-2*x^(2/3) + 2^(2/3)*(-1 + x^2)^(1/3)] + x^(8/3)*Log[x^(4/3) + x^ (2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)] + 2^(1/3)*x^(8/3)*Log[2*x^(4/3) + 2^(2/3)*x^(2/3)*(-1 + x^2)^(1/3) + 2^(1/3)*(-1 + x^2)^(2/3)]))/(4*x^3*( -1 + x^2)^(1/3))
Time = 1.98 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, 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.
| \(\displaystyle \int \frac {\sqrt [3]{x^3-x} \left (x^4-2\right )}{x^4 \left (x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x^3-x} \int -\frac {\sqrt [3]{x^2-1} \left (2-x^4\right )}{x^{11/3} \left (x^2+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x^3-x} \int \frac {\sqrt [3]{x^2-1} \left (2-x^4\right )}{x^{11/3} \left (x^2+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^3-x} \int \frac {\sqrt [3]{x^2-1} \left (2-x^4\right )}{x^3 \left (x^2+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \left (x^{2/3}+1\right )}{3 \left (x^{4/3}-x^{2/3}+1\right )}-\frac {2 \sqrt [3]{x^2-1}}{x}+\frac {2 \sqrt [3]{x^2-1}}{x^3}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1}}{3 \left (x^{2/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{x^3-x} \left (\frac {\arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\sqrt [3]{2} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log \left (x^2+1\right )}{6\ 2^{2/3}}+\frac {\left (x^2-1\right )^{4/3}}{4 x^{8/3}}+\frac {\sqrt [3]{x^2-1}}{x^{2/3}}+\frac {1}{4} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )+\frac {\log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{x^2-1}\right )}{2\ 2^{2/3}}\right )}{\sqrt [3]{x} \sqrt [3]{x^2-1}}\) |
(-3*(-x + x^3)^(1/3)*((-1 + x^2)^(1/3)/x^(2/3) + (-1 + x^2)^(4/3)/(4*x^(8/ 3)) + ArcTan[(1 + (2*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) + Arc Tan[(1 + (2*2^(1/3)*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]]/(3*2^(2/3)*Sqrt[3] ) + (2^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/( 3*Sqrt[3]) - Log[1 + x^2]/(6*2^(2/3)) + Log[x^(2/3) - (-1 + x^2)^(1/3)]/4 + Log[2^(1/3)*x^(2/3) - (-1 + x^2)^(1/3)]/(2*2^(2/3))))/(x^(1/3)*(-1 + x^2 )^(1/3))
3.27.39.3.1 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 = 2.70 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(\frac {\left (-15 x^{2}+3\right ) \left (x^{3}-x \right )^{\frac {1}{3}}+x^{3} \left (\left (2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{3}-x \right )^{\frac {1}{3}} x +\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right ) 2^{\frac {1}{3}}+2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right )}{4 x^{3}}\) | \(198\) |
1/4*((-15*x^2+3)*(x^3-x)^(1/3)+x^3*((2*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^ 3-x)^(1/3)))*3^(1/2)+ln((2^(2/3)*x^2+2^(1/3)*(x^3-x)^(1/3)*x+(x^3-x)^(2/3) )/x^2)-2*ln((-2^(1/3)*x+(x^3-x)^(1/3))/x))*2^(1/3)+2*arctan(1/3*3^(1/2)/x* (x+2*(x^3-x)^(1/3)))*3^(1/2)+ln((x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))/x^2)-2 *ln((-x+(x^3-x)^(1/3))/x)))/x^3
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (184) = 368\).
Time = 4.92 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=-\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{5} - 16 \, x^{3} + x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + 12 \, \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} + 4 \, x^{2} - 1\right )} {\left (x^{3} - x\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (71 \, x^{6} - 111 \, x^{4} + 33 \, x^{2} - 1\right )}\right )}}{6 \, {\left (109 \, x^{6} - 105 \, x^{4} + 3 \, x^{2} + 1\right )}}\right ) + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - x\right )}^{\frac {2}{3}} {\left (5 \, x^{2} - 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{4} - 16 \, x^{2} + 1\right )} + 24 \, {\left (2 \, x^{3} - x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{x^{4} + 2 \, x^{2} + 1}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{3} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{2} + 1\right )} + 6 \, {\left (x^{3} - x\right )}^{\frac {2}{3}}}{x^{2} + 1}\right ) + 12 \, \sqrt {3} x^{3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + 6 \, x^{3} \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + 18 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{2} - 1\right )}}{24 \, x^{3}} \]
-1/24*(4*4^(1/6)*sqrt(3)*(-1)^(1/3)*x^3*arctan(1/6*4^(1/6)*sqrt(3)*(6*4^(2 /3)*(-1)^(2/3)*(19*x^5 - 16*x^3 + x)*(x^3 - x)^(1/3) + 12*(-1)^(1/3)*(5*x^ 4 + 4*x^2 - 1)*(x^3 - x)^(2/3) + 4^(1/3)*(71*x^6 - 111*x^4 + 33*x^2 - 1))/ (109*x^6 - 105*x^4 + 3*x^2 + 1)) + 4^(2/3)*(-1)^(1/3)*x^3*log((6*4^(1/3)*( -1)^(2/3)*(x^3 - x)^(2/3)*(5*x^2 - 1) - 4^(2/3)*(-1)^(1/3)*(19*x^4 - 16*x^ 2 + 1) + 24*(2*x^3 - x)*(x^3 - x)^(1/3))/(x^4 + 2*x^2 + 1)) - 2*4^(2/3)*(- 1)^(1/3)*x^3*log(-(3*4^(2/3)*(-1)^(1/3)*(x^3 - x)^(1/3)*x + 4^(1/3)*(-1)^( 2/3)*(x^2 + 1) + 6*(x^3 - x)^(2/3))/(x^2 + 1)) + 12*sqrt(3)*x^3*arctan(-(4 4032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 27072047 93) - 10524305234*sqrt(3)*(x^3 - x)^(2/3))/(81835897185*x^2 - 1102302937)) + 6*x^3*log(-3*(x^3 - x)^(1/3)*x + 3*(x^3 - x)^(2/3) + 1) + 18*(x^3 - x)^ (1/3)*(5*x^2 - 1))/x^3
\[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=\int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} - 2\right )}{x^{4} \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{4} - 2\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{{\left (x^{2} + 1\right )} x^{4}} \,d x } \]
Time = 0.35 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=\frac {1}{2} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{4} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + \frac {1}{4} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right ) - 3 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
1/2*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x^2 + 1)^( 1/3))) + 1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1)^(1/3) + 1)) - 3/4* (-1/x^2 + 1)^(4/3) + 1/4*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-1/x^2 + 1)^(1/3) + (-1/x^2 + 1)^(2/3)) - 1/2*2^(1/3)*log(abs(-2^(1/3) + (-1/x^2 + 1)^(1/3)) ) - 3*(-1/x^2 + 1)^(1/3) + 1/4*log((-1/x^2 + 1)^(2/3) + (-1/x^2 + 1)^(1/3) + 1) - 1/2*log(abs((-1/x^2 + 1)^(1/3) - 1))
Timed out. \[ \int \frac {\sqrt [3]{-x+x^3} \left (-2+x^4\right )}{x^4 \left (1+x^2\right )} \, dx=\int \frac {{\left (x^3-x\right )}^{1/3}\,\left (x^4-2\right )}{x^4\,\left (x^2+1\right )} \,d x \]