Integrand size = 24, antiderivative size = 328 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x^2}{-x^2+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {1}{6} \log \left (x^2+\left (-x^2+x^4\right )^{2/3}\right )-\frac {1}{12} i \left (-i+\sqrt {3}\right ) \log \left (-x^2-i \sqrt {3} x^2+2 \left (-x^2+x^4\right )^{2/3}\right )+\frac {1}{12} i \left (i+\sqrt {3}\right ) \log \left (-x^2+i \sqrt {3} x^2+2 \left (-x^2+x^4\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} \sqrt {3} x \sqrt [3]{-x^2+x^4}-\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}}-\frac {\log \left (2 x^2+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} \sqrt {3} x \sqrt [3]{-x^2+x^4}+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}} \]
1/24*arctan(3^(1/2)*x^2/(-x^2+2^(1/3)*(x^4-x^2)^(2/3)))*2^(2/3)*3^(1/2)+1/ 6*ln(x^2+(x^4-x^2)^(2/3))-1/12*I*(-I+3^(1/2))*ln(-x^2-I*3^(1/2)*x^2+2*(x^4 -x^2)^(2/3))+1/12*I*(3^(1/2)+I)*ln(-x^2+I*3^(1/2)*x^2+2*(x^4-x^2)^(2/3))+1 /48*ln(-2*x^2+2^(2/3)*3^(1/2)*x*(x^4-x^2)^(1/3)-2^(1/3)*(x^4-x^2)^(2/3))*2 ^(2/3)-1/24*ln(2*x^2+2^(1/3)*(x^4-x^2)^(2/3))*2^(2/3)+1/48*ln(2*x^2+2^(2/3 )*3^(1/2)*x*(x^4-x^2)^(1/3)+2^(1/3)*(x^4-x^2)^(2/3))*2^(2/3)
Time = 0.69 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-1+x^2} \left (8 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \left (-1+x^2\right )^{2/3}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-\sqrt [3]{2} \left (-1+x^2\right )^{2/3}}\right )+8 \log \left (x^{2/3}+\left (-1+x^2\right )^{2/3}\right )+2^{2/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2}-\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )-2\ 2^{2/3} \log \left (2 x^{2/3}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )+2^{2/3} \log \left (2 x^{2/3}+2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )-4 \log \left (x^{4/3}-x^{2/3} \left (-1+x^2\right )^{2/3}+\left (-1+x^2\right )^{4/3}\right )\right )}{48 \sqrt [3]{x^2 \left (-1+x^2\right )}} \]
(x^(2/3)*(-1 + x^2)^(1/3)*(8*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) - 2 *(-1 + x^2)^(2/3))] - 2*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) - 2^(1/3)*(-1 + x^2)^(2/3))] + 8*Log[x^(2/3) + (-1 + x^2)^(2/3)] + 2^(2/3) *Log[-2*x^(2/3) + 2^(2/3)*Sqrt[3]*x^(1/3)*(-1 + x^2)^(1/3) - 2^(1/3)*(-1 + x^2)^(2/3)] - 2*2^(2/3)*Log[2*x^(2/3) + 2^(1/3)*(-1 + x^2)^(2/3)] + 2^(2/ 3)*Log[2*x^(2/3) + 2^(2/3)*Sqrt[3]*x^(1/3)*(-1 + x^2)^(1/3) + 2^(1/3)*(-1 + x^2)^(2/3)] - 4*Log[x^(4/3) - x^(2/3)*(-1 + x^2)^(2/3) + (-1 + x^2)^(4/3 )]))/(48*(x^2*(-1 + x^2))^(1/3))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 2.08 (sec) , antiderivative size = 737, normalized size of antiderivative = 2.25, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2467, 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.
| \(\displaystyle \int \frac {x^3}{\sqrt [3]{x^4-x^2} \left (x^6+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^2-1} \int \frac {x^{7/3}}{\sqrt [3]{x^2-1} \left (x^6+1\right )}dx}{\sqrt [3]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x^2-1} \int \frac {x^3}{\sqrt [3]{x^2-1} \left (x^6+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^4-x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x^2-1} \int \left (\frac {\sqrt [3]{x} \left (-x^{2/3}-1\right )}{9 \left (x^{4/3}-x^{2/3}+1\right ) \sqrt [3]{x^2-1}}+\frac {\sqrt [3]{x}}{9 \left (x^{2/3}+1\right ) \sqrt [3]{x^2-1}}+\frac {x \left (x^2+1\right )}{3 \sqrt [3]{x^2-1} \left (x^4-x^2+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^4-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x^2-1} \left (\frac {\sqrt [3]{1-x^2} x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^2,\frac {2 x^2}{1-i \sqrt {3}}\right )}{6 \sqrt {3} \left (\sqrt {3}+i\right ) \sqrt [3]{x^2-1}}-\frac {i \sqrt [3]{1-x^2} x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^2,\frac {2 x^2}{1-i \sqrt {3}}\right )}{12 \sqrt {3} \sqrt [3]{x^2-1}}-\frac {\sqrt [3]{1-x^2} x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {2 x^2}{1+i \sqrt {3}},x^2\right )}{6 \sqrt {3} \left (-\sqrt {3}+i\right ) \sqrt [3]{x^2-1}}+\frac {i \sqrt [3]{1-x^2} x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {2 x^2}{1+i \sqrt {3}},x^2\right )}{12 \sqrt {3} \sqrt [3]{x^2-1}}+\frac {\arctan \left (\frac {1-2^{2/3} \sqrt [3]{x^2-1}}{\sqrt {3}}\right )}{18 \sqrt [3]{2} \sqrt {3}}+\frac {5 \arctan \left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{36 \sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{9 \sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{18 \sqrt [3]{2} \sqrt {3}}+\frac {5 \log \left (-\left (\left (1-x^{2/3}\right ) \left (x^{2/3}+1\right )^2\right )\right )}{216 \sqrt [3]{2}}-\frac {\log \left (x^2+1\right )}{54 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{x^2-1}+\sqrt [3]{2}\right )}{36 \sqrt [3]{2}}-\frac {\log \left (1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}\right )}{27 \sqrt [3]{2}}+\frac {\log \left (\frac {2^{2/3} \left (1-x^{2/3}\right )^2}{\left (x^2-1\right )^{2/3}}+\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}+1\right )}{54 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{x^2-1}\right )}{36 \sqrt [3]{2}}-\frac {5 \log \left (-x^{2/3}+2^{2/3} \sqrt [3]{x^2-1}+1\right )}{72 \sqrt [3]{2}}\right )}{\sqrt [3]{x^4-x^2}}\) |
(3*x^(2/3)*(-1 + x^2)^(1/3)*(((-1/12*I)*x^(4/3)*(1 - x^2)^(1/3)*AppellF1[2 /3, 1/3, 1, 5/3, x^2, (2*x^2)/(1 - I*Sqrt[3])])/(Sqrt[3]*(-1 + x^2)^(1/3)) + (x^(4/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, (2*x^2)/(1 - I *Sqrt[3])])/(6*Sqrt[3]*(I + Sqrt[3])*(-1 + x^2)^(1/3)) + ((I/12)*x^(4/3)*( 1 - x^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (2*x^2)/(1 + I*Sqrt[3]), x^2])/( Sqrt[3]*(-1 + x^2)^(1/3)) - (x^(4/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (2*x^2)/(1 + I*Sqrt[3]), x^2])/(6*Sqrt[3]*(I - Sqrt[3])*(-1 + x^2)^( 1/3)) + (5*ArcTan[(1 - (2^(1/3)*(1 - x^(2/3)))/(-1 + x^2)^(1/3))/Sqrt[3]]) /(36*2^(1/3)*Sqrt[3]) + ArcTan[(1 + (2*2^(1/3)*(1 - x^(2/3)))/(-1 + x^2)^( 1/3))/Sqrt[3]]/(9*2^(1/3)*Sqrt[3]) - ArcTan[(1 + (2*2^(1/3)*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]]/(18*2^(1/3)*Sqrt[3]) + ArcTan[(1 - 2^(2/3)*(-1 + x^2 )^(1/3))/Sqrt[3]]/(18*2^(1/3)*Sqrt[3]) + (5*Log[-((1 - x^(2/3))*(1 + x^(2/ 3))^2)])/(216*2^(1/3)) - Log[1 + x^2]/(54*2^(1/3)) - Log[1 - (2^(1/3)*(1 - x^(2/3)))/(-1 + x^2)^(1/3)]/(27*2^(1/3)) + Log[1 + (2^(2/3)*(1 - x^(2/3)) ^2)/(-1 + x^2)^(2/3) + (2^(1/3)*(1 - x^(2/3)))/(-1 + x^2)^(1/3)]/(54*2^(1/ 3)) + Log[2^(1/3)*x^(2/3) - (-1 + x^2)^(1/3)]/(36*2^(1/3)) + Log[2^(1/3) + (-1 + x^2)^(1/3)]/(36*2^(1/3)) - (5*Log[1 - x^(2/3) + 2^(2/3)*(-1 + x^2)^ (1/3)])/(72*2^(1/3))))/(-x^2 + x^4)^(1/3)
3.30.12.3.1 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 = 29.37 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(\frac {\left (2 \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {1}{3}} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+x^{2}\right )}{3 x^{2}}\right ) \sqrt {3}-2 \ln \left (\frac {2^{\frac {2}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (\frac {-2^{\frac {2}{3}} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+2 \,2^{\frac {1}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {1}{3}} x^{2}-\left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}}{48}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (x^{2}-2 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}\right ) \sqrt {3}}{3 x^{2}}\right )}{6}-\frac {\ln \left (\frac {\left (\left (x^{4}-x^{2}\right )^{\frac {1}{3}}+1\right ) \left (x^{2}-\left (x^{4}-x^{2}\right )^{\frac {1}{3}}\right )}{x^{2}}\right )}{12}+\frac {\ln \left (\frac {x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{6}\) | \(216\) |
| trager | \(\text {Expression too large to display}\) | \(3016\) |
1/48*(2*arctan(1/3*3^(1/2)*(-2^(1/3)*(x^4-x^2)^(2/3)+x^2)/x^2)*3^(1/2)-2*l n((2^(2/3)*x^2+(x^4-x^2)^(2/3))/x^2)+ln((-2^(2/3)*(x^4-x^2)^(2/3)+2*2^(1/3 )*x^2+(x^4-x^2)^(1/3)*x^2-(x^4-x^2)^(1/3))/x^2))*2^(2/3)-1/6*3^(1/2)*arcta n(1/3*(x^2-2*(x^4-x^2)^(2/3))*3^(1/2)/x^2)-1/12*ln(((x^4-x^2)^(1/3)+1)*(x^ 2-(x^4-x^2)^(1/3))/x^2)+1/6*ln((x^2+(x^4-x^2)^(2/3))/x^2)
Time = 1.62 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.43 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {1}{72} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (24 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{8} - 2 \, x^{6} - 6 \, x^{4} - 2 \, x^{2} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}} - 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (x^{10} - 33 \, x^{8} + 110 \, x^{6} - 110 \, x^{4} + 33 \, x^{2} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} + \sqrt {6} 2^{\frac {1}{3}} {\left (x^{12} + 42 \, x^{10} - 417 \, x^{8} + 812 \, x^{6} - 417 \, x^{4} + 42 \, x^{2} + 1\right )}\right )}}{6 \, {\left (x^{12} - 102 \, x^{10} + 447 \, x^{8} - 628 \, x^{6} + 447 \, x^{4} - 102 \, x^{2} + 1\right )}}\right ) - \frac {1}{144} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {12 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{2} + 1\right )} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{8} - 32 \, x^{6} + 78 \, x^{4} - 32 \, x^{2} + 1\right )} + 6 \, {\left (x^{6} - 11 \, x^{4} + 11 \, x^{2} - 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}{x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1}\right ) + \frac {1}{72} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 12 \, {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}}}{x^{4} + 2 \, x^{2} + 1}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (x^{2} - 1\right )} - 2 \, \sqrt {3} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}}{3 \, {\left (x^{2} - 1\right )}}\right ) + \frac {1}{12} \, \log \left (\frac {x^{4} - x^{2} + 3 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )} + 3 \, {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}} + 1}{x^{4} - x^{2} + 1}\right ) \]
-1/72*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(1/6*2^(1/6)*(24*sqrt(6)*2^(2/3)*(- 1)^(2/3)*(x^8 - 2*x^6 - 6*x^4 - 2*x^2 + 1)*(x^4 - x^2)^(2/3) - 12*sqrt(6)* (-1)^(1/3)*(x^10 - 33*x^8 + 110*x^6 - 110*x^4 + 33*x^2 - 1)*(x^4 - x^2)^(1 /3) + sqrt(6)*2^(1/3)*(x^12 + 42*x^10 - 417*x^8 + 812*x^6 - 417*x^4 + 42*x ^2 + 1))/(x^12 - 102*x^10 + 447*x^8 - 628*x^6 + 447*x^4 - 102*x^2 + 1)) - 1/144*2^(2/3)*(-1)^(1/3)*log(-(12*2^(2/3)*(-1)^(1/3)*(x^4 - x^2)^(2/3)*(x^ 4 - 4*x^2 + 1) - 2^(1/3)*(-1)^(2/3)*(x^8 - 32*x^6 + 78*x^4 - 32*x^2 + 1) + 6*(x^6 - 11*x^4 + 11*x^2 - 1)*(x^4 - x^2)^(1/3))/(x^8 + 4*x^6 + 6*x^4 + 4 *x^2 + 1)) + 1/72*2^(2/3)*(-1)^(1/3)*log(-(6*2^(1/3)*(-1)^(2/3)*(x^4 - x^2 )^(1/3)*(x^2 - 1) - 2^(2/3)*(-1)^(1/3)*(x^4 + 2*x^2 + 1) + 12*(x^4 - x^2)^ (2/3))/(x^4 + 2*x^2 + 1)) - 1/6*sqrt(3)*arctan(-1/3*(sqrt(3)*(x^2 - 1) - 2 *sqrt(3)*(x^4 - x^2)^(1/3))/(x^2 - 1)) + 1/12*log((x^4 - x^2 + 3*(x^4 - x^ 2)^(1/3)*(x^2 - 1) + 3*(x^4 - x^2)^(2/3) + 1)/(x^4 - x^2 + 1))
\[ \int \frac {x^3}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {x^{3}}{\sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
\[ \int \frac {x^3}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{6} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {x^3}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{6} + 1\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {x^3}{\left (x^6+1\right )\,{\left (x^4-x^2\right )}^{1/3}} \,d x \]