Integrand size = 22, antiderivative size = 100 \[ \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{2 \sqrt [3]{1-x^3}}+\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]
1/2/(-x^3+1)^(1/3)-1/24*ln(x^3+1)*2^(2/3)+1/8*ln(2^(1/3)-(-x^3+1)^(1/3))*2 ^(2/3)+1/12*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)
Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.27 \[ \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{24} \left (\frac {12}{\sqrt [3]{1-x^3}}+2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+2\ 2^{2/3} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )-2^{2/3} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]
(12/(1 - x^3)^(1/3) + 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3 ))/Sqrt[3]] + 2*2^(2/3)*Log[-2 + 2^(2/3)*(1 - x^3)^(1/3)] - 2^(2/3)*Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2/3)])/24
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {946, 61, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 946 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{\left (1-x^3\right )^{4/3} \left (x^3+1\right )}dx^3\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3+\frac {3}{2 \sqrt [3]{1-x^3}}\right )\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {3 \int \frac {1}{\sqrt [3]{2}-\sqrt [3]{1-x^3}}d\sqrt [3]{1-x^3}}{2 \sqrt [3]{2}}+\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}\right )+\frac {3}{2 \sqrt [3]{1-x^3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )+\frac {3}{2 \sqrt [3]{1-x^3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (2^{2/3} \sqrt [3]{1-x^3}+1\right )}{\sqrt [3]{2}}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )+\frac {3}{2 \sqrt [3]{1-x^3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )+\frac {3}{2 \sqrt [3]{1-x^3}}\right )\) |
(3/(2*(1 - x^3)^(1/3)) + ((Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sq rt[3]])/2^(1/3) - Log[1 + x^3]/(2*2^(1/3)) + (3*Log[2^(1/3) - (1 - x^3)^(1 /3)])/(2*2^(1/3)))/2)/3
3.7.42.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Time = 8.42 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.22
| method | result | size |
| pseudoelliptic | \(-\frac {-2 \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) 2^{\frac {2}{3}} \sqrt {3}\, \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}-12}{24 \left (-x^{3}+1\right )^{\frac {1}{3}}}\) | \(122\) |
| trager | \(-\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{2 \left (x^{3}-1\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) \ln \left (\frac {-6 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-45 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) x^{3}+15 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}+63 \left (-x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+21 \left (-x^{3}+1\right )^{\frac {2}{3}}-14 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )-105 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{12}+\frac {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {15 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+72 {\operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+15 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right ) x^{3}+72 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}+126 \left (-x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+42 \left (-x^{3}+1\right )^{\frac {2}{3}}-35 \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )-168 \operatorname {RootOf}\left (\operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \operatorname {RootOf}\left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{2}\) | \(494\) |
| risch | \(\text {Expression too large to display}\) | \(669\) |
-1/24*(-2*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)*( -x^3+1)^(1/3)+2^(2/3)*ln((-x^3+1)^(2/3)+2^(1/3)*(-x^3+1)^(1/3)+2^(2/3))*(- x^3+1)^(1/3)-2*2^(2/3)*ln((-x^3+1)^(1/3)-2^(1/3))*(-x^3+1)^(1/3)-12)/(-x^3 +1)^(1/3)
Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.25 \[ \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {2 \, \sqrt {6} 2^{\frac {1}{6}} {\left (x^{3} - 1\right )} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} + 2 \, \sqrt {6} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + 2 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{24 \, {\left (x^{3} - 1\right )}} \]
1/24*(2*sqrt(6)*2^(1/6)*(x^3 - 1)*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3) + 2* sqrt(6)*(-x^3 + 1)^(1/3))) - 2^(2/3)*(x^3 - 1)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3)) + 2*2^(2/3)*(x^3 - 1)*log(-2^(1/3) + (-x^3 + 1)^(1/3)) - 12*(-x^3 + 1)^(2/3))/(x^3 - 1)
\[ \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\int \frac {x^{2}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \]
1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1 /3))) - 1/24*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^( 2/3)) + 1/12*2^(2/3)*log(-2^(1/3) + (-x^3 + 1)^(1/3)) + 1/2/(-x^3 + 1)^(1/ 3)
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) + \frac {1}{2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \]
1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1 /3))) - 1/24*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^( 2/3)) + 1/12*2^(2/3)*log(abs(-2^(1/3) + (-x^3 + 1)^(1/3))) + 1/2/(-x^3 + 1 )^(1/3)
Time = 8.45 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx=\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}}{4}\right )}{12}+\frac {1}{2\,{\left (1-x^3\right )}^{1/3}}+\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{24}-\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{24} \]