Integrand size = 17, antiderivative size = 177 \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx=\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left ((1-x) (1+x)^2\right )}{2 \sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
1/2*(-x^3+1)^(2/3)+1/2*x^2*hypergeom([1/3, 2/3],[5/3],x^3)-1/4*ln((1-x)*(1 +x)^2)*2^(2/3)-1/2*ln(x+(-x^3+1)^(1/3))+3/4*ln(-1+x+2^(2/3)*(-x^3+1)^(1/3) )*2^(2/3)+1/3*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-1/2*arcta n(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)
\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx=\int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx \]
Time = 0.38 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2578, 888, 2577, 769, 2574}
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.
| \(\displaystyle \int \frac {\left (1-x^3\right )^{2/3}}{x+1} \, dx\) |
| \(\Big \downarrow \) 2578 |
| \(\displaystyle \int \frac {x}{\sqrt [3]{1-x^3}}dx+\int \frac {1-x}{(x+1) \sqrt [3]{1-x^3}}dx+\frac {1}{2} \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \int \frac {1-x}{(x+1) \sqrt [3]{1-x^3}}dx+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{2} \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 2577 |
| \(\displaystyle -\int \frac {1}{\sqrt [3]{1-x^3}}dx+2 \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{2} \left (1-x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle 2 \int \frac {1}{(x+1) \sqrt [3]{1-x^3}}dx+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )\) |
| \(\Big \downarrow \) 2574 |
| \(\displaystyle \frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2 \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac {\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}}\right )+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )\) |
(1 - x^3)^(2/3)/2 + ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + (x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3])/2 - Log[x + (1 - x^3)^(1/3)]/2 + 2*(-1/2*(Sqrt[3]*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3] ])/2^(1/3) - Log[(1 - x)*(1 + x)^2]/(4*2^(1/3)) + (3*Log[-1 + x + 2^(2/3)* (1 - x^3)^(1/3)])/(4*2^(1/3)))
3.2.11.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[ Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b, 3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sq rt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2^(7/3 )*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^ (1/3)])/(2^(7/3)*Rt[b, 3]*c), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]
Int[((e_.) + (f_.)*(x_))/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)) , x_Symbol] :> Simp[f/d Int[1/(a + b*x^3)^(1/3), x], x] + Simp[(d*e - c*f )/d Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((a_) + (b_.)*(x_)^3)^(2/3)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[(a + b*x^3)^(2/3)/(2*d), x] + (Simp[1/d^2 Int[(a*d^2 + b*c^2*x)/((c + d*x)*(a + b*x^3)^(1/3)), x], x] - Simp[b*(c/d^2) Int[x/(a + b*x^3)^(1/3), x], x] ) /; FreeQ[{a, b, c, d}, x]
\[\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{1+x}d x\]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x + 1} \,d x } \]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{x + 1}\, dx \]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x + 1} \,d x } \]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x + 1} \,d x } \]
Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}}{x+1} \,d x \]
\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x} \, dx=\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{2}-\left (\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{4}+x^{3}-x -1}d x \right )-\left (\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}} x^{2}}{x^{4}+x^{3}-x -1}d x \right ) \]