\(\int \frac {(1-x^3)^{2/3}}{1+x^3} \, dx\) [113]

Optimal result

Integrand size = 19, antiderivative size = 132 \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right ) \]

[Out]

-1/6*ln(x^3+1)*2^(2/3)+1/2*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(2/3)-1/2*ln(x+(-x^3+1)^(1/3))+1/3*arctan(1/3*(1-2*
x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-1/3*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {399, 245, 384} \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left (x^3+1\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right ) \]

[In]

Int[(1 - x^3)^(2/3)/(1 + x^3),x]

[Out]

ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - (2^(2/3)*ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt
[3]])/Sqrt[3] - Log[1 + x^3]/(3*2^(1/3)) + Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/2^(1/3) - Log[x + (1 - x^3)^(1/
3)]/2

Rule 245

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]]/(Sq
rt[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]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx-\int \frac {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 {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.55 \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\frac {1}{6} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2 \log \left (x+\sqrt [3]{1-x^3}\right )+2\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+\log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )-2^{2/3} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[(1 - x^3)^(2/3)/(1 + x^3),x]

[Out]

(2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3)^(1/3))] - 2*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2^(2/3)*(1
- x^3)^(1/3))] - 2*Log[x + (1 - x^3)^(1/3)] + 2*2^(2/3)*Log[2*x + 2^(2/3)*(1 - x^3)^(1/3)] + Log[x^2 - x*(1 -
x^3)^(1/3) + (1 - x^3)^(2/3)] - 2^(2/3)*Log[-2*x^2 + 2^(2/3)*x*(1 - x^3)^(1/3) - 2^(1/3)*(1 - x^3)^(2/3)])/6

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.36

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{3}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{3}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{6}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right )}{3}+\frac {\ln \left (\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}-x \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{2}}{x^{2}}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{3}\)	\(179\)

[In]

int((-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)

[Out]

-1/3*ln((x+(-x^3+1)^(1/3))/x)+1/3*2^(2/3)*ln((2^(1/3)*x+(-x^3+1)^(1/3))/x)-1/6*2^(2/3)*ln((2^(2/3)*x^2-2^(1/3)
*(-x^3+1)^(1/3)*x+(-x^3+1)^(2/3))/x^2)+1/3*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(-2^(2/3)*(-x^3+1)^(1/3)+x)/x)+1
/6*ln(((-x^3+1)^(2/3)-x*(-x^3+1)^(1/3)+x^2)/x^2)-1/3*3^(1/2)*arctan(1/3*(-2*(-x^3+1)^(1/3)+x)*3^(1/2)/x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.45 \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\frac {1}{3} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \cdot 4^{\frac {1}{3}} \log \left (\frac {4^{\frac {2}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

integrate((-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")

[Out]

-1/3*4^(1/3)*sqrt(3)*arctan(-1/3*(sqrt(3)*x - 4^(1/3)*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 1/3*sqrt(3)*arctan(-1/3*(
sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 1/3*4^(1/3)*log((4^(2/3)*x + 2*(-x^3 + 1)^(1/3))/x) - 1/6*4^(1/3)
*log((2*4^(1/3)*x^2 - 4^(2/3)*(-x^3 + 1)^(1/3)*x + 2*(-x^3 + 1)^(2/3))/x^2) - 1/3*log((x + (-x^3 + 1)^(1/3))/x
) + 1/6*log((x^2 - (-x^3 + 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2)

Sympy [F]

\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

[In]

integrate((-x**3+1)**(2/3)/(x**3+1),x)

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(2/3)/((x + 1)*(x**2 - x + 1)), x)

Maxima [F]

\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{3} + 1} \,d x } \]

[In]

integrate((-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate((-x^3 + 1)^(2/3)/(x^3 + 1), x)

Giac [F]

\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{3} + 1} \,d x } \]

[In]

integrate((-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate((-x^3 + 1)^(2/3)/(x^3 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}}{x^3+1} \,d x \]

[In]

int((1 - x^3)^(2/3)/(x^3 + 1),x)

[Out]

int((1 - x^3)^(2/3)/(x^3 + 1), x)