\(\int \frac {1}{(1+x^3) \sqrt [3]{-x^2+x^3}} \, dx\) [2322]

Optimal result

Integrand size = 21, antiderivative size = 180 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(571\) vs. \(2(180)=360\).

Time = 0.20 (sec) , antiderivative size = 571, normalized size of antiderivative = 3.17, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2081, 6857, 93} \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log (-x-1)}{6 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{-1} x-1\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (-(-1)^{2/3} x-1\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}} \]

[In]

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

[Out]

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

Rule 93

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

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (1+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {1}{3 (-1-x) \sqrt [3]{-1+x} x^{2/3}}-\frac {1}{3 \sqrt [3]{-1+x} x^{2/3} \left (-1+\sqrt [3]{-1} x\right )}-\frac {1}{3 \sqrt [3]{-1+x} x^{2/3} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{6 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\sqrt [3]{-1} x\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1-(-1)^{2/3} x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {\sqrt [3]{-1+x} x^{2/3} \left (2^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )+\log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )\right )-4 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 \sqrt [3]{(-1+x) x^2}} \]

[In]

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

[Out]

((-1 + x)^(1/3)*x^(2/3)*(2^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(2^(2/3)*(-1 + x)^(1/3) + x^(1/3))] - 2*L
og[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] + Log[2^(1/3)*(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3)*x^(1/3) + 2*x^(2/
3)]) - 4*RootSum[1 - #1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1 & ]))/(12*((-1 + x)
*x^2)^(1/3))

Maple [N/A] (verified)

Time = 5.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(-\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{6}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{12}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right )}{6}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{3}\)	\(140\)
trager	\(\text {Expression too large to display}\)	\(2155\)

[In]

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

[Out]

-1/6*2^(2/3)*ln((-2^(1/3)*x+((-1+x)*x^2)^(1/3))/x)+1/12*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*((-1+x)*x^2)^(1/3)*x+(
(-1+x)*x^2)^(2/3))/x^2)-1/6*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*((-1+x)*x^2)^(1/3)+x)/x)-1/3*sum(ln((-
_R*x+((-1+x)*x^2)^(1/3))/x)/_R,_R=RootOf(_Z^6-_Z^3+1))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.32 (sec) , antiderivative size = 575, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x + i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x - i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (-\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x - 2 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (-i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (-i \, \sqrt {3} - 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{2} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

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

[Out]

1/12*2^(2/3)*(I*sqrt(3) - 1)^(1/3)*(sqrt(-3) - 1)*log(((sqrt(3)*2^(1/3)*(I*sqrt(-3)*x + I*x) + 2^(1/3)*(sqrt(-
3)*x + x))*(I*sqrt(3) - 1)^(2/3) + 8*(x^3 - x^2)^(1/3))/x) - 1/12*2^(2/3)*(I*sqrt(3) - 1)^(1/3)*(sqrt(-3) + 1)
*log(((sqrt(3)*2^(1/3)*(-I*sqrt(-3)*x + I*x) - 2^(1/3)*(sqrt(-3)*x - x))*(I*sqrt(3) - 1)^(2/3) + 8*(x^3 - x^2)
^(1/3))/x) - 1/12*2^(2/3)*(-I*sqrt(3) - 1)^(1/3)*(sqrt(-3) + 1)*log(((sqrt(3)*2^(1/3)*(I*sqrt(-3)*x - I*x) - 2
^(1/3)*(sqrt(-3)*x - x))*(-I*sqrt(3) - 1)^(2/3) + 8*(x^3 - x^2)^(1/3))/x) + 1/12*2^(2/3)*(-I*sqrt(3) - 1)^(1/3
)*(sqrt(-3) - 1)*log(((sqrt(3)*2^(1/3)*(-I*sqrt(-3)*x - I*x) + 2^(1/3)*(sqrt(-3)*x + x))*(-I*sqrt(3) - 1)^(2/3
) + 8*(x^3 - x^2)^(1/3))/x) - 1/6*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(-1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x - 2*sqrt(6
)*(-1)^(1/3)*(x^3 - x^2)^(1/3))/x) + 1/6*2^(2/3)*(-1)^(1/3)*log(-(2^(1/3)*(-1)^(2/3)*x - (x^3 - x^2)^(1/3))/x)
+ 1/6*2^(2/3)*(I*sqrt(3) - 1)^(1/3)*log(((-I*sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*(I*sqrt(3) - 1)^(2/3) + 4*(x^3 -
x^2)^(1/3))/x) + 1/6*2^(2/3)*(-I*sqrt(3) - 1)^(1/3)*log(((I*sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*(-I*sqrt(3) - 1)^(2
/3) + 4*(x^3 - x^2)^(1/3))/x) - 1/12*2^(2/3)*(-1)^(1/3)*log(-(2^(2/3)*(-1)^(1/3)*x^2 - 2^(1/3)*(-1)^(2/3)*(x^3
- x^2)^(1/3)*x - (x^3 - x^2)^(2/3))/x^2)

Sympy [N/A]

Not integrable

Time = 0.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 34.57 (sec) , antiderivative size = 967, normalized size of antiderivative = 5.37 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x + 1)^(1/3))) + 1/3*(sqrt(3)*cos(4/9*pi)^5 -
10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi) +
10*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqrt(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)
*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi
))) + 1/3*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^
4 - 5*cos(2/9*pi)^4*sin(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2/9*pi)^2 - sqr
t(3)*sin(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(2/9*pi) + 2*(-1/x + 1)^(1/3))
/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) - 1/3*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 +
5*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1/9*pi)^2*sin(1/9*pi)^3 + sin(1/9*p
i)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1
)*cos(1/9*pi) - 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) + 1/6*(5*sqrt(3)*cos(4/9*pi)^4*sin(4/
9*pi) - 10*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*pi)^3*sin(
4/9*pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos(4/9*pi)^2 - sin(4/9*pi)^2)*l
og((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3) + 1) + 1/6*(5*sqrt(3)*cos(2/9*pi
)^4*sin(2/9*pi) - 10*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*pi)^5 + cos(2/9*pi)^5 - 10*cos(2/9*
pi)^3*sin(2/9*pi)^2 + 5*cos(2/9*pi)*sin(2/9*pi)^4 + 2*sqrt(3)*cos(2/9*pi)*sin(2/9*pi) + cos(2/9*pi)^2 - sin(2/
9*pi)^2)*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3) + 1) + 1/6*(5*sqrt(3)*
cos(1/9*pi)^4*sin(1/9*pi) - 10*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + sqrt(3)*sin(1/9*pi)^5 - cos(1/9*pi)^5 + 1
0*cos(1/9*pi)^3*sin(1/9*pi)^2 - 5*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) + cos(1/9*pi)^
2 - sin(1/9*pi)^2)*log((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi))*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3) + 1) + 1/12*2
^(2/3)*log(2^(2/3) + 2^(1/3)*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3)) - 1/6*2^(2/3)*log(abs(-2^(1/3) + (-1/x + 1)^
(1/3)))

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{\left (x^3+1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]

[In]

int(1/((x^3 + 1)*(x^3 - x^2)^(1/3)),x)

[Out]

int(1/((x^3 + 1)*(x^3 - x^2)^(1/3)), x)