Integrand size = 19, antiderivative size = 88 \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \] Output:
-1/6*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)-1/ 12*ln(x^3+1)*2^(2/3)+1/4*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(2/3)
Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2 \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \] Input:
Integrate[1/((1 - x^3)^(1/3)*(1 + x^3)),x]
Output:
-1/6*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2^(2/3)*(1 - x^3)^(1/3))] - 2*Log[ 2*x + 2^(2/3)*(1 - x^3)^(1/3)] + Log[-2*x^2 + 2^(2/3)*x*(1 - x^3)^(1/3) - 2^(1/3)*(1 - x^3)^(2/3)])/2^(1/3)
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {901}
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.
\(\displaystyle \int \frac {1}{\sqrt [3]{1-x^3} \left (x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 901 |
\(\displaystyle -\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}\) |
Input:
Int[1/((1 - x^3)^(1/3)*(1 + x^3)),x]
Output:
-(ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3])) - Log[1 + x^3]/(6*2^(1/3)) + Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/(2*2^(1/3) )
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[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]
Time = 4.57 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {2}{3}} \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right )+\ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{6}\) | \(95\) |
trager | \(\text {Expression too large to display}\) | \(616\) |
Input:
int(1/(-x^3+1)^(1/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
1/6*2^(2/3)*(3^(1/2)*arctan(1/3*3^(1/2)*(-2^(2/3)*(-x^3+1)^(1/3)+x)/x)+ln( (2^(1/3)*x+(-x^3+1)^(1/3))/x)-1/2*ln((2^(2/3)*x^2-2^(1/3)*(-x^3+1)^(1/3)*x +(-x^3+1)^(2/3))/x^2))
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (67) = 134\).
Time = 1.58 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.80 \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{3} \cdot 2^{\frac {1}{6}} \sqrt {\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (6 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )} + 12 \, {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1}\right ) + \frac {1}{18} \cdot 2^{\frac {2}{3}} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) - \frac {1}{36} \cdot 2^{\frac {2}{3}} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) \] Input:
integrate(1/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="fricas")
Output:
-1/3*2^(1/6)*sqrt(1/6)*arctan(2^(1/6)*sqrt(1/6)*(6*2^(2/3)*(5*x^7 + 4*x^4 - x)*(-x^3 + 1)^(2/3) - 2^(1/3)*(71*x^9 - 111*x^6 + 33*x^3 - 1) + 12*(19*x ^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3))/(109*x^9 - 105*x^6 + 3*x^3 + 1)) + 1/ 18*2^(2/3)*log((6*2^(1/3)*(-x^3 + 1)^(1/3)*x^2 + 2^(2/3)*(x^3 + 1) + 6*(-x ^3 + 1)^(2/3)*x)/(x^3 + 1)) - 1/36*2^(2/3)*log((3*2^(2/3)*(5*x^4 - x)*(-x^ 3 + 1)^(2/3) + 2^(1/3)*(19*x^6 - 16*x^3 + 1) - 12*(2*x^5 - x^2)*(-x^3 + 1) ^(1/3))/(x^6 + 2*x^3 + 1))
\[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(1/(-x**3+1)**(1/3)/(x**3+1),x)
Output:
Integral(1/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)
\[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="maxima")
Output:
integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)), x)
\[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="giac")
Output:
integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{{\left (1-x^3\right )}^{1/3}\,\left (x^3+1\right )} \,d x \] Input:
int(1/((1 - x^3)^(1/3)*(x^3 + 1)),x)
Output:
int(1/((1 - x^3)^(1/3)*(x^3 + 1)), x)
\[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{\left (-x^{3}+1\right )^{\frac {1}{3}} x^{3}+\left (-x^{3}+1\right )^{\frac {1}{3}}}d x \] Input:
int(1/(-x^3+1)^(1/3)/(x^3+1),x)
Output:
int(1/(( - x**3 + 1)**(1/3)*x**3 + ( - x**3 + 1)**(1/3)),x)