Integrand size = 22, antiderivative size = 105 \[ \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {\left (1-x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \] Output:
-1/2*(-x^3+1)^(2/3)/x^2+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 = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{12} \left (-\frac {6 \left (1-x^3\right )^{2/3}}{x^2}+2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+2^{2/3} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[1/(x^3*(1 - x^3)^(1/3)*(1 + x^3)),x]
Output:
((-6*(1 - x^3)^(2/3))/x^2 + 2*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2^(2 /3)*(1 - x^3)^(1/3))] - 2*2^(2/3)*Log[2*x + 2^(2/3)*(1 - x^3)^(1/3)] + 2^( 2/3)*Log[-2*x^2 + 2^(2/3)*x*(1 - x^3)^(1/3) - 2^(1/3)*(1 - x^3)^(2/3)])/12
Time = 0.37 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {980, 27, 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.
| \(\displaystyle \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 980 |
| \(\displaystyle \frac {1}{2} \int -\frac {2}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx-\frac {\left (1-x^3\right )^{2/3}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {1}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx-\frac {\left (1-x^3\right )^{2/3}}{2 x^2}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}-\frac {\left (1-x^3\right )^{2/3}}{2 x^2}\) |
Input:
Int[1/(x^3*(1 - x^3)^(1/3)*(1 + x^3)),x]
Output:
-1/2*(1 - x^3)^(2/3)/x^2 + 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^n*(m + 1)) Int[(e*x)^(m + n)*( a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Time = 27.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.20
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{2}-2 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-6 \left (-x^{3}+1\right )^{\frac {2}{3}}}{12 x^{2}}\) | \(126\) |
| risch | \(\text {Expression too large to display}\) | \(929\) |
| trager | \(\text {Expression too large to display}\) | \(1158\) |
Input:
int(1/x^3/(-x^3+1)^(1/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
1/12*(-2*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(-2^(2/3)*(-x^3+1)^(1/3)+x)/x) *x^2-2*2^(2/3)*ln((2^(1/3)*x+(-x^3+1)^(1/3))/x)*x^2+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)*x^2-6*(-x^3+1)^(2/3))/x^2
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (81) = 162\).
Time = 1.83 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.57 \[ \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {12 \cdot 2^{\frac {1}{6}} \sqrt {\frac {1}{6}} x^{2} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (6 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )} + 12 \, {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1}\right ) - 2 \cdot 2^{\frac {2}{3}} x^{2} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 2^{\frac {2}{3}} x^{2} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 18 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{36 \, x^{2}} \] Input:
integrate(1/x^3/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="fricas")
Output:
1/36*(12*2^(1/6)*sqrt(1/6)*x^2*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)) - 2*2^(2/3)*x^2*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)) + 2^(2/3)*x^2*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)) - 18*(-x^3 + 1)^(2/3))/x^2
\[ \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^{3} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(1/x**3/(-x**3+1)**(1/3)/(x**3+1),x)
Output:
Integral(1/(x**3*(-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)
\[ \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="maxima")
Output:
integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^3), x)
\[ \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="giac")
Output:
integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^3), x)
Timed out. \[ \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^3\,{\left (1-x^3\right )}^{1/3}\,\left (x^3+1\right )} \,d x \] Input:
int(1/(x^3*(1 - x^3)^(1/3)*(x^3 + 1)),x)
Output:
int(1/(x^3*(1 - x^3)^(1/3)*(x^3 + 1)), x)
\[ \int \frac {1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{\left (-x^{3}+1\right )^{\frac {1}{3}} x^{6}+\left (-x^{3}+1\right )^{\frac {1}{3}} x^{3}}d x \] Input:
int(1/x^3/(-x^3+1)^(1/3)/(x^3+1),x)
Output:
int(1/(( - x**3 + 1)**(1/3)*x**6 + ( - x**3 + 1)**(1/3)*x**3),x)