Integrand size = 22, antiderivative size = 124 \[ \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {\sqrt [3]{1-x^3}}{4 x^4}+\frac {\sqrt [3]{1-x^3}}{4 x}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]
-1/4*(-x^3+1)^(1/3)/x^4+1/4*(-x^3+1)^(1/3)/x+1/12*ln(x^3+1)*2^(1/3)-1/4*ln (-2^(1/3)*x-(-x^3+1)^(1/3))*2^(1/3)-1/6*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1) ^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)
Time = 0.45 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{12} \left (-\frac {3 \left (1-x^3\right )^{4/3}}{x^4}-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2 \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+\sqrt [3]{2} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]
((-3*(1 - x^3)^(4/3))/x^4 - 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2^(2 /3)*(1 - x^3)^(1/3))] - 2*2^(1/3)*Log[2*x + 2^(2/3)*(1 - x^3)^(1/3)] + 2^( 1/3)*Log[-2*x^2 + 2^(2/3)*x*(1 - x^3)^(1/3) - 2^(1/3)*(1 - x^3)^(2/3)])/12
Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {980, 25, 1053, 27, 992}
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.
\(\displaystyle \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 980 |
\(\displaystyle \frac {1}{4} \int -\frac {1-3 x^3}{x^2 \left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx-\frac {\sqrt [3]{1-x^3}}{4 x^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{4} \int \frac {1-3 x^3}{x^2 \left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx-\frac {\sqrt [3]{1-x^3}}{4 x^4}\) |
\(\Big \downarrow \) 1053 |
\(\displaystyle \frac {1}{4} \left (\int \frac {4 x}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx+\frac {\sqrt [3]{1-x^3}}{x}\right )-\frac {\sqrt [3]{1-x^3}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (4 \int \frac {x}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx+\frac {\sqrt [3]{1-x^3}}{x}\right )-\frac {\sqrt [3]{1-x^3}}{4 x^4}\) |
\(\Big \downarrow \) 992 |
\(\displaystyle \frac {1}{4} \left (4 \left (-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2\ 2^{2/3}}\right )+\frac {\sqrt [3]{1-x^3}}{x}\right )-\frac {\sqrt [3]{1-x^3}}{4 x^4}\) |
-1/4*(1 - x^3)^(1/3)/x^4 + ((1 - x^3)^(1/3)/x + 4*(-(ArcTan[(1 - (2*2^(1/3 )*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3])) + Log[1 + x^3]/(6*2^(2/3 )) - Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/(2*2^(2/3))))/4
3.7.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((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^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Time = 22.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(\frac {-2 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{4}+\left (3 x^{3}-3\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} x^{4} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right )+\ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{12 x^{4}}\) | \(127\) |
trager | \(\text {Expression too large to display}\) | \(1169\) |
risch | \(\text {Expression too large to display}\) | \(1394\) |
1/12*(-2*2^(1/3)*ln((2^(1/3)*x+(-x^3+1)^(1/3))/x)*x^4+(3*x^3-3)*(-x^3+1)^( 1/3)+2^(1/3)*x^4*(2*3^(1/2)*arctan(1/3*3^(1/2)*(-2^(2/3)*(-x^3+1)^(1/3)+x) /x)+ln((2^(2/3)*x^2-2^(1/3)*(-x^3+1)^(1/3)*x+(-x^3+1)^(2/3))/x^2)))/x^4
Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (95) = 190\).
Time = 1.41 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.52 \[ \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{4} \arctan \left (-\frac {4^{\frac {1}{6}} {\left (6 \cdot 4^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 12 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{4} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )} - 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{4} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 18 \, {\left (x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{72 \, x^{4}} \]
-1/72*(4*4^(1/6)*sqrt(3)*(-1)^(1/3)*x^4*arctan(-1/6*4^(1/6)*(6*4^(2/3)*sqr t(3)*(-1)^(2/3)*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3) - 12*sqrt(3)*(-1) ^(1/3)*(5*x^7 + 4*x^4 - x)*(-x^3 + 1)^(2/3) - 4^(1/3)*sqrt(3)*(71*x^9 - 11 1*x^6 + 33*x^3 - 1))/(109*x^9 - 105*x^6 + 3*x^3 + 1)) - 2*4^(2/3)*(-1)^(1/ 3)*x^4*log(-(3*4^(2/3)*(-1)^(1/3)*(-x^3 + 1)^(1/3)*x^2 - 4^(1/3)*(-1)^(2/3 )*(x^3 + 1) - 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 4^(2/3)*(-1)^(1/3)*x^4*lo g((6*4^(1/3)*(-1)^(2/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) - 4^(2/3)*(-1)^(1/3)* (19*x^6 - 16*x^3 + 1) - 24*(2*x^5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) - 18*(x^3 - 1)*(-x^3 + 1)^(1/3))/x^4
\[ \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^{5} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
\[ \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{5}} \,d x } \]
\[ \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{5}} \,d x } \]
Timed out. \[ \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {1}{x^5\,{\left (1-x^3\right )}^{2/3}\,\left (x^3+1\right )} \,d x \]