Integrand size = 22, antiderivative size = 98 \[ \int \frac {x^5}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
-1/2*(-x^3+1)^(2/3)+1/12*ln(x^3+1)*2^(2/3)-1/4*ln(2^(1/3)-(-x^3+1)^(1/3))* 2^(2/3)-1/6*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)
Time = 0.16 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.29 \[ \int \frac {x^5}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\frac {1}{12} \left (-6 \left (1-x^3\right )^{2/3}-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-2\ 2^{2/3} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )+2^{2/3} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]
(-6*(1 - x^3)^(2/3) - 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3 ))/Sqrt[3]] - 2*2^(2/3)*Log[-2 + 2^(2/3)*(1 - x^3)^(1/3)] + 2^(2/3)*Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2/3)])/12
Time = 0.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {948, 90, 67, 16, 1082, 217}
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.
\(\displaystyle \int \frac {x^5}{\sqrt [3]{1-x^3} \left (x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {x^3}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{3} \left (-\int \frac {1}{\sqrt [3]{1-x^3} \left (x^3+1\right )}dx^3-\frac {3}{2} \left (1-x^3\right )^{2/3}\right )\) |
\(\Big \downarrow \) 67 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \int \frac {1}{\sqrt [3]{2}-\sqrt [3]{1-x^3}}d\sqrt [3]{1-x^3}}{2 \sqrt [3]{2}}-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}-\frac {3}{2} \left (1-x^3\right )^{2/3}+\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}-\frac {3}{2} \left (1-x^3\right )^{2/3}+\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \int \frac {1}{-x^6-3}d\left (2^{2/3} \sqrt [3]{1-x^3}+1\right )}{\sqrt [3]{2}}-\frac {3}{2} \left (1-x^3\right )^{2/3}+\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} \left (-\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {3}{2} \left (1-x^3\right )^{2/3}+\frac {\log \left (x^3+1\right )}{2 \sqrt [3]{2}}-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\right )\) |
((-3*(1 - x^3)^(2/3))/2 - (Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sq rt[3]])/2^(1/3) + Log[1 + x^3]/(2*2^(1/3)) - (3*Log[2^(1/3) - (1 - x^3)^(1 /3)])/(2*2^(1/3)))/3
3.7.8.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Time = 9.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(-\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{2}-\frac {2^{\frac {2}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )}{6}+\frac {2^{\frac {2}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )}{12}-\frac {\arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) 2^{\frac {2}{3}} \sqrt {3}}{6}\) | \(95\) |
trager | \(\text {Expression too large to display}\) | \(668\) |
risch | \(\text {Expression too large to display}\) | \(778\) |
-1/2*(-x^3+1)^(2/3)-1/6*2^(2/3)*ln((-x^3+1)^(1/3)-2^(1/3))+1/12*2^(2/3)*ln ((-x^3+1)^(2/3)+2^(1/3)*(-x^3+1)^(1/3)+2^(2/3))-1/6*arctan(1/3*(1+2^(2/3)* (-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)
Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.28 \[ \int \frac {x^5}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (2 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} \]
-1/6*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(1/6*2^(1/6)*(2*sqrt(6)*(-1)^(1/3)*( -x^3 + 1)^(1/3) - sqrt(6)*2^(1/3))) - 1/12*2^(2/3)*(-1)^(1/3)*log(2^(1/3)* (-1)^(2/3)*(-x^3 + 1)^(1/3) - 2^(2/3)*(-1)^(1/3) + (-x^3 + 1)^(2/3)) + 1/6 *2^(2/3)*(-1)^(1/3)*log(-2^(1/3)*(-1)^(2/3) + (-x^3 + 1)^(1/3)) - 1/2*(-x^ 3 + 1)^(2/3)
\[ \int \frac {x^5}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {x^{5}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} \]
-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1 /3))) + 1/12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^( 2/3)) - 1/6*2^(2/3)*log(-2^(1/3) + (-x^3 + 1)^(1/3)) - 1/2*(-x^3 + 1)^(2/3 )
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} \]
-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1 /3))) + 1/12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^( 2/3)) - 1/6*2^(2/3)*log(abs(-2^(1/3) + (-x^3 + 1)^(1/3))) - 1/2*(-x^3 + 1) ^(2/3)
Time = 8.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.13 \[ \int \frac {x^5}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-2^{1/3}\right )}{6}-\frac {{\left (1-x^3\right )}^{2/3}}{2}-\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}+\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \]