Integrand size = 22, antiderivative size = 95 \[ \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\sqrt [3]{1-x^3}+\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \] Output:
-(-x^3+1)^(1/3)+1/6*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(1/3) *3^(1/2)+1/12*ln(x^3+1)*2^(1/3)-1/4*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(1/3)
Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.33 \[ \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{12} \left (-12 \sqrt [3]{1-x^3}+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-2 \sqrt [3]{2} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )+\sqrt [3]{2} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[x^5/((1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
(-12*(1 - x^3)^(1/3) + 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/ 3))/Sqrt[3]] - 2*2^(1/3)*Log[-2 + 2^(2/3)*(1 - x^3)^(1/3)] + 2^(1/3)*Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2/3)])/12
Time = 0.37 (sec) , antiderivative size = 99, 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, 69, 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.
\(\displaystyle \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx^3\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{3} \left (-\int \frac {1}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx^3-3 \sqrt [3]{1-x^3}\right )\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \int \frac {1}{\sqrt [3]{2}-\sqrt [3]{1-x^3}}d\sqrt [3]{1-x^3}}{2\ 2^{2/3}}+\frac {3 \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}}{2 \sqrt [3]{2}}-3 \sqrt [3]{1-x^3}+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \int \frac {1}{x^6+\sqrt [3]{2} \sqrt [3]{1-x^3}+2^{2/3}}d\sqrt [3]{1-x^3}}{2 \sqrt [3]{2}}-3 \sqrt [3]{1-x^3}+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{3} \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (2^{2/3} \sqrt [3]{1-x^3}+1\right )}{2^{2/3}}-3 \sqrt [3]{1-x^3}+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3}}-3 \sqrt [3]{1-x^3}+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
Input:
Int[x^5/((1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
(-3*(1 - x^3)^(1/3) + (Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3 ]])/2^(2/3) + Log[1 + x^3]/(2*2^(2/3)) - (3*Log[2^(1/3) - (1 - x^3)^(1/3)] )/(2*2^(2/3)))/3
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_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) 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 = 9.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(-\left (-x^{3}+1\right )^{\frac {1}{3}}-\frac {2^{\frac {1}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )}{6}+\frac {2^{\frac {1}{3}} \ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )}{12}+\frac {\arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6}\) | \(95\) |
trager | \(\text {Expression too large to display}\) | \(711\) |
risch | \(\text {Expression too large to display}\) | \(1061\) |
Input:
int(x^5/(-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
-(-x^3+1)^(1/3)-1/6*2^(1/3)*ln((-x^3+1)^(1/3)-2^(1/3))+1/12*2^(1/3)*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^(1/3)*3^(1/2)
Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}}\right )}\right ) + \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2 \cdot 4^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {2}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \] Input:
integrate(x^5/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")
Output:
1/2*4^(1/6)*sqrt(1/3)*arctan(1/2*4^(1/6)*sqrt(1/3)*(4^(2/3)*(-x^3 + 1)^(1/ 3) + 4^(1/3))) + 1/24*4^(2/3)*log(4^(2/3)*(-x^3 + 1)^(1/3) + 2*(-x^3 + 1)^ (2/3) + 2*4^(1/3)) - 1/12*4^(2/3)*log(-4^(2/3) + 2*(-x^3 + 1)^(1/3)) - (-x ^3 + 1)^(1/3)
\[ \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{5}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(x**5/(-x**3+1)**(2/3)/(x**3+1),x)
Output:
Integral(x**5/((-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x )
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02 \[ \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \] Input:
integrate(x^5/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")
Output:
1/6*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/ 3))) + 1/12*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2 /3)) - 1/6*2^(1/3)*log(-2^(1/3) + (-x^3 + 1)^(1/3)) - (-x^3 + 1)^(1/3)
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.03 \[ \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \] Input:
integrate(x^5/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")
Output:
1/6*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/ 3))) + 1/12*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2 /3)) - 1/6*2^(1/3)*log(abs(-2^(1/3) + (-x^3 + 1)^(1/3))) - (-x^3 + 1)^(1/3 )
Time = 3.69 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.19 \[ \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{2}-\frac {2^{1/3}}{2}\right )}{6}-{\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,\ln \left (3\,{\left (1-x^3\right )}^{1/3}-\frac {3\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}+\frac {2^{1/3}\,\ln \left (\frac {3\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+3\,{\left (1-x^3\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \] Input:
int(x^5/((1 - x^3)^(2/3)*(x^3 + 1)),x)
Output:
(2^(1/3)*log((3*2^(1/3)*(3^(1/2)*1i + 1))/2 + 3*(1 - x^3)^(1/3))*(3^(1/2)* 1i + 1))/12 - (1 - x^3)^(1/3) - (2^(1/3)*log(3*(1 - x^3)^(1/3) - (3*2^(1/3 )*(3^(1/2)*1i - 1))/2)*(3^(1/2)*1i - 1))/12 - (2^(1/3)*log((1 - x^3)^(1/3) /2 - 2^(1/3)/2))/6
\[ \int \frac {x^5}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{5}}{\left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}+\left (-x^{3}+1\right )^{\frac {2}{3}}}d x \] Input:
int(x^5/(-x^3+1)^(2/3)/(x^3+1),x)
Output:
int(x**5/(( - x**3 + 1)**(2/3)*x**3 + ( - x**3 + 1)**(2/3)),x)