Integrand size = 22, antiderivative size = 125 \[ \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\sqrt [3]{1-x^3}+\frac {1}{4} \left (1-x^3\right )^{4/3}-\frac {1}{7} \left (1-x^3\right )^{7/3}+\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \] Output:
-(-x^3+1)^(1/3)+1/4*(-x^3+1)^(4/3)-1/7*(-x^3+1)^(7/3)+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 = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10 \[ \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{84} \left (3 \sqrt [3]{1-x^3} \left (-25+x^3-4 x^6\right )+14 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-14 \sqrt [3]{2} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )+7 \sqrt [3]{2} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \] Input:
Integrate[x^11/((1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
(3*(1 - x^3)^(1/3)*(-25 + x^3 - 4*x^6) + 14*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^ (2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 14*2^(1/3)*Log[-2 + 2^(2/3)*(1 - x^3)^(1 /3)] + 7*2^(1/3)*Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2/3) ])/84
Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {948, 99, 2009}
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.
\(\displaystyle \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {x^9}{\left (1-x^3\right )^{2/3} \left (x^3+1\right )}dx^3\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{3} \int \left (\left (1-x^3\right )^{4/3}-\sqrt [3]{1-x^3}-\frac {1}{\left (x^3+1\right ) \left (1-x^3\right )^{2/3}}+\frac {1}{\left (1-x^3\right )^{2/3}}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3}}-\frac {3}{7} \left (1-x^3\right )^{7/3}+\frac {3}{4} \left (1-x^3\right )^{4/3}-3 \sqrt [3]{1-x^3}+\frac {\log \left (x^3+1\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\right )\) |
Input:
Int[x^11/((1 - x^3)^(2/3)*(1 + x^3)),x]
Output:
(-3*(1 - x^3)^(1/3) + (3*(1 - x^3)^(4/3))/4 - (3*(1 - x^3)^(7/3))/7 + (Sqr t[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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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]]
Time = 12.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(\frac {\left (-4 x^{6}+x^{3}-25\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{28}+\frac {2^{\frac {1}{3}} \left (2 \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}+\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )-2 \ln \left (\left (-x^{3}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )\right )}{12}\) | \(100\) |
trager | \(\text {Expression too large to display}\) | \(722\) |
risch | \(\text {Expression too large to display}\) | \(1579\) |
Input:
int(x^11/(-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)
Output:
1/28*(-4*x^6+x^3-25)*(-x^3+1)^(1/3)+1/12*2^(1/3)*(2*arctan(1/3*(1+2^(2/3)* (-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+ln((-x^3+1)^(2/3)+2^(1/3)*(-x^3+1)^(1/3)+ 2^(2/3))-2*ln((-x^3+1)^(1/3)-2^(1/3)))
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} \arctan \left (\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}}\right )}\right ) + \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2 \cdot 4^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {2}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - \frac {1}{28} \, {\left (4 \, x^{6} - x^{3} + 25\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \] Input:
integrate(x^11/(-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)) - 1/2 8*(4*x^6 - x^3 + 25)*(-x^3 + 1)^(1/3)
\[ \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{11}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \] Input:
integrate(x**11/(-x**3+1)**(2/3)/(x**3+1),x)
Output:
Integral(x**11/((-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x)
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.95 \[ \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{7} \, {\left (-x^{3} + 1\right )}^{\frac {7}{3}} + \frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{4} \, {\left (-x^{3} + 1\right )}^{\frac {4}{3}} + \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \] Input:
integrate(x^11/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")
Output:
-1/7*(-x^3 + 1)^(7/3) + 1/6*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^ (1/3) + 2*(-x^3 + 1)^(1/3))) + 1/4*(-x^3 + 1)^(4/3) + 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 = 127, normalized size of antiderivative = 1.02 \[ \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=-\frac {1}{7} \, {\left (x^{3} - 1\right )}^{2} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{6} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \frac {1}{4} \, {\left (-x^{3} + 1\right )}^{\frac {4}{3}} + \frac {1}{12} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \] Input:
integrate(x^11/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")
Output:
-1/7*(x^3 - 1)^2*(-x^3 + 1)^(1/3) + 1/6*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3) *2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/3))) + 1/4*(-x^3 + 1)^(4/3) + 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 = 3.67 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08 \[ \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\frac {{\left (1-x^3\right )}^{4/3}}{4}-{\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,\ln \left (3\,2^{1/3}-3\,{\left (1-x^3\right )}^{1/3}\right )}{6}-\frac {{\left (1-x^3\right )}^{7/3}}{7}-\frac {2^{1/3}\,\ln \left (3\,{\left (1-x^3\right )}^{1/3}-\frac {3\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}+\frac {2^{1/3}\,\ln \left (\frac {3\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+3\,{\left (1-x^3\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \] Input:
int(x^11/((1 - x^3)^(2/3)*(x^3 + 1)),x)
Output:
(1 - x^3)^(4/3)/4 - (1 - x^3)^(1/3) - (2^(1/3)*log(3*2^(1/3) - 3*(1 - x^3) ^(1/3)))/6 - (1 - x^3)^(7/3)/7 - (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((3*2^(1/3)*(3^ (1/2)*1i + 1))/2 + 3*(1 - x^3)^(1/3))*(3^(1/2)*1i + 1))/12
\[ \int \frac {x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx=\int \frac {x^{11}}{\left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}+\left (-x^{3}+1\right )^{\frac {2}{3}}}d x \] Input:
int(x^11/(-x^3+1)^(2/3)/(x^3+1),x)
Output:
int(x**11/(( - x**3 + 1)**(2/3)*x**3 + ( - x**3 + 1)**(2/3)),x)