3.21.11 \(\int \frac {(-1+x^3) (1+x^6)^{2/3} (1-x^3+x^6)}{x^6 (1+x^3)} \, dx\) [2011]

3.21.11.1 Optimal result

Integrand size = 35, antiderivative size = 142 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx=\frac {\left (1+x^6\right )^{2/3} \left (2-15 x^3+2 x^6\right )}{10 x^5}-2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^6}}\right )-2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^6}-\sqrt [3]{2} \left (1+x^6\right )^{2/3}\right )}{\sqrt [3]{2}} \]

output

1/10*(x^6+1)^(2/3)*(2*x^6-15*x^3+2)/x^5-2^(2/3)*3^(1/2)*arctan(3^(1/2)*x/( 
-x+2^(2/3)*(x^6+1)^(1/3)))-2^(2/3)*ln(2*x+2^(2/3)*(x^6+1)^(1/3))+1/2*ln(-2 
*x^2+2^(2/3)*x*(x^6+1)^(1/3)-2^(1/3)*(x^6+1)^(2/3))*2^(2/3)
 

3.21.11.2 Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx=\frac {\left (1+x^6\right )^{2/3} \left (2-15 x^3+2 x^6\right )}{10 x^5}-2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^6}}\right )-2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^6}-\sqrt [3]{2} \left (1+x^6\right )^{2/3}\right )}{\sqrt [3]{2}} \]

input

Integrate[((-1 + x^3)*(1 + x^6)^(2/3)*(1 - x^3 + x^6))/(x^6*(1 + x^3)),x]
 

output

((1 + x^6)^(2/3)*(2 - 15*x^3 + 2*x^6))/(10*x^5) - 2^(2/3)*Sqrt[3]*ArcTan[( 
Sqrt[3]*x)/(-x + 2^(2/3)*(1 + x^6)^(1/3))] - 2^(2/3)*Log[2*x + 2^(2/3)*(1 
+ x^6)^(1/3)] + Log[-2*x^2 + 2^(2/3)*x*(1 + x^6)^(1/3) - 2^(1/3)*(1 + x^6) 
^(2/3)]/2^(1/3)
 

3.21.11.3 Rubi [F]

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.

\(\displaystyle \int \frac {\left (x^3-1\right ) \left (x^6+1\right )^{2/3} \left (x^6-x^3+1\right )}{x^6 \left (x^3+1\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (-\frac {2 \left (x^6+1\right )^{2/3}}{x+1}-\frac {\left (x^6+1\right )^{2/3}}{x^6}+\left (x^6+1\right )^{2/3}+\frac {3 \left (x^6+1\right )^{2/3}}{x^3}+\frac {2 \left (x^6+1\right )^{2/3} (x-2)}{x^2-x+1}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {\left (x^6+1\right )^{2/3}}{x+1}dx+2 \left (1+i \sqrt {3}\right ) \int \frac {\left (x^6+1\right )^{2/3}}{2 x-i \sqrt {3}-1}dx+2 \left (1-i \sqrt {3}\right ) \int \frac {\left (x^6+1\right )^{2/3}}{2 x+i \sqrt {3}-1}dx+x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},-x^6\right )+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},-x^6\right )}{5 x^5}-\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{2 x^2}\)

input

Int[((-1 + x^3)*(1 + x^6)^(2/3)*(1 - x^3 + x^6))/(x^6*(1 + x^3)),x]
 

output

$Aborted
 

3.21.11.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

3.21.11.4 Maple [A] (verified)

Time = 9.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {-10 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{6}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (2 x^{6}-15 x^{3}+2\right ) \left (x^{6}+1\right )^{\frac {2}{3}}+5 \,2^{\frac {2}{3}} x^{5} \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x -2^{\frac {2}{3}} \left (x^{6}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} x \left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{10 x^{5}}\)	\(123\)

input

int((x^3-1)*(x^6+1)^(2/3)*(x^6-x^3+1)/x^6/(x^3+1),x,method=_RETURNVERBOSE)
 

output

1/10*(-10*2^(2/3)*ln((2^(1/3)*x+(x^6+1)^(1/3))/x)*x^5+(2*x^6-15*x^3+2)*(x^ 
6+1)^(2/3)+5*2^(2/3)*x^5*(-2*arctan(1/3*3^(1/2)*(x-2^(2/3)*(x^6+1)^(1/3))/ 
x)*3^(1/2)+ln((2^(2/3)*x^2-2^(1/3)*x*(x^6+1)^(1/3)+(x^6+1)^(2/3))/x^2)))/x 
^5
 

3.21.11.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (113) = 226\).

Time = 32.33 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.42 \[ \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx=\frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (x^{13} - 2 \, x^{10} - 6 \, x^{7} - 2 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{14} - 14 \, x^{11} + 6 \, x^{8} - 14 \, x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (x^{18} - 30 \, x^{15} + 51 \, x^{12} - 52 \, x^{9} + 51 \, x^{6} - 30 \, x^{3} + 1\right )}}{3 \, {\left (x^{18} + 6 \, x^{15} - 93 \, x^{12} + 20 \, x^{9} - 93 \, x^{6} + 6 \, x^{3} + 1\right )}}\right ) + 10 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - 5 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{7} - 4 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \left (-4\right )^{\frac {2}{3}} {\left (x^{12} - 14 \, x^{9} + 6 \, x^{6} - 14 \, x^{3} + 1\right )} + 24 \, {\left (x^{8} - x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) + 3 \, {\left (2 \, x^{6} - 15 \, x^{3} + 2\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]

input

integrate((x^3-1)*(x^6+1)^(2/3)*(x^6-x^3+1)/x^6/(x^3+1),x, algorithm="fric 
as")
 

output

1/30*(10*sqrt(3)*(-4)^(1/3)*x^5*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(x^13 - 2 
*x^10 - 6*x^7 - 2*x^4 + x)*(x^6 + 1)^(2/3) + 6*sqrt(3)*(-4)^(1/3)*(x^14 - 
14*x^11 + 6*x^8 - 14*x^5 + x^2)*(x^6 + 1)^(1/3) - sqrt(3)*(x^18 - 30*x^15 
+ 51*x^12 - 52*x^9 + 51*x^6 - 30*x^3 + 1))/(x^18 + 6*x^15 - 93*x^12 + 20*x 
^9 - 93*x^6 + 6*x^3 + 1)) + 10*(-4)^(1/3)*x^5*log((3*(-4)^(2/3)*(x^6 + 1)^ 
(1/3)*x^2 + 6*(x^6 + 1)^(2/3)*x - (-4)^(1/3)*(x^6 + 2*x^3 + 1))/(x^6 + 2*x 
^3 + 1)) - 5*(-4)^(1/3)*x^5*log((6*(-4)^(1/3)*(x^7 - 4*x^4 + x)*(x^6 + 1)^ 
(2/3) + (-4)^(2/3)*(x^12 - 14*x^9 + 6*x^6 - 14*x^3 + 1) + 24*(x^8 - x^5 + 
x^2)*(x^6 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 1)) + 3*(2*x^6 - 15* 
x^3 + 2)*(x^6 + 1)^(2/3))/x^5
 

3.21.11.6 Sympy [F]

\[ \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx=\int \frac {\left (\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (x^{2} + x + 1\right ) \left (x^{6} - x^{3} + 1\right )}{x^{6} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

input

integrate((x**3-1)*(x**6+1)**(2/3)*(x**6-x**3+1)/x**6/(x**3+1),x)
 

output

Integral(((x**2 + 1)*(x**4 - x**2 + 1))**(2/3)*(x - 1)*(x**2 + x + 1)*(x** 
6 - x**3 + 1)/(x**6*(x + 1)*(x**2 - x + 1)), x)
 

3.21.11.7 Maxima [F]

\[ \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{6}} \,d x } \]

input

integrate((x^3-1)*(x^6+1)^(2/3)*(x^6-x^3+1)/x^6/(x^3+1),x, algorithm="maxi 
ma")
 

output

integrate((x^6 - x^3 + 1)*(x^6 + 1)^(2/3)*(x^3 - 1)/((x^3 + 1)*x^6), x)
 

3.21.11.8 Giac [F]

\[ \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{6}} \,d x } \]

input

integrate((x^3-1)*(x^6+1)^(2/3)*(x^6-x^3+1)/x^6/(x^3+1),x, algorithm="giac 
")
 

output

integrate((x^6 - x^3 + 1)*(x^6 + 1)^(2/3)*(x^3 - 1)/((x^3 + 1)*x^6), x)
 

3.21.11.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^3\right ) \left (1+x^6\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (1+x^3\right )} \, dx=\int \frac {\left (x^3-1\right )\,{\left (x^6+1\right )}^{2/3}\,\left (x^6-x^3+1\right )}{x^6\,\left (x^3+1\right )} \,d x \]

input

int(((x^3 - 1)*(x^6 + 1)^(2/3)*(x^6 - x^3 + 1))/(x^6*(x^3 + 1)),x)
 

output

int(((x^3 - 1)*(x^6 + 1)^(2/3)*(x^6 - x^3 + 1))/(x^6*(x^3 + 1)), x)