\(\int \frac {x^3 (2+x^3)}{(1+x^3)^{2/3} (1+x^3+x^6)} \, dx\) [7]

Optimal result

Integrand size = 28, antiderivative size = 85 \[ \int \frac {x^3 \left (2+x^3\right )}{\left (1+x^3\right )^{2/3} \left (1+x^3+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2-2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (x^2+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (x^4-x^2 \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \] Output:

-1/3*arctan(3^(1/2)*x^2/(x^2-2*(x^3+1)^(1/3)))*3^(1/2)-1/3*ln(x^2+(x^3+1)^ 
(1/3))+1/6*ln(x^4-x^2*(x^3+1)^(1/3)+(x^3+1)^(2/3))
 

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \left (2+x^3\right )}{\left (1+x^3\right )^{2/3} \left (1+x^3+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2-2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (x^2+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (x^4-x^2 \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \] Input:

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

Output:

-(ArcTan[(Sqrt[3]*x^2)/(x^2 - 2*(1 + x^3)^(1/3))]/Sqrt[3]) - Log[x^2 + (1 
+ x^3)^(1/3)]/3 + Log[x^4 - x^2*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.65 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7279, 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.

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (-\sqrt {3}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-x^3,-\frac {2 x^3}{1-i \sqrt {3}}\right )}{\sqrt {3}+i}+\frac {\left (\sqrt {3}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-x^3,-\frac {2 x^3}{1+i \sqrt {3}}\right )}{-\sqrt {3}+i}+x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-x^3\right )\)

Input:

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

Output:

((I - Sqrt[3])*x*AppellF1[1/3, 2/3, 1, 4/3, -x^3, (-2*x^3)/(1 - I*Sqrt[3]) 
])/(I + Sqrt[3]) + ((I + Sqrt[3])*x*AppellF1[1/3, 2/3, 1, 4/3, -x^3, (-2*x 
^3)/(1 + I*Sqrt[3])])/(I - Sqrt[3]) + x*Hypergeometric2F1[1/3, 2/3, 4/3, - 
x^3]
 

Defintions of rubi rules used

rule 2009

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

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.63 (sec) , antiderivative size = 473, normalized size of antiderivative = 5.56

method	result	size
trager	\(-\frac {\ln \left (-\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{6}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}-3 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+x^{6}+3 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-\left (x^{3}+1\right )^{\frac {1}{3}} x^{4}-2 \left (x^{3}+1\right )^{\frac {2}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-1}{x^{6}+x^{3}+1}\right )}{3}+\frac {\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{6}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+6 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+3 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-\left (x^{3}+1\right )^{\frac {1}{3}} x^{4}+\left (x^{3}+1\right )^{\frac {2}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}+1}\right )}{3}-\ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{6}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}+6 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+3 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-\left (x^{3}+1\right )^{\frac {1}{3}} x^{4}+\left (x^{3}+1\right )^{\frac {2}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )\)	\(473\)

Input:

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

Output:

-1/3*ln(-(9*RootOf(9*_Z^2-3*_Z+1)^2*x^6-6*RootOf(9*_Z^2-3*_Z+1)*x^6-3*(x^3 
+1)^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^4+x^6+3*(x^3+1)^(2/3)*RootOf(9*_Z^2-3*_Z 
+1)*x^2-(x^3+1)^(1/3)*x^4-2*(x^3+1)^(2/3)*x^2+3*RootOf(9*_Z^2-3*_Z+1)*x^3- 
x^3+3*RootOf(9*_Z^2-3*_Z+1)-1)/(x^6+x^3+1))+1/3*ln((9*RootOf(9*_Z^2-3*_Z+1 
)^2*x^6-3*RootOf(9*_Z^2-3*_Z+1)*x^6+6*(x^3+1)^(1/3)*RootOf(9*_Z^2-3*_Z+1)* 
x^4+3*(x^3+1)^(2/3)*RootOf(9*_Z^2-3*_Z+1)*x^2-(x^3+1)^(1/3)*x^4+(x^3+1)^(2 
/3)*x^2-3*RootOf(9*_Z^2-3*_Z+1)*x^3+x^3-3*RootOf(9*_Z^2-3*_Z+1)+1)/(x^6+x^ 
3+1))-ln((9*RootOf(9*_Z^2-3*_Z+1)^2*x^6-3*RootOf(9*_Z^2-3*_Z+1)*x^6+6*(x^3 
+1)^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^4+3*(x^3+1)^(2/3)*RootOf(9*_Z^2-3*_Z+1)* 
x^2-(x^3+1)^(1/3)*x^4+(x^3+1)^(2/3)*x^2-3*RootOf(9*_Z^2-3*_Z+1)*x^3+x^3-3* 
RootOf(9*_Z^2-3*_Z+1)+1)/(x^6+x^3+1))*RootOf(9*_Z^2-3*_Z+1)
 

Fricas [A] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \left (2+x^3\right )}{\left (1+x^3\right )^{2/3} \left (1+x^3+x^6\right )} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x^{2} - 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {1}{6} \, \log \left (\frac {x^{6} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{4} + x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x^{2} + 1}{x^{6} + x^{3} + 1}\right ) \] Input:

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

Output:

-1/3*sqrt(3)*arctan(-1/3*(sqrt(3)*x^2 - 2*sqrt(3)*(x^3 + 1)^(1/3))/x^2) - 
1/6*log((x^6 + 3*(x^3 + 1)^(1/3)*x^4 + x^3 + 3*(x^3 + 1)^(2/3)*x^2 + 1)/(x 
^6 + x^3 + 1))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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