\(\int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} (-4+5 x^3-4 x^6+x^9)} \, dx\) [561]

Optimal result

Integrand size = 38, antiderivative size = 43 \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\frac {1}{3} \arctan \left (\frac {1-x^3}{\sqrt [4]{1+x^6}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {-1+x^3}{\sqrt [4]{1+x^6}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 8.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\frac {1}{3} \arctan \left (\frac {1-x^3}{\sqrt [4]{1+x^6}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {-1+x^3}{\sqrt [4]{1+x^6}}\right ) \] Input:

Integrate[(2 + x^3 + x^6)/(x*(1 + x^6)^(1/4)*(-4 + 5*x^3 - 4*x^6 + x^9)),x 
]
 

Output:

ArcTan[(1 - x^3)/(1 + x^6)^(1/4)]/3 - ArcTanh[(-1 + x^3)/(1 + x^6)^(1/4)]/ 
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.

\(\displaystyle \int \frac {x^6+x^3+2}{x \sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )} \, dx\)

\(\Big \downarrow \) 7282

\(\displaystyle \frac {1}{3} \int -\frac {x^6+x^3+2}{x^3 \sqrt [4]{x^6+1} \left (-x^9+4 x^6-5 x^3+4\right )}dx^3\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{3} \int \frac {x^6+x^3+2}{x^3 \sqrt [4]{x^6+1} \left (-x^9+4 x^6-5 x^3+4\right )}dx^3\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{3} \int \left (\frac {-x^6+2 x^3-7}{2 \sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )}+\frac {1}{2 x^3 \sqrt [4]{x^6+1}}\right )dx^3\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {7}{2} \int \frac {1}{\sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )}dx^3-\int \frac {x^3}{\sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )}dx^3+\frac {1}{2} \int \frac {x^6}{\sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )}dx^3-\frac {1}{2} \arctan \left (\sqrt [4]{x^6+1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{x^6+1}\right )\right )\)

Input:

Int[(2 + x^3 + x^6)/(x*(1 + x^6)^(1/4)*(-4 + 5*x^3 - 4*x^6 + x^9)),x]
 

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 88.39 (sec) , antiderivative size = 346, normalized size of antiderivative = 8.05

method	result	size
trager	\(-\frac {\ln \left (-\frac {x^{12}+2 \left (x^{6}+1\right )^{\frac {1}{4}} x^{9}-4 x^{9}+2 \sqrt {x^{6}+1}\, x^{6}-6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x^{3}+7 x^{6}-4 x^{3} \sqrt {x^{6}+1}+6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-2 \left (x^{6}+1\right )^{\frac {3}{4}}-4 x^{3}+2 \sqrt {x^{6}+1}-2 \left (x^{6}+1\right )^{\frac {1}{4}}+2}{x^{3} \left (x^{9}-4 x^{6}+5 x^{3}-4\right )}\right )}{6}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{12}-2 \left (x^{6}+1\right )^{\frac {1}{4}} x^{9}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{9}-2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x^{3}+4 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{6}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+2 \left (x^{6}+1\right )^{\frac {1}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3} \left (x^{9}-4 x^{6}+5 x^{3}-4\right )}\right )}{6}\)	\(346\)

Input:

int((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x,method=_RETURNVERBOS 
E)
 

Output:

-1/6*ln(-(x^12+2*(x^6+1)^(1/4)*x^9-4*x^9+2*(x^6+1)^(1/2)*x^6-6*(x^6+1)^(1/ 
4)*x^6+2*(x^6+1)^(3/4)*x^3+7*x^6-4*x^3*(x^6+1)^(1/2)+6*(x^6+1)^(1/4)*x^3-2 
*(x^6+1)^(3/4)-4*x^3+2*(x^6+1)^(1/2)-2*(x^6+1)^(1/4)+2)/x^3/(x^9-4*x^6+5*x 
^3-4))-1/6*RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^12-2*(x^6+1)^(1/4)*x^9-4*R 
ootOf(_Z^2+1)*x^9-2*(x^6+1)^(1/2)*RootOf(_Z^2+1)*x^6+6*(x^6+1)^(1/4)*x^6+7 
*RootOf(_Z^2+1)*x^6+2*(x^6+1)^(3/4)*x^3+4*(x^6+1)^(1/2)*RootOf(_Z^2+1)*x^3 
-6*(x^6+1)^(1/4)*x^3-4*RootOf(_Z^2+1)*x^3-2*(x^6+1)^(3/4)-2*(x^6+1)^(1/2)* 
RootOf(_Z^2+1)+2*(x^6+1)^(1/4)+2*RootOf(_Z^2+1))/x^3/(x^9-4*x^6+5*x^3-4))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (34) = 68\).

Time = 6.70 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.88 \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\frac {1}{6} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )} + {\left (x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}\right )}}{x^{12} - 4 \, x^{9} + 5 \, x^{6} - 4 \, x^{3}}\right ) + \frac {1}{6} \, \log \left (-\frac {x^{12} - 4 \, x^{9} + 7 \, x^{6} - 4 \, x^{3} - 2 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )} + 2 \, {\left (x^{6} - 2 \, x^{3} + 1\right )} \sqrt {x^{6} + 1} - 2 \, {\left (x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 2}{x^{12} - 4 \, x^{9} + 5 \, x^{6} - 4 \, x^{3}}\right ) \] Input:

integrate((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x, algorithm="fr 
icas")
 

Output:

1/6*arctan(2*((x^6 + 1)^(3/4)*(x^3 - 1) + (x^9 - 3*x^6 + 3*x^3 - 1)*(x^6 + 
 1)^(1/4))/(x^12 - 4*x^9 + 5*x^6 - 4*x^3)) + 1/6*log(-(x^12 - 4*x^9 + 7*x^ 
6 - 4*x^3 - 2*(x^6 + 1)^(3/4)*(x^3 - 1) + 2*(x^6 - 2*x^3 + 1)*sqrt(x^6 + 1 
) - 2*(x^9 - 3*x^6 + 3*x^3 - 1)*(x^6 + 1)^(1/4) + 2)/(x^12 - 4*x^9 + 5*x^6 
 - 4*x^3))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\text {Timed out} \] Input:

integrate((x**6+x**3+2)/x/(x**6+1)**(1/4)/(x**9-4*x**6+5*x**3-4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\int { \frac {x^{6} + x^{3} + 2}{{\left (x^{9} - 4 \, x^{6} + 5 \, x^{3} - 4\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x} \,d x } \] Input:

integrate((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x, algorithm="ma 
xima")
 

Output:

integrate((x^6 + x^3 + 2)/((x^9 - 4*x^6 + 5*x^3 - 4)*(x^6 + 1)^(1/4)*x), x 
)
 

Giac [F]

\[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\int { \frac {x^{6} + x^{3} + 2}{{\left (x^{9} - 4 \, x^{6} + 5 \, x^{3} - 4\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x} \,d x } \] Input:

integrate((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x, algorithm="gi 
ac")
 

Output:

integrate((x^6 + x^3 + 2)/((x^9 - 4*x^6 + 5*x^3 - 4)*(x^6 + 1)^(1/4)*x), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\int \frac {x^6+x^3+2}{x\,{\left (x^6+1\right )}^{1/4}\,\left (x^9-4\,x^6+5\,x^3-4\right )} \,d x \] Input:

int((x^3 + x^6 + 2)/(x*(x^6 + 1)^(1/4)*(5*x^3 - 4*x^6 + x^9 - 4)),x)
 

Output:

int((x^3 + x^6 + 2)/(x*(x^6 + 1)^(1/4)*(5*x^3 - 4*x^6 + x^9 - 4)), x)
 

Reduce [F]

\[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\int \frac {x^{5}}{\left (x^{6}+1\right )^{\frac {1}{4}} x^{9}-4 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+5 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-4 \left (x^{6}+1\right )^{\frac {1}{4}}}d x +\int \frac {x^{2}}{\left (x^{6}+1\right )^{\frac {1}{4}} x^{9}-4 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+5 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-4 \left (x^{6}+1\right )^{\frac {1}{4}}}d x +2 \left (\int \frac {1}{\left (x^{6}+1\right )^{\frac {1}{4}} x^{10}-4 \left (x^{6}+1\right )^{\frac {1}{4}} x^{7}+5 \left (x^{6}+1\right )^{\frac {1}{4}} x^{4}-4 \left (x^{6}+1\right )^{\frac {1}{4}} x}d x \right ) \] Input:

int((x^6+x^3+2)/x/(x^6+1)^(1/4)/(x^9-4*x^6+5*x^3-4),x)
 

Output:

int(x**5/((x**6 + 1)**(1/4)*x**9 - 4*(x**6 + 1)**(1/4)*x**6 + 5*(x**6 + 1) 
**(1/4)*x**3 - 4*(x**6 + 1)**(1/4)),x) + int(x**2/((x**6 + 1)**(1/4)*x**9 
- 4*(x**6 + 1)**(1/4)*x**6 + 5*(x**6 + 1)**(1/4)*x**3 - 4*(x**6 + 1)**(1/4 
)),x) + 2*int(1/((x**6 + 1)**(1/4)*x**10 - 4*(x**6 + 1)**(1/4)*x**7 + 5*(x 
**6 + 1)**(1/4)*x**4 - 4*(x**6 + 1)**(1/4)*x),x)