\(\int \frac {(-4+x^3) (1-x^3+x^4)}{x^2 (-1+x^3)^{3/4} (-1+x^3+x^4)} \, dx\) [1318]

Optimal result

Integrand size = 38, antiderivative size = 95 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\frac {4 \sqrt [4]{-1+x^3}}{x}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 3.65 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\frac {4 \sqrt [4]{-1+x^3}}{x}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x}{\left (x^3-1\right )^{3/4}}-\frac {2}{\left (x^3-1\right )^{3/4}}+\frac {4}{\left (x^3-1\right )^{3/4} x^2}+\frac {2 \left (x^3-4 x^2+x-1\right )}{\left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \frac {1}{\left (x^3-1\right )^{3/4} \left (-x^4-x^3+1\right )}dx+2 \int \frac {x}{\left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )}dx+2 \int \frac {x^3}{\left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )}dx-8 \int \frac {x^2}{\left (x^3-1\right )^{3/4} \left (x^4+x^3-1\right )}dx-\frac {2 \left (1-x^3\right )^{3/4} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},x^3\right )}{\left (x^3-1\right )^{3/4}}-\frac {4 \left (1-x^3\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{4},\frac {2}{3},x^3\right )}{\left (x^3-1\right )^{3/4} x}+\frac {\left (1-x^3\right )^{3/4} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},x^3\right )}{2 \left (x^3-1\right )^{3/4}}\)

Input:

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

Output:

$Aborted
 

Maple [C] (verified)

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

Time = 5.20 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.32

method	result	size
trager	\(\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}-1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}-1}\right )\)	\(220\)
risch	\(\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}+\frac {\left (2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {3}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x^{7}+3 x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{3}-1\right ) \left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x^{7}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{9}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x^{4}+3 x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {3}{4}} x^{3}+2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{3}-1\right ) \left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )\right ) {\left (\left (x^{3}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{3}-1\right )^{\frac {3}{4}}}\)	\(587\)

Input:

int((x^3-4)*(x^4-x^3+1)/x^2/(x^3-1)^(3/4)/(x^4+x^3-1),x,method=_RETURNVERB 
OSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (78) = 156\).

Time = 15.19 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.03 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=-\frac {4 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x}{x^{4} - x^{3} + 1}\right ) + \sqrt {2} x \log \left (\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} + x^{3} - 1}\right ) - \sqrt {2} x \log \left (\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} + x^{3} - 1}\right ) - 8 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{2 \, x} \] Input:

integrate((x^3-4)*(x^4-x^3+1)/x^2/(x^3-1)^(3/4)/(x^4+x^3-1),x, algorithm=" 
fricas")
 

Output:

-1/2*(4*sqrt(2)*x*arctan((sqrt(2)*(x^3 - 1)^(1/4)*x^3 + sqrt(2)*(x^3 - 1)^ 
(3/4)*x)/(x^4 - x^3 + 1)) + sqrt(2)*x*log((x^4 + 2*sqrt(2)*(x^3 - 1)^(1/4) 
*x^3 + x^3 + 4*sqrt(x^3 - 1)*x^2 + 2*sqrt(2)*(x^3 - 1)^(3/4)*x - 1)/(x^4 + 
 x^3 - 1)) - sqrt(2)*x*log((x^4 - 2*sqrt(2)*(x^3 - 1)^(1/4)*x^3 + x^3 + 4* 
sqrt(x^3 - 1)*x^2 - 2*sqrt(2)*(x^3 - 1)^(3/4)*x - 1)/(x^4 + x^3 - 1)) - 8* 
(x^3 - 1)^(1/4))/x
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((x^3-4)*(x^4-x^3+1)/x^2/(x^3-1)^(3/4)/(x^4+x^3-1),x, algorithm=" 
maxima")
 

Output:

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

Giac [F]

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

integrate((x^3-4)*(x^4-x^3+1)/x^2/(x^3-1)^(3/4)/(x^4+x^3-1),x, algorithm=" 
giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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