\(\int \frac {x (-3+x^4)}{(1+x^4)^{2/3} (1+x^3+x^4)} \, dx\) [988]

Optimal result

Integrand size = 26, antiderivative size = 75 \[ \int \frac {x \left (-3+x^4\right )}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^4}}\right )+\log \left (x+\sqrt [3]{1+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96 \[ \int \frac {x \left (-3+x^4\right )}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1+x^4}}\right )+\log \left (x+\sqrt [3]{1+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{\left (x^4+1\right )^{2/3} \left (x^4+x^3+1\right )}dx-4 \int \frac {x}{\left (x^4+1\right )^{2/3} \left (x^4+x^3+1\right )}dx+\int \frac {x^3}{\left (x^4+1\right )^{2/3} \left (x^4+x^3+1\right )}dx-\frac {3^{3/4} \sqrt {2-\sqrt {3}} \left (1-\sqrt [3]{x^4+1}\right ) \sqrt {\frac {\left (x^4+1\right )^{2/3}+\sqrt [3]{x^4+1}+1}{\left (-\sqrt [3]{x^4+1}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{x^4+1}+\sqrt {3}+1}{-\sqrt [3]{x^4+1}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{2 x^2 \sqrt {-\frac {1-\sqrt [3]{x^4+1}}{\left (-\sqrt [3]{x^4+1}-\sqrt {3}+1\right )^2}}}-x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {2}{3},\frac {5}{4},-x^4\right )\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.92 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\ln \left (\frac {x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}-x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )\)	\(69\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+1}{x^{4}+x^{3}+1}\right )-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x -2 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}-1}{x^{4}+x^{3}+1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x -2 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}-1}{x^{4}+x^{3}+1}\right )\)	\(287\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.36 \[ \int \frac {x \left (-3+x^4\right )}{\left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )} \, dx=-\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} + 1\right )}}{x^{4} - 8 \, x^{3} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} + x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + x^{3} + 1}\right ) \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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