\(\int \frac {(-1+x^4)^{2/3} (3+x^4) (-1+x^3+x^4)}{x^6 (-1-x^3+x^4)} \, dx\) [1393]

Optimal result

Integrand size = 38, antiderivative size = 100 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+x^3+x^4\right )}{x^6 \left (-1-x^3+x^4\right )} \, dx=\frac {3 \left (-1+x^4\right )^{2/3} \left (-1+5 x^3+x^4\right )}{5 x^5}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^4}}\right )+2 \log \left (-x+\sqrt [3]{-1+x^4}\right )-\log \left (x^2+x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \] Output:

3/5*(x^4-1)^(2/3)*(x^4+5*x^3-1)/x^5-2*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^4-1 
)^(1/3)))+2*ln(-x+(x^4-1)^(1/3))-ln(x^2+x*(x^4-1)^(1/3)+(x^4-1)^(2/3))
 

Mathematica [A] (verified)

Time = 2.43 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+x^3+x^4\right )}{x^6 \left (-1-x^3+x^4\right )} \, dx=\frac {3 \left (-1+x^4\right )^{2/3} \left (-1+5 x^3+x^4\right )}{5 x^5}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^4}}\right )+2 \log \left (-x+\sqrt [3]{-1+x^4}\right )-\log \left (x^2+x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \] Input:

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

Output:

(3*(-1 + x^4)^(2/3)*(-1 + 5*x^3 + x^4))/(5*x^5) - 2*Sqrt[3]*ArcTan[(Sqrt[3 
]*x)/(x + 2*(-1 + x^4)^(1/3))] + 2*Log[-x + (-1 + x^4)^(1/3)] - Log[x^2 + 
x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]
 

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -6 \int \frac {\left (x^4-1\right )^{2/3}}{x^4-x^3-1}dx+8 \int \frac {x \left (x^4-1\right )^{2/3}}{x^4-x^3-1}dx-\frac {4 \sqrt {2} 3^{3/4} \left (\sqrt [3]{x^4-1}+1\right ) \sqrt {\frac {\left (x^4-1\right )^{2/3}-\sqrt [3]{x^4-1}+1}{\left (\sqrt [3]{x^4-1}+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{x^4-1}-\sqrt {3}+1}{\sqrt [3]{x^4-1}+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{x^4-1}+1}{\left (\sqrt [3]{x^4-1}+\sqrt {3}+1\right )^2}} x^2}+\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{x^4-1}+1\right ) \sqrt {\frac {\left (x^4-1\right )^{2/3}-\sqrt [3]{x^4-1}+1}{\left (\sqrt [3]{x^4-1}+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{x^4-1}-\sqrt {3}+1}{\sqrt [3]{x^4-1}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{x^4-1}+1}{\left (\sqrt [3]{x^4-1}+\sqrt {3}+1\right )^2}} x^2}-\frac {\left (x^4-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{\left (1-x^4\right )^{2/3} x}-\frac {3 \left (x^4-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 \left (1-x^4\right )^{2/3} x^5}-\frac {12 x^2}{\sqrt [3]{x^4-1}+\sqrt {3}+1}+\frac {3 \left (x^4-1\right )^{2/3}}{x^2}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 5.82 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(\frac {-5 \ln \left (\frac {x^{2}+x \left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+10 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{4}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{5}+10 \ln \left (\frac {-x +\left (x^{4}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+3 \left (x^{4}-1\right )^{\frac {2}{3}} \left (x^{4}+5 x^{3}-1\right )}{5 x^{5}}\)	\(105\)
risch	\(\frac {3 x^{7}-3 x^{3}+\frac {3}{5} x^{8}-\frac {6}{5} x^{4}+\frac {3}{5}}{x^{5} \left (x^{4}-1\right )^{\frac {1}{3}}}+2 \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+2 \left (x^{4}-1\right )^{\frac {2}{3}} x -\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{4}-x^{3}-1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -2 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-\left (x^{4}-1\right )^{\frac {2}{3}} x -\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{4}-x^{3}-1}\right )\)	\(295\)
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}} \left (x^{4}+5 x^{3}-1\right )}{5 x^{5}}+192 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (\frac {-1385865216 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{4}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{4}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x [phone] \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+222105 x^{4}+431087 \left (x^{4}-1\right )^{\frac {2}{3}} x +431087 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+340561 x^{3}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-15695904 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-222105}{x^{4}-x^{3}-1}\right )-192 \ln \left (-\frac {1385865216 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{4}-2598497280 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{4}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x [phone] \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}+8690496 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+91770 x^{4}+255269 \left (x^{4}-1\right )^{\frac {2}{3}} x +255269 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+31920 x^{3}-1385865216 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-44568096 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-91770}{x^{4}-x^{3}-1}\right ) \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-2 \ln \left (-\frac {1385865216 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{4}-2598497280 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{4}[phone] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x [phone] \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}+8690496 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+91770 x^{4}+255269 \left (x^{4}-1\right )^{\frac {2}{3}} x +255269 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+31920 x^{3}-1385865216 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-44568096 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-91770}{x^{4}-x^{3}-1}\right )\)	\(618\)

Input:

int((x^4-1)^(2/3)*(x^4+3)*(x^4+x^3-1)/x^6/(x^4-x^3-1),x,method=_RETURNVERB 
OSE)
 

Output:

1/5*(-5*ln((x^2+x*(x^4-1)^(1/3)+(x^4-1)^(2/3))/x^2)*x^5+10*3^(1/2)*arctan( 
1/3*(x+2*(x^4-1)^(1/3))*3^(1/2)/x)*x^5+10*ln((-x+(x^4-1)^(1/3))/x)*x^5+3*( 
x^4-1)^(2/3)*(x^4+5*x^3-1))/x^5
 

Fricas [A] (verification not implemented)

Time = 3.00 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.47 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+x^3+x^4\right )}{x^6 \left (-1-x^3+x^4\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {14106128635054532 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 89654043956484782 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (35416555940707109 \, x^{4} + 2357401720008016 \, x^{3} - 35416555940707109\right )}}{3 \, {\left (51678794422160641 \, x^{4} + 201291873609016 \, x^{3} - 51678794422160641\right )}}\right ) - 5 \, x^{5} \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - x^{3} - 1}\right ) - 3 \, {\left (x^{4} + 5 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \] Input:

integrate((x^4-1)^(2/3)*(x^4+3)*(x^4+x^3-1)/x^6/(x^4-x^3-1),x, algorithm=" 
fricas")
 

Output:

-1/5*(10*sqrt(3)*x^5*arctan(-1/3*(14106128635054532*sqrt(3)*(x^4 - 1)^(1/3 
)*x^2 - 89654043956484782*sqrt(3)*(x^4 - 1)^(2/3)*x - sqrt(3)*(35416555940 
707109*x^4 + 2357401720008016*x^3 - 35416555940707109))/(51678794422160641 
*x^4 + 201291873609016*x^3 - 51678794422160641)) - 5*x^5*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)) - 3*(x^4 
 + 5*x^3 - 1)*(x^4 - 1)^(2/3))/x^5
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((x^4-1)^(2/3)*(x^4+3)*(x^4+x^3-1)/x^6/(x^4-x^3-1),x, algorithm=" 
maxima")
 

Output:

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

Giac [F]

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

integrate((x^4-1)^(2/3)*(x^4+3)*(x^4+x^3-1)/x^6/(x^4-x^3-1),x, algorithm=" 
giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+x^3+x^4\right )}{x^6 \left (-1-x^3+x^4\right )} \, dx=\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}} x^{4}+15 \left (x^{4}-1\right )^{\frac {2}{3}} x^{3}-3 \left (x^{4}-1\right )^{\frac {2}{3}}+30 \left (\int \frac {\left (x^{4}-1\right )^{\frac {2}{3}}}{x^{8}-x^{7}-2 x^{4}+x^{3}+1}d x \right ) x^{5}+10 \left (\int \frac {\left (x^{4}-1\right )^{\frac {2}{3}} x^{4}}{x^{8}-x^{7}-2 x^{4}+x^{3}+1}d x \right ) x^{5}}{5 x^{5}} \] Input:

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

Output:

(3*(x**4 - 1)**(2/3)*x**4 + 15*(x**4 - 1)**(2/3)*x**3 - 3*(x**4 - 1)**(2/3 
) + 30*int((x**4 - 1)**(2/3)/(x**8 - x**7 - 2*x**4 + x**3 + 1),x)*x**5 + 1 
0*int(((x**4 - 1)**(2/3)*x**4)/(x**8 - x**7 - 2*x**4 + x**3 + 1),x)*x**5)/ 
(5*x**5)