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

Optimal result

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

[Out]

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))

Rubi [F]

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

[In]

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

[Out]

(3*(-1 + x^4)^(2/3))/x^2 - (12*x^2)/(1 + Sqrt[3] + (-1 + x^4)^(1/3)) + (6*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + (-1 +
x^4)^(1/3))*Sqrt[(1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]*EllipticE[ArcS
in[(1 - Sqrt[3] + (-1 + x^4)^(1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))], -7 - 4*Sqrt[3]])/(x^2*Sqrt[(1 + (-1 + x
^4)^(1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]) - (4*Sqrt[2]*3^(3/4)*(1 + (-1 + x^4)^(1/3))*Sqrt[(1 - (-1 + x^
4)^(1/3) + (-1 + x^4)^(2/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] + (-1 + x^4)^(1
/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))], -7 - 4*Sqrt[3]])/(x^2*Sqrt[(1 + (-1 + x^4)^(1/3))/(1 + Sqrt[3] + (-1 +
x^4)^(1/3))^2]) - (3*(-1 + x^4)^(2/3)*Hypergeometric2F1[-5/4, -2/3, -1/4, x^4])/(5*x^5*(1 - x^4)^(2/3)) - ((-
1 + x^4)^(2/3)*Hypergeometric2F1[-2/3, -1/4, 3/4, x^4])/(x*(1 - x^4)^(2/3)) - 6*Defer[Int][(-1 + x^4)^(2/3)/(-
1 - x^3 + x^4), x] + 8*Defer[Int][(x*(-1 + x^4)^(2/3))/(-1 - x^3 + x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \left (-1+x^4\right )^{2/3}}{x^6}-\frac {6 \left (-1+x^4\right )^{2/3}}{x^3}+\frac {\left (-1+x^4\right )^{2/3}}{x^2}+\frac {2 (-3+4 x) \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4}\right ) \, dx \\ & = 2 \int \frac {(-3+4 x) \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+3 \int \frac {\left (-1+x^4\right )^{2/3}}{x^6} \, dx-6 \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1+x^4\right )^{2/3}}{x^2} \, dx \\ & = 2 \int \left (-\frac {3 \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4}+\frac {4 x \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4}\right ) \, dx-3 \text {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )+\frac {\left (-1+x^4\right )^{2/3} \int \frac {\left (1-x^4\right )^{2/3}}{x^2} \, dx}{\left (1-x^4\right )^{2/3}}+\frac {\left (3 \left (-1+x^4\right )^{2/3}\right ) \int \frac {\left (1-x^4\right )^{2/3}}{x^6} \, dx}{\left (1-x^4\right )^{2/3}} \\ & = \frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \left (1-x^4\right )^{2/3}}-4 \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )-6 \int \frac {\left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+8 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx \\ & = \frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \left (1-x^4\right )^{2/3}}-6 \int \frac {\left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+8 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx-\frac {\left (6 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2} \\ & = \frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \left (1-x^4\right )^{2/3}}-6 \int \frac {\left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+8 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx-\frac {\left (6 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}-\frac {\left (6 \left (-1+\sqrt {3}\right ) \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2} \\ & = \frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {12 x^2}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}+\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}-\frac {4 \sqrt {2} 3^{3/4} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right ),-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \left (1-x^4\right )^{2/3}}-6 \int \frac {\left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx+8 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1-x^3+x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+x^3+x^4\right )}{x^6 \left (-1-x^3+x^4\right )} \, dx=\frac {3 \left (-1+x^4\right )^{2/3} \left (-1+5 x^3+x^4\right )}{5 x^5}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^4}}\right )+2 \log \left (-x+\sqrt [3]{-1+x^4}\right )-\log \left (x^2+x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]

[In]

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

[Out]

(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*L
og[-x + (-1 + x^4)^(1/3)] - Log[x^2 + x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]

Maple [A] (verified)

Time = 5.35 (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	\(\frac {\frac {3}{5} x^{8}+3 x^{7}-\frac {6}{5} x^{4}-3 x^{3}+\frac {3}{5}}{x^{5} \left (x^{4}-1\right )^{\frac {1}{3}}}+2 \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\left (x^{4}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-2 \left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{4}-x^{3}-1}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\left (x^{4}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+\left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{4}-x^{3}-1}\right )\)	\(290\)
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}-2598497280 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}-15695904 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{4}-65890176 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -65890176 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-62825856 \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}-1385865216 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}[phone] \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] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} 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] \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} 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 )\)	\(619\)

[In]

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

[Out]

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)

none

Time = 3.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.47 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+x^3+x^4\right )}{x^6 \left (-1-x^3+x^4\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {14106128635054532 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 89654043956484782 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (35416555940707109 \, x^{4} + 2357401720008016 \, x^{3} - 35416555940707109\right )}}{3 \, {\left (51678794422160641 \, x^{4} + 201291873609016 \, x^{3} - 51678794422160641\right )}}\right ) - 5 \, x^{5} \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - x^{3} - 1}\right ) - 3 \, {\left (x^{4} + 5 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]

[In]

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

[Out]

-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)*(35416555940707109*x^4 + 2357401720008016*x^3 - 35416555940707109))/(516787944221606
41*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. \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1+x^3+x^4\right )}{x^6 \left (-1-x^3+x^4\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

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