\(\int \frac {(-1+x^4)^{2/3} (3+x^4)}{x^3 (-1+x^3+x^4)} \, dx\) [1236]

Optimal result

Integrand size = 28, antiderivative size = 90 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^3 \left (-1+x^3+x^4\right )} \, dx=\frac {3 \left (-1+x^4\right )^{2/3}}{2 x^2}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^4}}\right )+\log \left (x+\sqrt [3]{-1+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(3*(-1 + x^4)^(2/3))/(2*x^2) - (6*x^2)/(1 + Sqrt[3] + (-1 + x^4)^(1/3)) + (3*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[A
rcSin[(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]) - (2*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*Defer[Int][(-1 + x^4)^(2/3)/(-1 + x^3 + x^4), x] + 4*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^3}+\frac {(3+4 x) \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4}\right ) \, dx \\ & = -\left (3 \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \frac {(3+4 x) \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx \\ & = -\left (\frac {3}{2} \text {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+\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 \\ & = \frac {3 \left (-1+x^4\right )^{2/3}}{2 x^2}-2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )+3 \int \frac {\left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx+4 \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}}{2 x^2}+3 \int \frac {\left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx+4 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx-\frac {\left (3 \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}}{2 x^2}+3 \int \frac {\left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx+4 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx-\frac {\left (3 \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 (3 \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}}{2 x^2}-\frac {6 x^2}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}+\frac {3 \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 {2 \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}}}+3 \int \frac {\left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx+4 \int \frac {x \left (-1+x^4\right )^{2/3}}{-1+x^3+x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^3 \left (-1+x^3+x^4\right )} \, dx=\frac {3 \left (-1+x^4\right )^{2/3}}{2 x^2}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^4}}\right )+\log \left (x+\sqrt [3]{-1+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]

[In]

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

[Out]

(3*(-1 + x^4)^(2/3))/(2*x^2) + 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

Maple [A] (verified)

Time = 7.43 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{4}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \ln \left (\frac {x +\left (x^{4}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}-x \left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{4}-1\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(94\)
risch	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\left (x^{4}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+2 \left (x^{4}-1\right )^{\frac {2}{3}} x -2 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+x^{3}+1}{x^{4}+x^{3}-1}\right )-\ln \left (-\frac {\left (x^{4}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-\left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-1}{x^{4}+x^{3}-1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {\left (x^{4}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-\left (x^{4}-1\right )^{\frac {2}{3}} x +\left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-1}{x^{4}+x^{3}-1}\right )\)	\(293\)
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{2 x^{2}}-12 \ln \left (\frac {38450304 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-72094320 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-3013524 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+9414864 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -9414864 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-2810712 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+15302 x^{4}+769857 \left (x^{4}-1\right )^{\frac {2}{3}} x -769857 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+17488 x^{3}-38450304 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+3013524 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-15302}{x^{4}+x^{3}-1}\right ) \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+12 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {38450304 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-72094320 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+9421908 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}-9414864 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +9414864 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-9205008 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+533445 x^{4}-14715 \left (x^{4}-1\right )^{\frac {2}{3}} x +14715 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-248941 x^{3}-38450304 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-9421908 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-533445}{x^{4}+x^{3}-1}\right )-\ln \left (\frac {38450304 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-72094320 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-3013524 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+9414864 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -9414864 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}-2810712 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+15302 x^{4}+769857 \left (x^{4}-1\right )^{\frac {2}{3}} x -769857 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+17488 x^{3}-38450304 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+3013524 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-15302}{x^{4}+x^{3}-1}\right )\)	\(600\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 3.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^3 \left (-1+x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {33798185694614068 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 35774000716806898 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18215948833549379 \, x^{4} - 16570144372161104 \, x^{3} - 18215948833549379\right )}}{18912305915671589 \, x^{4} + 15948583382382344 \, x^{3} - 18912305915671589}\right ) + x^{2} \log \left (\frac {x^{4} + x^{3} + 3 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} + x^{3} - 1}\right ) + 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]

[In]

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

[Out]

1/2*(2*sqrt(3)*x^2*arctan(-(33798185694614068*sqrt(3)*(x^4 - 1)^(1/3)*x^2 - 35774000716806898*sqrt(3)*(x^4 - 1
)^(2/3)*x + sqrt(3)*(18215948833549379*x^4 - 16570144372161104*x^3 - 18215948833549379))/(18912305915671589*x^
4 + 15948583382382344*x^3 - 18912305915671589)) + x^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)) + 3*(x^4 - 1)^(2/3))/x^2

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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