\(\int \frac {\sqrt [3]{-1+x^4} (3+x^4)}{x^2 (-1+x^3+x^4)} \, dx\) [1235]

Optimal result

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

[Out]

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

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

[In]

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

[Out]

(3*(-1 + x^4)^(1/3)*Hypergeometric2F1[-1/3, -1/4, 3/4, x^4])/(x*(1 - x^4)^(1/3)) + 3*Defer[Int][(x*(-1 + x^4)^
(1/3))/(-1 + x^3 + x^4), x] + 4*Defer[Int][(x^2*(-1 + x^4)^(1/3))/(-1 + x^3 + x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \sqrt [3]{-1+x^4}}{x^2}+\frac {x (3+4 x) \sqrt [3]{-1+x^4}}{-1+x^3+x^4}\right ) \, dx \\ & = -\left (3 \int \frac {\sqrt [3]{-1+x^4}}{x^2} \, dx\right )+\int \frac {x (3+4 x) \sqrt [3]{-1+x^4}}{-1+x^3+x^4} \, dx \\ & = -\frac {\left (3 \sqrt [3]{-1+x^4}\right ) \int \frac {\sqrt [3]{1-x^4}}{x^2} \, dx}{\sqrt [3]{1-x^4}}+\int \left (\frac {3 x \sqrt [3]{-1+x^4}}{-1+x^3+x^4}+\frac {4 x^2 \sqrt [3]{-1+x^4}}{-1+x^3+x^4}\right ) \, dx \\ & = \frac {3 \sqrt [3]{-1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \sqrt [3]{1-x^4}}+3 \int \frac {x \sqrt [3]{-1+x^4}}{-1+x^3+x^4} \, dx+4 \int \frac {x^2 \sqrt [3]{-1+x^4}}{-1+x^3+x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 9.75 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{4}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x -2 \ln \left (\frac {x +\left (x^{4}-1\right )^{\frac {1}{3}}}{x}\right ) x +\ln \left (\frac {x^{2}-x \left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{4}-1\right )^{\frac {1}{3}}}{2 x}\)	\(87\)
risch	\(\frac {3 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-x^{8}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-2 \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-\left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+2 x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x +2 \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x -1}{\left (x^{4}+x^{3}-1\right ) \left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )+\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^{7}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}+x^{8}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}-x^{7}-\left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-2 \left (x^{8}-2 x^{4}+1\right )^{\frac {2}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x +x^{3}+\left (x^{8}-2 x^{4}+1\right )^{\frac {1}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{\left (x^{4}+x^{3}-1\right ) \left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\right ) {\left (\left (x^{4}-1\right )^{2}\right )}^{\frac {1}{3}}}{\left (x^{4}-1\right )^{\frac {2}{3}}}\)	\(505\)
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}+6 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (-\frac {-9612576 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{4}[phone] \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}-1506762 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{4}+4707432 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -4707432 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-1405356 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \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}+9612576 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+1506762 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+15302}{x^{4}+x^{3}-1}\right )-6 \ln \left (\frac {9612576 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{4}-18023580 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}-4710954 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{4}+4707432 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -4707432 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+4602504 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \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}-9612576 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+4710954 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-533445}{x^{4}+x^{3}-1}\right ) \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+\ln \left (\frac {9612576 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{4}-18023580 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{3}-4710954 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{4}+4707432 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -4707432 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+4602504 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \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}-9612576 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+4710954 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-533445}{x^{4}+x^{3}-1}\right )\)	\(599\)

[In]

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

Fricas [A] (verification not implemented)

none

Time = 2.73 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1+x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (-\frac {33798185694614068 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 35774000716806898 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18215948833549379 \, x^{4} - 16570144372161104 \, x^{3} - 18215948833549379\right )}}{18912305915671589 \, x^{4} + 15948583382382344 \, x^{3} - 18912305915671589}\right ) - x \log \left (\frac {x^{4} + x^{3} + 3 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} + x^{3} - 1}\right ) + 6 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{2 \, x} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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