\(\int \frac {(-3+x^4) (1+x^4)^{2/3}}{x^3 (1-x^3+x^4)} \, dx\) [1234]

Optimal result

Integrand size = 30, antiderivative size = 90 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+\log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]

[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]

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

[In]

Int[((-3 + x^4)*(1 + x^4)^(2/3))/(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[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)]) + (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 = 1.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+\log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]

[In]

Integrate[((-3 + x^4)*(1 + x^4)^(2/3))/(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 = 6.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{4}+1\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(95\)
risch	\(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x +2 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}+1}{x^{4}-x^{3}+1}\right )-\ln \left (\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x -x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-1}{x^{4}-x^{3}+1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x -x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-1}{x^{4}-x^{3}+1}\right )\)	\(299\)
trager	\(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{2 x^{2}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-10989 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-2067 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2746 x^{4}+8010 \left (x^{4}+1\right )^{\frac {2}{3}} x +8010 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+4119 x^{3}+10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-10989 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2746}{x^{4}-x^{3}+1}\right )-3 \ln \left (\frac {10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+17859 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-11673 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+7554 x^{4}+11217 \left (x^{4}+1\right )^{\frac {2}{3}} x +11217 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+2518 x^{3}+10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+17859 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+7554}{x^{4}-x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-\ln \left (\frac {10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+17859 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-11673 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+7554 x^{4}+11217 \left (x^{4}+1\right )^{\frac {2}{3}} x +11217 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+2518 x^{3}+10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+17859 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+7554}{x^{4}-x^{3}+1}\right )\)	\(606\)

[In]

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

[Out]

1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^4+1)^(1/3)))*x^2+2*ln((-x+(x^4+1)^(1/3))/x)*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 = 1.85 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.49 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {13034 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 686 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (37 \, x^{4} + 6137 \, x^{3} + 37\right )}}{3 \, {\left (x^{4} + 6859 \, x^{3} + 1\right )}}\right ) - x^{2} \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} - x^{3} + 1}\right ) - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]

[In]

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

[Out]

-1/2*(2*sqrt(3)*x^2*arctan(-1/3*(13034*sqrt(3)*(x^4 + 1)^(1/3)*x^2 - 686*sqrt(3)*(x^4 + 1)^(2/3)*x + sqrt(3)*(
37*x^4 + 6137*x^3 + 37))/(x^4 + 6859*x^3 + 1)) - 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. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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