\(\int \frac {(-3+x^4) (1+x^4)^{2/3} (1+2 x^3+x^4)}{x^6 (1-x^3+x^4)} \, dx\) [1475]

Optimal result

Integrand size = 40, antiderivative size = 104 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+2 x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3} \left (2+15 x^3+2 x^4\right )}{10 x^5}-3 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+3 \log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {3}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]

[Out]

3/10*(x^4+1)^(2/3)*(2*x^4+15*x^3+2)/x^5-3*3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^4+1)^(1/3)))+3*ln(-x+(x^4+1)^(1/3))
-3/2*ln(x^2+x*(x^4+1)^(1/3)+(x^4+1)^(2/3))

Rubi [F]

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

[In]

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

[Out]

(9*(1 + x^4)^(2/3))/(2*x^2) + (18*x^2)/(1 - Sqrt[3] - (1 + x^4)^(1/3)) - (9*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)]) + (6*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*Hypergeometric2F1[-5/4, -2/3, -1/4, -x^4])/(5*x^5) - Hypergeometric2F1[-2/3, -1/4, 3/4, -x^4]/x
- 9*Defer[Int][(1 + x^4)^(2/3)/(1 - x^3 + x^4), x] + 12*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 {9 \left (1+x^4\right )^{2/3}}{x^3}+\frac {\left (1+x^4\right )^{2/3}}{x^2}+\frac {3 (-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^6} \, dx\right )+3 \int \frac {(-3+4 x) \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx-9 \int \frac {\left (1+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^4\right )^{2/3}}{x^2} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}+3 \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 {9}{2} \text {Subst}\left (\int \frac {\left (1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right ) \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}-6 \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^2}} \, dx,x,x^2\right )-9 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+12 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}-9 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+12 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx-\frac {\left (9 \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 {9 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}-9 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+12 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+\frac {\left (9 \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 (9 \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 {9 \left (1+x^4\right )^{2/3}}{2 x^2}+\frac {18 x^2}{1-\sqrt {3}-\sqrt [3]{1+x^4}}-\frac {9 \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 {6 \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 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{5 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}-9 \int \frac {\left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx+12 \int \frac {x \left (1+x^4\right )^{2/3}}{1-x^3+x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.55 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+2 x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3} \left (2+15 x^3+2 x^4\right )}{10 x^5}-3 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+3 \log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {3}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]

[In]

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

[Out]

(3*(1 + x^4)^(2/3)*(2 + 15*x^3 + 2*x^4))/(10*x^5) - 3*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^(1/3))] + 3*
Log[-x + (1 + x^4)^(1/3)] - (3*Log[x^2 + x*(1 + x^4)^(1/3) + (1 + x^4)^(2/3)])/2

Maple [A] (verified)

Time = 4.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {3 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{4}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{5}-\frac {3 \ln \left (\frac {x^{2}+x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}}{2}+3 \ln \left (\frac {-x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\frac {3 x^{4} \left (x^{4}+1\right )^{\frac {2}{3}}}{5}+\frac {9 x^{3} \left (x^{4}+1\right )^{\frac {2}{3}}}{2}+\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{5}}{x^{5}}\)	\(119\)
risch	\(\frac {\frac {3}{5} x^{8}+\frac {6}{5} x^{4}+\frac {3}{5}+\frac {9}{2} x^{7}+\frac {9}{2} x^{3}}{x^{5} \left (x^{4}+1\right )^{\frac {1}{3}}}+3 \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}-2 \left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{4}-x^{3}+1}\right )+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 \left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}+\left (x^{4}+1\right )^{\frac {2}{3}} x +x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{4}-x^{3}+1}\right )\)	\(292\)
trager	\(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}} \left (2 x^{4}+15 x^{3}+2\right )}{10 x^{5}}+3 \ln \left (-\frac {-12357 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+24714 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+684 \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 -33651 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+27465 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+5036 x^{4}-8010 \left (x^{4}+1\right )^{\frac {2}{3}} x -3207 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+7554 x^{3}-12357 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+684 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5036}{x^{4}-x^{3}+1}\right )+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {-22662 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+45324 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-26781 \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 +24030 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+684 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-4119 x^{4}+11217 \left (x^{4}+1\right )^{\frac {2}{3}} x -3207 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-1373 x^{3}-22662 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-26781 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4119}{x^{4}-x^{3}+1}\right )\)	\(418\)

[In]

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

[Out]

3/10*(10*3^(1/2)*arctan(1/3*(x+2*(x^4+1)^(1/3))*3^(1/2)/x)*x^5-5*ln((x^2+x*(x^4+1)^(1/3)+(x^4+1)^(2/3))/x^2)*x
^5+10*ln((-x+(x^4+1)^(1/3))/x)*x^5+2*x^4*(x^4+1)^(2/3)+15*x^3*(x^4+1)^(2/3)+2*(x^4+1)^(2/3))/x^5

Fricas [A] (verification not implemented)

none

Time = 1.94 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.40 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+2 x^3+x^4\right )}{x^6 \left (1-x^3+x^4\right )} \, dx=-\frac {3 \, {\left (10 \, \sqrt {3} x^{5} \arctan \left (-\frac {13034 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 686 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (37 \, x^{4} + 6137 \, x^{3} + 37\right )}}{3 \, {\left (x^{4} + 6859 \, x^{3} + 1\right )}}\right ) - 5 \, x^{5} \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} - x^{3} + 1}\right ) - {\left (2 \, x^{4} + 15 \, x^{3} + 2\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}\right )}}{10 \, x^{5}} \]

[In]

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

[Out]

-3/10*(10*sqrt(3)*x^5*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)) - 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)) - (2*x^4 + 15*x^3 + 2)*(x^4 + 1)^(2/3))/x^5

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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