\(\int \frac {(-3+x^4) (1-x^3+x^4) (1+x^3+x^4)^{2/3}}{x^6 (1+x^4)} \, dx\) [1750]

Optimal result

Integrand size = 38, antiderivative size = 118 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right ) \left (1+x^3+x^4\right )^{2/3}}{x^6 \left (1+x^4\right )} \, dx=\frac {3 \left (1+x^3+x^4\right )^{2/3} \left (2-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^4}}\right )-\log \left (-x+\sqrt [3]{1+x^3+x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^4}+\left (1+x^3+x^4\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \left (1+x^3+x^4\right )^{2/3}}{x^6}+\frac {3 \left (1+x^3+x^4\right )^{2/3}}{x^3}+\frac {\left (1+x^3+x^4\right )^{2/3}}{x^2}-\frac {4 x \left (1+x^3+x^4\right )^{2/3}}{1+x^4}\right ) \, dx \\ & = -\left (3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx\right )+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx-4 \int \frac {x \left (1+x^3+x^4\right )^{2/3}}{1+x^4} \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx \\ & = -\left (3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx\right )+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx-4 \int \left (-\frac {i x \left (1+x^3+x^4\right )^{2/3}}{2 \left (-i+x^2\right )}+\frac {i x \left (1+x^3+x^4\right )^{2/3}}{2 \left (i+x^2\right )}\right ) \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx \\ & = 2 i \int \frac {x \left (1+x^3+x^4\right )^{2/3}}{-i+x^2} \, dx-2 i \int \frac {x \left (1+x^3+x^4\right )^{2/3}}{i+x^2} \, dx-3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx \\ & = 2 i \int \left (-\frac {\left (1+x^3+x^4\right )^{2/3}}{2 \left (\sqrt [4]{-1}-x\right )}+\frac {\left (1+x^3+x^4\right )^{2/3}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx-2 i \int \left (-\frac {\left (1+x^3+x^4\right )^{2/3}}{2 \left (-(-1)^{3/4}-x\right )}+\frac {\left (1+x^3+x^4\right )^{2/3}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx-3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx \\ & = -\left (i \int \frac {\left (1+x^3+x^4\right )^{2/3}}{\sqrt [4]{-1}-x} \, dx\right )+i \int \frac {\left (1+x^3+x^4\right )^{2/3}}{-(-1)^{3/4}-x} \, dx+i \int \frac {\left (1+x^3+x^4\right )^{2/3}}{\sqrt [4]{-1}+x} \, dx-i \int \frac {\left (1+x^3+x^4\right )^{2/3}}{-(-1)^{3/4}+x} \, dx-3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^6} \, dx+3 \int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^3+x^4\right )^{2/3}}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right ) \left (1+x^3+x^4\right )^{2/3}}{x^6 \left (1+x^4\right )} \, dx=\frac {3 \left (1+x^3+x^4\right )^{2/3} \left (2-3 x^3+2 x^4\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^4}}\right )-\log \left (-x+\sqrt [3]{1+x^3+x^4}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^4}+\left (1+x^3+x^4\right )^{2/3}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.72 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(\frac {5 \ln \left (\frac {x^{2}+x \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}}+\left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-10 \ln \left (\frac {-x +\left (x^{4}+x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (6 x^{4}-9 x^{3}+6\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}\)	\(121\)
risch	\(\frac {\frac {3}{5} x^{8}-\frac {3}{10} x^{7}+\frac {6}{5} x^{4}-\frac {9}{10} x^{6}-\frac {3}{10} x^{3}+\frac {3}{5}}{x^{5} \left (x^{4}+x^{3}+1\right )^{\frac {1}{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}+3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 x^{4}-3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2}{x^{4}+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}+3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{4}+x^{3}+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}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{4}+1}\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^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{4}+x^{3}+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}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{4}+1}\right )\)	\(442\)
trager	\(\frac {3 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} \left (2 x^{4}-3 x^{3}+2\right )}{10 x^{5}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {35271 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-70542 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+45738 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+37851 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +37851 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+61365 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+10467 x^{4}+32261 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +32261 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+24423 x^{3}+35271 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+45738 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+10467}{x^{4}+1}\right )-3 \ln \left (-\frac {-35271 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+70542 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+69252 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+37851 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +37851 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+14337 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-29632 x^{4}-44878 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x -44878 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-37040 x^{3}-35271 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+69252 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-29632}{x^{4}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+\ln \left (-\frac {-35271 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}+70542 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+69252 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+37851 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x +37851 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+14337 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-29632 x^{4}-44878 \left (x^{4}+x^{3}+1\right )^{\frac {2}{3}} x -44878 \left (x^{4}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-37040 x^{3}-35271 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+69252 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-29632}{x^{4}+1}\right )\)	\(642\)

[In]

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

[Out]

1/10*(5*ln((x^2+x*(x^4+x^3+1)^(1/3)+(x^4+x^3+1)^(2/3))/x^2)*x^5-10*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^4+x^3+
1)^(1/3)))*x^5-10*ln((-x+(x^4+x^3+1)^(1/3))/x)*x^5+(6*x^4-9*x^3+6)*(x^4+x^3+1)^(2/3))/x^5

Fricas [A] (verification not implemented)

none

Time = 1.75 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.30 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right ) \left (1+x^3+x^4\right )^{2/3}}{x^6 \left (1+x^4\right )} \, dx=\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {7043582 \, \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 984256 \, \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (145408 \, x^{4} + 3029663 \, x^{3} + 145408\right )}}{32768 \, x^{4} + 12041757 \, x^{3} + 32768}\right ) - 5 \, x^{5} \log \left (\frac {x^{4} + 3 \, {\left (x^{4} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + 1}\right ) + 3 \, {\left (2 \, x^{4} - 3 \, x^{3} + 2\right )} {\left (x^{4} + x^{3} + 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \]

[In]

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

[Out]

1/10*(10*sqrt(3)*x^5*arctan(-(7043582*sqrt(3)*(x^4 + x^3 + 1)^(1/3)*x^2 - 984256*sqrt(3)*(x^4 + x^3 + 1)^(2/3)
*x + sqrt(3)*(145408*x^4 + 3029663*x^3 + 145408))/(32768*x^4 + 12041757*x^3 + 32768)) - 5*x^5*log((x^4 + 3*(x^
4 + x^3 + 1)^(1/3)*x^2 - 3*(x^4 + x^3 + 1)^(2/3)*x + 1)/(x^4 + 1)) + 3*(2*x^4 - 3*x^3 + 2)*(x^4 + x^3 + 1)^(2/
3))/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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