\(\int \frac {(-1-x^4+2 x^6) \sqrt [3]{x-x^5+x^7}}{(1+x^2-x^4+x^6)^2} \, dx\) [2043]

Optimal result

Integrand size = 42, antiderivative size = 146 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=-\frac {x \sqrt [3]{x-x^5+x^7}}{2 \left (1+x^2-x^4+x^6\right )}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{x-x^5+x^7}}{-2 x+\sqrt [3]{x-x^5+x^7}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (x+\sqrt [3]{x-x^5+x^7}\right )-\frac {1}{12} \log \left (x^2-x \sqrt [3]{x-x^5+x^7}+\left (x-x^5+x^7\right )^{2/3}\right ) \]

[Out]

-x*(x^7-x^5+x)^(1/3)/(2*x^6-2*x^4+2*x^2+2)-1/6*arctan(3^(1/2)*(x^7-x^5+x)^(1/3)/(-2*x+(x^7-x^5+x)^(1/3)))*3^(1
/2)+1/6*ln(x+(x^7-x^5+x)^(1/3))-1/12*ln(x^2-x*(x^7-x^5+x)^(1/3)+(x^7-x^5+x)^(2/3))

Rubi [F]

\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx \]

[In]

Int[((-1 - x^4 + 2*x^6)*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6)^2,x]

[Out]

(-9*(x - x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 - x^6 + x^9)^(1/3))/(1 + x^3 - x^6 + x^9)^2, x], x, x^
(2/3)])/(2*x^(1/3)*(1 - x^4 + x^6)^(1/3)) - (3*(x - x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][(x^4*(1 - x^6 + x
^9)^(1/3))/(1 + x^3 - x^6 + x^9)^2, x], x, x^(2/3)])/(x^(1/3)*(1 - x^4 + x^6)^(1/3)) + (3*(x - x^5 + x^7)^(1/3
)*Defer[Subst][Defer[Int][(x^7*(1 - x^6 + x^9)^(1/3))/(1 + x^3 - x^6 + x^9)^2, x], x, x^(2/3)])/(2*x^(1/3)*(1
- x^4 + x^6)^(1/3)) + (3*(x - x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][(x*(1 - x^6 + x^9)^(1/3))/(1 + x^3 - x^
6 + x^9), x], x, x^(2/3)])/(x^(1/3)*(1 - x^4 + x^6)^(1/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x-x^5+x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{1-x^4+x^6} \left (-1-x^4+2 x^6\right )}{\left (1+x^2-x^4+x^6\right )^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^{12}+x^{18}} \left (-1-x^{12}+2 x^{18}\right )}{\left (1+x^6-x^{12}+x^{18}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9} \left (-1-x^6+2 x^9\right )}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \left (\frac {x \left (-3-2 x^3+x^6\right ) \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}+\frac {2 x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \left (-3-2 x^3+x^6\right ) \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \left (-\frac {3 x \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}-\frac {2 x^4 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}+\frac {x^7 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ & = \frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x^7 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}-\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}+\frac {\left (3 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{1+x^3-x^6+x^9} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^4+x^6}}-\frac {\left (9 \sqrt [3]{x-x^5+x^7}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1-x^6+x^9}}{\left (1+x^3-x^6+x^9\right )^2} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^4+x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{x-x^5+x^7} \left (-\frac {6 x^{4/3}}{1+x^2-x^4+x^6}+\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{1-x^4+x^6}}\right )}{\sqrt [3]{1-x^4+x^6}}+\frac {2 \log \left (x^{2/3}+\sqrt [3]{1-x^4+x^6}\right )}{\sqrt [3]{1-x^4+x^6}}-\frac {\log \left (x^{4/3}-x^{2/3} \sqrt [3]{1-x^4+x^6}+\left (1-x^4+x^6\right )^{2/3}\right )}{\sqrt [3]{1-x^4+x^6}}\right )}{12 \sqrt [3]{x}} \]

[In]

Integrate[((-1 - x^4 + 2*x^6)*(x - x^5 + x^7)^(1/3))/(1 + x^2 - x^4 + x^6)^2,x]

[Out]

((x - x^5 + x^7)^(1/3)*((-6*x^(4/3))/(1 + x^2 - x^4 + x^6) + (2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) - 2*
(1 - x^4 + x^6)^(1/3))])/(1 - x^4 + x^6)^(1/3) + (2*Log[x^(2/3) + (1 - x^4 + x^6)^(1/3)])/(1 - x^4 + x^6)^(1/3
) - Log[x^(4/3) - x^(2/3)*(1 - x^4 + x^6)^(1/3) + (1 - x^4 + x^6)^(2/3)]/(1 - x^4 + x^6)^(1/3)))/(12*x^(1/3))

Maple [A] (verified)

Time = 12.06 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.39

method	result	size
pseudoelliptic	\(-\frac {\left (\left (-2 x^{6}+2 x^{4}-2 x^{2}-2\right ) \ln \left (\frac {x +{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}}{x}\right )+6 {\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}} x +\left (x^{6}-x^{4}+x^{2}+1\right ) \left (2 \arctan \left (\frac {\left (-2 {\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) \sqrt {3}+\ln \left (\frac {{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {2}{3}}-{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )\right )\right ) x}{12 \left (x +{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}\right ) \left ({\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {2}{3}}+x \left (x -{\left (x \left (x^{6}-x^{4}+1\right )\right )}^{\frac {1}{3}}\right )\right )}\)	\(203\)
trager	\(\text {Expression too large to display}\)	\(739\)
risch	\(\text {Expression too large to display}\)	\(1528\)

[In]

int((2*x^6-x^4-1)*(x^7-x^5+x)^(1/3)/(x^6-x^4+x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

-1/12*((-2*x^6+2*x^4-2*x^2-2)*ln((x+(x*(x^6-x^4+1))^(1/3))/x)+6*(x*(x^6-x^4+1))^(1/3)*x+(x^6-x^4+x^2+1)*(2*arc
tan(1/3*(-2*(x*(x^6-x^4+1))^(1/3)+x)*3^(1/2)/x)*3^(1/2)+ln(((x*(x^6-x^4+1))^(2/3)-(x*(x^6-x^4+1))^(1/3)*x+x^2)
/x^2)))*x/(x+(x*(x^6-x^4+1))^(1/3))/((x*(x^6-x^4+1))^(2/3)+x*(x-(x*(x^6-x^4+1))^(1/3)))

Fricas [A] (verification not implemented)

none

Time = 1.80 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=-\frac {2 \, \sqrt {3} {\left (x^{6} - x^{4} + x^{2} + 1\right )} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} - x^{4} - x^{2} + 1\right )} - 2 \, \sqrt {3} {\left (x^{7} - x^{5} + x\right )}^{\frac {2}{3}}}{x^{6} - x^{4} + x^{2} + 1}\right ) - {\left (x^{6} - x^{4} + x^{2} + 1\right )} \log \left (\frac {x^{6} - x^{4} + x^{2} + 3 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{6} - x^{4} + x^{2} + 1}\right ) + 6 \, {\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} x}{12 \, {\left (x^{6} - x^{4} + x^{2} + 1\right )}} \]

[In]

integrate((2*x^6-x^4-1)*(x^7-x^5+x)^(1/3)/(x^6-x^4+x^2+1)^2,x, algorithm="fricas")

[Out]

-1/12*(2*sqrt(3)*(x^6 - x^4 + x^2 + 1)*arctan(-(2*sqrt(3)*(x^7 - x^5 + x)^(1/3)*x + sqrt(3)*(x^6 - x^4 - x^2 +
1) - 2*sqrt(3)*(x^7 - x^5 + x)^(2/3))/(x^6 - x^4 + x^2 + 1)) - (x^6 - x^4 + x^2 + 1)*log((x^6 - x^4 + x^2 + 3
*(x^7 - x^5 + x)^(1/3)*x + 3*(x^7 - x^5 + x)^(2/3) + 1)/(x^6 - x^4 + x^2 + 1)) + 6*(x^7 - x^5 + x)^(1/3)*x)/(x
^6 - x^4 + x^2 + 1)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((2*x**6-x**4-1)*(x**7-x**5+x)**(1/3)/(x**6-x**4+x**2+1)**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int { \frac {{\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - x^{4} - 1\right )}}{{\left (x^{6} - x^{4} + x^{2} + 1\right )}^{2}} \,d x } \]

[In]

integrate((2*x^6-x^4-1)*(x^7-x^5+x)^(1/3)/(x^6-x^4+x^2+1)^2,x, algorithm="maxima")

[Out]

integrate((x^7 - x^5 + x)^(1/3)*(2*x^6 - x^4 - 1)/(x^6 - x^4 + x^2 + 1)^2, x)

Giac [F]

\[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int { \frac {{\left (x^{7} - x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} - x^{4} - 1\right )}}{{\left (x^{6} - x^{4} + x^{2} + 1\right )}^{2}} \,d x } \]

[In]

integrate((2*x^6-x^4-1)*(x^7-x^5+x)^(1/3)/(x^6-x^4+x^2+1)^2,x, algorithm="giac")

[Out]

integrate((x^7 - x^5 + x)^(1/3)*(2*x^6 - x^4 - 1)/(x^6 - x^4 + x^2 + 1)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1-x^4+2 x^6\right ) \sqrt [3]{x-x^5+x^7}}{\left (1+x^2-x^4+x^6\right )^2} \, dx=\int -\frac {\left (-2\,x^6+x^4+1\right )\,{\left (x^7-x^5+x\right )}^{1/3}}{{\left (x^6-x^4+x^2+1\right )}^2} \,d x \]

[In]

int(-((x^4 - 2*x^6 + 1)*(x - x^5 + x^7)^(1/3))/(x^2 - x^4 + x^6 + 1)^2,x)

[Out]

int(-((x^4 - 2*x^6 + 1)*(x - x^5 + x^7)^(1/3))/(x^2 - x^4 + x^6 + 1)^2, x)