\(\int \frac {(-1+x^4+2 x^6) \sqrt [3]{x+x^5+x^7}}{(1+x^4+x^6) (1-x^2+x^4+x^6)} \, dx\) [1277]

Optimal result

Integrand size = 48, antiderivative size = 92 \[ \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx=\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^5+x^7}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{x+x^5+x^7}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{x+x^5+x^7}+\left (x+x^5+x^7\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(3*(x + x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][x/(1 + x^6 + x^9)^(2/3), x], x, x^(2/3)])/(x^(1/3)*(1 + x^4 +
x^6)^(1/3)) - (9*(x + x^5 + x^7)^(1/3)*Defer[Subst][Defer[Int][x/((1 + x^6 + x^9)^(2/3)*(1 - x^3 + x^6 + x^9)
), 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)^(2/3)*(1 - x^3 + x^6 + x^9)), 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)^(2/3)*(1 - x^3 + x^6 + x^9)), x], x, x^(2/3)])/(2*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} \left (-1+x^4+2 x^6\right )}{\left (1+x^4+x^6\right )^{2/3} \left (1-x^2+x^4+x^6\right )} \, 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 \left (-1+x^{12}+2 x^{18}\right )}{\left (1+x^{12}+x^{18}\right )^{2/3} \left (1-x^6+x^{12}+x^{18}\right )} \, 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 \left (-1+x^6+2 x^9\right )}{\left (1+x^6+x^9\right )^{2/3} \left (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 \left (\frac {2 x}{\left (1+x^6+x^9\right )^{2/3}}+\frac {x \left (-3+2 x^3-x^6\right )}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}\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 )}{\left (1+x^6+x^9\right )^{2/3} \left (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 (1+x^6+x^9\right )^{2/3}} \, 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}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}+\frac {2 x^4}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}-\frac {x^7}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}\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 (1+x^6+x^9\right )^{2/3}} \, 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}{\left (1+x^6+x^9\right )^{2/3} \left (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 (1+x^6+x^9\right )^{2/3}} \, 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^4}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, 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}{\left (1+x^6+x^9\right )^{2/3} \left (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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.49 \[ \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx=\frac {\sqrt [3]{x+x^5+x^7} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^4+x^6}}\right )+2 \log \left (-x^{2/3}+\sqrt [3]{1+x^4+x^6}\right )-\log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^4+x^6}+\left (1+x^4+x^6\right )^{2/3}\right )\right )}{4 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 8.70 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {{\left (x \left (x^{6}+x^{4}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )}{2}-\frac {\ln \left (\frac {{\left (x \left (x^{6}+x^{4}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{6}+x^{4}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{4}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{6}+x^{4}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{2}\)	\(92\)
trager	\(\text {Expression too large to display}\)	\(738\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 1.73 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} + x^{4} + x^{2} + 1\right )} + 2 \, \sqrt {3} {\left (x^{7} + x^{5} + x\right )}^{\frac {2}{3}}}{x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{6} + x^{4} - x^{2} + 3 \, {\left (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{7} + x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{6} + x^{4} - x^{2} + 1}\right ) \]

[In]

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

[Out]

1/2*sqrt(3)*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)) + 1/4*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))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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