\(\int \frac {-1+x^6}{(1+x^6) \sqrt [3]{1+a^3 x^3+x^6}} \, dx\) [1826]

Optimal result

Integrand size = 29, antiderivative size = 124 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{1+a^3 x^3+x^6}}\right )}{\sqrt {3} a}+\frac {\log \left (-a x+\sqrt [3]{1+a^3 x^3+x^6}\right )}{3 a}-\frac {\log \left (a^2 x^2+a x \sqrt [3]{1+a^3 x^3+x^6}+\left (1+a^3 x^3+x^6\right )^{2/3}\right )}{6 a} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(x*(1 + (2*x^3)/(a^3 - Sqrt[-4 + a^6]))^(1/3)*(1 + (2*x^3)/(a^3 + Sqrt[-4 + a^6]))^(1/3)*AppellF1[1/3, 1/3, 1/
3, 4/3, (-2*x^3)/(a^3 - Sqrt[-4 + a^6]), (-2*x^3)/(a^3 + Sqrt[-4 + a^6])])/(1 + a^3*x^3 + x^6)^(1/3) - (I/3)*D
efer[Int][1/((I - x)*(1 + a^3*x^3 + x^6)^(1/3)), x] - (I/3)*Defer[Int][1/((I + x)*(1 + a^3*x^3 + x^6)^(1/3)),
x] - (Sqrt[1 - I*Sqrt[3]]*Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] - Sqrt[2]*x)*(1 + a^3*x^3 + x^6)^(1/3)), x])/3 -
(Sqrt[1 + I*Sqrt[3]]*Defer[Int][1/((Sqrt[1 + I*Sqrt[3]] - Sqrt[2]*x)*(1 + a^3*x^3 + x^6)^(1/3)), x])/3 - (Sqrt
[1 - I*Sqrt[3]]*Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] + Sqrt[2]*x)*(1 + a^3*x^3 + x^6)^(1/3)), x])/3 - (Sqrt[1 +
I*Sqrt[3]]*Defer[Int][1/((Sqrt[1 + I*Sqrt[3]] + Sqrt[2]*x)*(1 + a^3*x^3 + x^6)^(1/3)), x])/3

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {2}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx\right )+\int \frac {1}{\sqrt [3]{1+a^3 x^3+x^6}} \, dx \\ & = -\left (2 \int \left (\frac {1}{3 \left (1+x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}}+\frac {2-x^2}{3 \left (1-x^2+x^4\right ) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx\right )+\frac {\left (\sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}}\right ) \int \frac {1}{\sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}}} \, dx}{\sqrt [3]{1+a^3 x^3+x^6}} \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {2}{3} \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {2}{3} \int \frac {2-x^2}{\left (1-x^2+x^4\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {2}{3} \int \left (\frac {i}{2 (i-x) \sqrt [3]{1+a^3 x^3+x^6}}+\frac {i}{2 (i+x) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx-\frac {2}{3} \int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {1}{3} i \int \frac {1}{(i-x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} i \int \frac {1}{(i+x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {1}{3} i \int \frac {1}{(i-x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} i \int \frac {1}{(i+x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}}+\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}}+\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}}\right ) \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{a^3-\sqrt {-4+a^6}}} \sqrt [3]{1+\frac {2 x^3}{a^3+\sqrt {-4+a^6}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{a^3-\sqrt {-4+a^6}},-\frac {2 x^3}{a^3+\sqrt {-4+a^6}}\right )}{\sqrt [3]{1+a^3 x^3+x^6}}-\frac {1}{3} i \int \frac {1}{(i-x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} i \int \frac {1}{(i+x) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx-\frac {1}{3} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{1+a^3 x^3+x^6}}\right )-2 \log \left (a \left (a x-\sqrt [3]{1+a^3 x^3+x^6}\right )\right )+\log \left (a^2 x^2+a x \sqrt [3]{1+a^3 x^3+x^6}+\left (1+a^3 x^3+x^6\right )^{2/3}\right )}{6 a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (a x +2 \left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}\right )}{3 a x}\right )+2 \ln \left (\frac {-a x +\left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {a^{2} x^{2}+a x \left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}+\left (a^{3} x^{3}+x^{6}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{6 a}\)	\(116\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 2.73 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.19 \[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {1}{3}} a^{2} x^{2} - 2 \, \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {2}{3}} a x + \sqrt {3} {\left (a^{3} x^{3} + x^{6} + 1\right )}}{9 \, a^{3} x^{3} + x^{6} + 1}\right ) - \log \left (\frac {x^{6} + 3 \, {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {1}{3}} a^{2} x^{2} - 3 \, {\left (a^{3} x^{3} + x^{6} + 1\right )}^{\frac {2}{3}} a x + 1}{x^{6} + 1}\right )}{6 \, a} \]

[In]

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

[Out]

-1/6*(2*sqrt(3)*arctan(-(4*sqrt(3)*(a^3*x^3 + x^6 + 1)^(1/3)*a^2*x^2 - 2*sqrt(3)*(a^3*x^3 + x^6 + 1)^(2/3)*a*x
+ sqrt(3)*(a^3*x^3 + x^6 + 1))/(9*a^3*x^3 + x^6 + 1)) - log((x^6 + 3*(a^3*x^3 + x^6 + 1)^(1/3)*a^2*x^2 - 3*(a
^3*x^3 + x^6 + 1)^(2/3)*a*x + 1)/(x^6 + 1)))/a

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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