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

Optimal. Leaf size=288 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^2}-x}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^2}+x}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}-x}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt [3]{x^4+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^4+x^2}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^2+\left (x^4+x^2\right )^{2/3}}{x \sqrt [3]{x^4+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^4+x^2\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^4+x^2}}\right )}{6 \sqrt [3]{2}} \]

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Rubi [F]  time = 2.46, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/6*(x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - x), x], x, x^(1/3)])/(x^2 + x^4)^(1
/3) - (x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + x), x], x, x^(1/3)])/(6*(x^2 + x^4
)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(1/9)*x)
, x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1
+ x^6)^(2/3)/(1 + (-1)^(1/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^
(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(2/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1
- I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + (-1)^(2/9)*x), x], x, x^(1/
3)])/(6*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(1/3)*
x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/
(1 + (-1)^(1/3)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Su
bst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(4/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*
x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + (-1)^(4/9)*x), x], x, x^(1/3)])/(6*(x^2 +
x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(5/9
)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int
][(1 + x^6)^(2/3)/(1 + (-1)^(5/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x^2)^(1/3)*Defer[
Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(2/3)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x
^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + (-1)^(2/3)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) -
((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(7/9)*x), x], x, x
^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2
/3)/(1 + (-1)^(7/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defe
r[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(8/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[
3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + (-1)^(8/9)*x), x], x, x^(1/3)])/(6*(x
^2 + x^4)^(1/3))

Rubi steps

\begin {align*} \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1+x^6}{x^{2/3} \sqrt [3]{1+x^2} \left (-1+x^6\right )} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {\left (1+x^2\right )^{2/3} \left (1-x^2+x^4\right )}{x^{2/3} \left (-1+x^6\right )} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3} \left (1-x^6+x^{12}\right )}{-1+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {\left (1+x^6\right )^{2/3}}{18 (1-x)}-\frac {\left (1+x^6\right )^{2/3}}{18 (1+x)}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-\sqrt [9]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+\sqrt [9]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{2/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{2/9} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1-\sqrt [3]{-1} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{4/9} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{4/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{5/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{5/9} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{2/3} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{2/3} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{7/9} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{7/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{8/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{8/9} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+\sqrt [3]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{2/3} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{2/3} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-\sqrt [9]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+\sqrt [9]{-1} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{4/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{4/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{7/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{7/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{2/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{2/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{5/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{5/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1-(-1)^{8/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^6\right )^{2/3}}{1+(-1)^{8/9} x} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}\\ \end {align*}

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Mathematica [F]  time = 0.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.80, size = 288, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^4}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^4}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt [3]{x^2+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^2+x^4}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^2+\left (x^2+x^4\right )^{2/3}}{x \sqrt [3]{x^2+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^2+x^4\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^2+x^4}}\right )}{6 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 2.99, size = 456, normalized size = 1.58 \begin {gather*} \frac {1}{12} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (4 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} + 8 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} - \sqrt {6} 2^{\frac {1}{3}} {\left (x^{5} - 8 \, x^{4} - 2 \, x^{3} - 8 \, x^{2} + x\right )}\right )}}{6 \, {\left (x^{5} + 8 \, x^{4} - 2 \, x^{3} + 8 \, x^{2} + x\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {4 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2 \, x^{2} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} + 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + x + 1\right )} - 4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + x\right )} - \sqrt {3} {\left (x^{5} - 4 \, x^{4} + x^{3} - 4 \, x^{2} + x\right )}}{3 \, {\left (x^{5} + 4 \, x^{4} + x^{3} + 4 \, x^{2} + x\right )}}\right ) + \frac {1}{3} \, \log \left (\frac {x^{3} - x^{2} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x^{2} + x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{3} - x^{2} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - x^{2} + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(1/6*2^(1/6)*(4*sqrt(6)*2^(2/3)*(-1)^(2/3)*(x^4 + x^2)^(2/3)*(x^2 + 2*x
+ 1) + 8*sqrt(6)*(-1)^(1/3)*(x^4 + x^2)^(1/3)*(x^3 - 2*x^2 + x) - sqrt(6)*2^(1/3)*(x^5 - 8*x^4 - 2*x^3 - 8*x^2
+ x))/(x^5 + 8*x^4 - 2*x^3 + 8*x^2 + x)) + 1/12*2^(2/3)*(-1)^(1/3)*log(-(4*2^(1/3)*(-1)^(2/3)*(x^4 + x^2)^(1/
3)*x - 2^(2/3)*(-1)^(1/3)*(x^3 + 2*x^2 + x) + 4*(x^4 + x^2)^(2/3))/(x^3 - 2*x^2 + x)) - 1/24*2^(2/3)*(-1)^(1/3
)*log((2^(1/3)*(-1)^(2/3)*(x^3 - 2*x^2 + x) + 2*2^(2/3)*(-1)^(1/3)*(x^4 + x^2)^(2/3) + 4*(x^4 + x^2)^(1/3)*x)/
(x^3 - 2*x^2 + x)) - 1/3*sqrt(3)*arctan(1/3*(4*sqrt(3)*(x^4 + x^2)^(2/3)*(x^2 + x + 1) - 4*sqrt(3)*(x^4 + x^2)
^(1/3)*(x^3 - x^2 + x) - sqrt(3)*(x^5 - 4*x^4 + x^3 - 4*x^2 + x))/(x^5 + 4*x^4 + x^3 + 4*x^2 + x)) + 1/3*log((
x^3 - x^2 + 2*(x^4 + x^2)^(1/3)*x + x - 2*(x^4 + x^2)^(2/3))/(x^3 + x^2 + x)) + 1/6*log((x^3 - x^2 + 2*(x^4 +
x^2)^(1/3)*x + x - 2*(x^4 + x^2)^(2/3))/(x^3 - x^2 + x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 71.31, size = 5994, normalized size = 20.81

method	result	size
trager	\(\text {Expression too large to display}\)	\(5994\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6+1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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