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

Optimal. Leaf size=154 \[ -\frac {\left (x^4+x^2\right )^{2/3}}{x \left (x^2+1\right )}+\frac {\tan ^{-1}\left (\frac {3^{2/3} x \sqrt [3]{x^4+x^2}}{\sqrt [3]{3} x^2-\left (x^4+x^2\right )^{2/3}}\right )}{3^{2/3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{3} x}{\sqrt [3]{x^4+x^2}}\right )}{3 \sqrt [6]{3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{3} x^2+\frac {\left (x^4+x^2\right )^{2/3}}{\sqrt [6]{3}}}{x \sqrt [3]{x^4+x^2}}\right )}{3 \sqrt [6]{3}} \]

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Rubi [F]  time = 2.21, 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]

(-2*x*(1 + x^2)^(1/3)*AppellF1[1/6, 1/3, 1, 7/6, -x^2, (2*x^2)/(1 - I*Sqrt[3])])/(x^2 + x^4)^(1/3) - (2*x*(1 +
x^2)^(1/3)*AppellF1[1/6, 1/3, 1, 7/6, -x^2, (2*x^2)/(1 + I*Sqrt[3])])/(x^2 + x^4)^(1/3) + (3*x*(1 + x^2)^(1/3
)*Hypergeometric2F1[1/6, 1/3, 7/6, -x^2])/(x^2 + x^4)^(1/3) - ((I/3)*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defe
r[Int][1/((I - x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(x^2 + x^4)^(1/3) - ((I/3)*x^(2/3)*(1 + x^2)^(1/3)*Defer[
Subst][Defer[Int][1/((I + x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(x^2 + x^4)^(1/3) - (Sqrt[1 - I*Sqrt[3]]*x^(2/
3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] - Sqrt[2]*x)*(1 + x^6)^(1/3)), x], x, x^(1/
3)])/(3*(x^2 + x^4)^(1/3)) - (Sqrt[1 + I*Sqrt[3]]*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((Sqrt[1 +
I*Sqrt[3]] - Sqrt[2]*x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(x^2 + x^4)^(1/3)) - (Sqrt[1 - I*Sqrt[3]]*x^(2/
3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] + Sqrt[2]*x)*(1 + x^6)^(1/3)), x], x, x^(1/
3)])/(3*(x^2 + x^4)^(1/3)) - (Sqrt[1 + I*Sqrt[3]]*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((Sqrt[1 +
I*Sqrt[3]] + Sqrt[2]*x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(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 (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-1+x^{18}}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )} \, 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 {1}{\sqrt [3]{1+x^6}}-\frac {2}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )}\right ) \, 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 \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{9 \left (1+x^2\right ) \sqrt [3]{1+x^6}}+\frac {2-x^2}{9 \left (1-x^2+x^4\right ) \sqrt [3]{1+x^6}}+\frac {2-x^6}{3 \sqrt [3]{1+x^6} \left (1-x^6+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2-x^2}{\left (1-x^2+x^4\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2-x^6}{\sqrt [3]{1+x^6} \left (1-x^6+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {i}{2 (i-x) \sqrt [3]{1+x^6}}+\frac {i}{2 (i+x) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\sqrt [3]{1+x^6} \left (-1-i \sqrt {3}+2 x^6\right )}+\frac {-1+i \sqrt {3}}{\sqrt [3]{1+x^6} \left (-1+i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (-1-i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (-1+i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};-x^2,\frac {2 x^2}{1-i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};-x^2,\frac {2 x^2}{1+i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\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+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+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\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+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+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}\\ &=-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};-x^2,\frac {2 x^2}{1-i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};\frac {1}{3},1;\frac {7}{6};-x^2,\frac {2 x^2}{1+i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \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 {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1-i \sqrt {3}} \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 {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1-i \sqrt {3}} \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 {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1+i \sqrt {3}} \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 {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1+i \sqrt {3}} \sqrt [3]{x^2+x^4}}\\ \end {align*}

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Mathematica [F]  time = 1.02, 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 = 0.55, size = 154, normalized size = 1.00 \begin {gather*} -\frac {\left (x^4+x^2\right )^{2/3}}{x \left (x^2+1\right )}+\frac {\tan ^{-1}\left (\frac {3^{2/3} x \sqrt [3]{x^4+x^2}}{\sqrt [3]{3} x^2-\left (x^4+x^2\right )^{2/3}}\right )}{3^{2/3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [6]{3} x}{\sqrt [3]{x^4+x^2}}\right )}{3 \sqrt [6]{3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{3} x^2+\frac {\left (x^4+x^2\right )^{2/3}}{\sqrt [6]{3}}}{x \sqrt [3]{x^4+x^2}}\right )}{3 \sqrt [6]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 3.07, size = 1749, normalized size = 11.36

result too large to display

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]

1/36*(2*3^(5/6)*(x^3 + x)*log((2*3^(5/6)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + 13*x) - 6*(x^4 + x^2)^(2/3)*(26*
x^2 + sqrt(3)*(15*x^2 - 26*x + 15) - 45*x + 26) + 3*3^(1/3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 + 15*x) + 6*(x^
4 + x^2)^(1/3)*(3^(2/3)*(15*x^3 - 26*x^2 + 15*x) + 3^(1/6)*(26*x^3 - 45*x^2 + 26*x)))/(x^5 - x^3 + x)) - 2*3^(
5/6)*(x^3 + x)*log(-(2*3^(5/6)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + 13*x) + 6*(x^4 + x^2)^(2/3)*(26*x^2 - sqrt
(3)*(15*x^2 - 26*x + 15) - 45*x + 26) - 3*3^(1/3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 + 15*x) - 6*(x^4 + x^2)^(
1/3)*(3^(2/3)*(15*x^3 - 26*x^2 + 15*x) - 3^(1/6)*(26*x^3 - 45*x^2 + 26*x)))/(x^5 - x^3 + x)) - 3^(5/6)*(x^3 +
x)*log(12*(1351*3^(2/3)*(x^5 - x^3 + x) + 2*(x^4 + x^2)^(2/3)*(3^(5/6)*(1351*x^2 - 2340*x + 1351) + 3*3^(1/3)*
(780*x^2 - 1351*x + 780)) + 6*(x^4 + x^2)^(1/3)*(1351*x^3 - 2340*x^2 + sqrt(3)*(780*x^3 - 1351*x^2 + 780*x) +
1351*x) + 2340*3^(1/6)*(x^5 - x^3 + x))/(x^5 - x^3 + x)) + 3^(5/6)*(x^3 + x)*log(12*(1351*3^(2/3)*(x^5 - x^3 +
x) - 2*(x^4 + x^2)^(2/3)*(3^(5/6)*(1351*x^2 - 2340*x + 1351) - 3*3^(1/3)*(780*x^2 - 1351*x + 780)) + 6*(x^4 +
x^2)^(1/3)*(1351*x^3 - 2340*x^2 - sqrt(3)*(780*x^3 - 1351*x^2 + 780*x) + 1351*x) - 2340*3^(1/6)*(x^5 - x^3 +
x))/(x^5 - x^3 + x)) - 12*3^(1/3)*(x^3 + x)*arctan(1/9*(72*x^8 + 72*x^2 + 2*sqrt(3)*(2*3^(2/3)*(13*x^9 - 45*x^
8 - 260*x^7 - 540*x^6 - 663*x^5 - 540*x^4 - 260*x^3 - 45*x^2 + 13*x) - 12*(x^4 + x^2)^(2/3)*(3^(5/6)*(15*x^5 -
104*x^4 - 15*x^3 - 104*x^2 + 15*x) - 2*3^(1/3)*(13*x^5 - 90*x^4 - 13*x^3 - 90*x^2 + 13*x)) + 6*(26*x^7 - 135*
x^6 - 312*x^5 - 405*x^4 - 312*x^3 - 135*x^2 - 3*sqrt(3)*(5*x^7 - 26*x^6 - 60*x^5 - 78*x^4 - 60*x^3 - 26*x^2 +
5*x) + 26*x)*(x^4 + x^2)^(1/3) - 3*3^(1/6)*(15*x^9 - 52*x^8 - 300*x^7 - 624*x^6 - 765*x^5 - 624*x^4 - 300*x^3
- 52*x^2 + 15*x))*sqrt((1351*3^(2/3)*(x^5 - x^3 + x) + 2*(x^4 + x^2)^(2/3)*(3^(5/6)*(1351*x^2 - 2340*x + 1351)
+ 3*3^(1/3)*(780*x^2 - 1351*x + 780)) + 6*(x^4 + x^2)^(1/3)*(1351*x^3 - 2340*x^2 + sqrt(3)*(780*x^3 - 1351*x^
2 + 780*x) + 1351*x) + 2340*3^(1/6)*(x^5 - x^3 + x))/(x^5 - x^3 + x)) + 12*(x^4 + x^2)^(2/3)*(3^(2/3)*(x^6 - 1
2*x^4 - 12*x^2 + 1) + 9*3^(1/6)*(x^5 + 3*x^3 + x)) + 3*sqrt(3)*(x^9 + 46*x^7 + 99*x^5 + 46*x^3 + x) + 12*(x^4
+ x^2)^(1/3)*(3^(5/6)*(x^7 + 12*x^5 + 12*x^3 + x) + 3*3^(1/3)*(5*x^6 + 7*x^4 + 5*x^2)))/(x^9 - 50*x^7 - 93*x^5
- 50*x^3 + x)) - 12*3^(1/3)*(x^3 + x)*arctan(1/9*(72*x^8 + 72*x^2 + 2*sqrt(3)*(2*3^(2/3)*(13*x^9 - 45*x^8 - 2
60*x^7 - 540*x^6 - 663*x^5 - 540*x^4 - 260*x^3 - 45*x^2 + 13*x) + 12*(x^4 + x^2)^(2/3)*(3^(5/6)*(15*x^5 - 104*
x^4 - 15*x^3 - 104*x^2 + 15*x) + 2*3^(1/3)*(13*x^5 - 90*x^4 - 13*x^3 - 90*x^2 + 13*x)) + 6*(26*x^7 - 135*x^6 -
312*x^5 - 405*x^4 - 312*x^3 - 135*x^2 + 3*sqrt(3)*(5*x^7 - 26*x^6 - 60*x^5 - 78*x^4 - 60*x^3 - 26*x^2 + 5*x)
+ 26*x)*(x^4 + x^2)^(1/3) + 3*3^(1/6)*(15*x^9 - 52*x^8 - 300*x^7 - 624*x^6 - 765*x^5 - 624*x^4 - 300*x^3 - 52*
x^2 + 15*x))*sqrt((1351*3^(2/3)*(x^5 - x^3 + x) - 2*(x^4 + x^2)^(2/3)*(3^(5/6)*(1351*x^2 - 2340*x + 1351) - 3*
3^(1/3)*(780*x^2 - 1351*x + 780)) + 6*(x^4 + x^2)^(1/3)*(1351*x^3 - 2340*x^2 - sqrt(3)*(780*x^3 - 1351*x^2 + 7
80*x) + 1351*x) - 2340*3^(1/6)*(x^5 - x^3 + x))/(x^5 - x^3 + x)) + 12*(x^4 + x^2)^(2/3)*(3^(2/3)*(x^6 - 12*x^4
- 12*x^2 + 1) - 9*3^(1/6)*(x^5 + 3*x^3 + x)) - 3*sqrt(3)*(x^9 + 46*x^7 + 99*x^5 + 46*x^3 + x) - 12*(x^4 + x^2
)^(1/3)*(3^(5/6)*(x^7 + 12*x^5 + 12*x^3 + x) - 3*3^(1/3)*(5*x^6 + 7*x^4 + 5*x^2)))/(x^9 - 50*x^7 - 93*x^5 - 50
*x^3 + x)) - 36*(x^4 + x^2)^(2/3))/(x^3 + 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 = 33.33, size = 7488, normalized size = 48.62 \begin {gather*} \text {output too large to display} \end {gather*}

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)

[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 = -0.01 \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 - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 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 - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)/((x**2*(x**2 + 1))**(1/3)*(x**2 + 1)*(x**4 - x**2 + 1))
, x)

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