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

Optimal. Leaf size=196 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^5+x}+x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^5+x}-2 x\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5+x}-x}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^5+x}+x}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {1}{6} \log \left (-\sqrt [3]{x^5+x} x+\left (x^5+x\right )^{2/3}+x^2\right )-\frac {\log \left (2^{2/3} \sqrt [3]{x^5+x} x+\sqrt [3]{2} \left (x^5+x\right )^{2/3}+2 x^2\right )}{12 \sqrt [3]{2}} \]

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Rubi [F]  time = 1.50, 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+x^5} \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-((x*(1 + x^4)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (-2*x^4)/(1 - I*Sqrt[3]), -x^4])/(x + x^5)^(1/3)) - (x*(1 + x^
4)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (-2*x^4)/(1 + I*Sqrt[3]), -x^4])/(x + x^5)^(1/3) - ((I - Sqrt[3])*x^3*(1 +
x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^4, (-2*x^4)/(1 - I*Sqrt[3])])/(4*(I + Sqrt[3])*(x + x^5)^(1/3)) - ((
I + Sqrt[3])*x^3*(1 + x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^4, (-2*x^4)/(1 + I*Sqrt[3])])/(4*(I - Sqrt[3])*
(x + x^5)^(1/3)) + (3*x*(1 + x^4)^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -x^4])/(2*(x + x^5)^(1/3)) + (x^(1/3)
*(1 + x^4)^(1/3)*Defer[Subst][Defer[Int][1/((-1 + x)*(1 + x^6)^(1/3)), x], x, x^(2/3)])/(3*(x + x^5)^(1/3)) -
((1 - I*Sqrt[3])*x^(1/3)*(1 + x^4)^(1/3)*Defer[Subst][Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*(1 + x^6)^(1/3)), x]
, x, x^(2/3)])/(3*(x + x^5)^(1/3)) - ((1 + I*Sqrt[3])*x^(1/3)*(1 + x^4)^(1/3)*Defer[Subst][Defer[Int][1/((1 +
I*Sqrt[3] + 2*x)*(1 + x^6)^(1/3)), x], x, x^(2/3)])/(3*(x + x^5)^(1/3))

Rubi steps

\begin {align*} \int \frac {1+x^6}{\sqrt [3]{x+x^5} \left (-1+x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \int \frac {1+x^6}{\sqrt [3]{x} \sqrt [3]{1+x^4} \left (-1+x^6\right )} \, dx}{\sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^9}{\sqrt [3]{1+x^6} \left (-1+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^6}}+\frac {2}{\sqrt [3]{1+x^6} \left (-1+x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (-1+x^9\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{9 (-1+x) \sqrt [3]{1+x^6}}+\frac {-2-x}{9 \left (1+x+x^2\right ) \sqrt [3]{1+x^6}}+\frac {-2-x^3}{3 \sqrt [3]{1+x^6} \left (1+x^3+x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {-2-x^3}{\sqrt [3]{1+x^6} \left (1+x^3+x^6\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-i+\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\sqrt [3]{1+x^6} \left (1-i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {i+\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\sqrt [3]{1+x^6} \left (1+i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^6} \left (1-i \sqrt {3}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^6} \left (1+i \sqrt {3}+2 x^6\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=-\frac {x \sqrt [3]{1+x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^4}{1-i \sqrt {3}},-x^4\right )}{\sqrt [3]{x+x^5}}-\frac {x \sqrt [3]{1+x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^4}{1+i \sqrt {3}},-x^4\right )}{\sqrt [3]{x+x^5}}+\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=-\frac {x \sqrt [3]{1+x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^4}{1-i \sqrt {3}},-x^4\right )}{\sqrt [3]{x+x^5}}-\frac {x \sqrt [3]{1+x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^4}{1+i \sqrt {3}},-x^4\right )}{\sqrt [3]{x+x^5}}-\frac {\left (i-\sqrt {3}\right ) x^3 \sqrt [3]{1+x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{4 \left (i+\sqrt {3}\right ) \sqrt [3]{x+x^5}}-\frac {\left (i+\sqrt {3}\right ) x^3 \sqrt [3]{1+x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{4 \left (i-\sqrt {3}\right ) \sqrt [3]{x+x^5}}+\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^6}{\sqrt [3]{x+x^5} \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.13, size = 196, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x+x^5}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^5}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {1}{3} \log \left (x+\sqrt [3]{x+x^5}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^5}\right )}{6 \sqrt [3]{2}}+\frac {1}{6} \log \left (x^2-x \sqrt [3]{x+x^5}+\left (x+x^5\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^5}+\sqrt [3]{2} \left (x+x^5\right )^{2/3}\right )}{12 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^6)/((x + x^5)^(1/3)*(-1 + x^6)),x]

[Out]

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

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fricas [B]  time = 4.66, size = 368, normalized size = 1.88 \begin {gather*} -\frac {1}{36} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} {\left (x^{8} + 14 \, x^{6} + 6 \, x^{4} + 14 \, x^{2} + 1\right )} {\left (x^{5} + x\right )}^{\frac {2}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (x^{12} - 24 \, x^{10} - 57 \, x^{8} - 56 \, x^{6} - 57 \, x^{4} - 24 \, x^{2} + 1\right )} - 24 \, \sqrt {6} {\left (x^{9} - x^{7} - x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{12} + 48 \, x^{10} + 15 \, x^{8} + 88 \, x^{6} + 15 \, x^{4} + 48 \, x^{2} + 1\right )}}\right ) - \frac {1}{72} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{8} + 14 \, x^{6} + 6 \, x^{4} + 14 \, x^{2} + 1\right )} + 12 \cdot 2^{\frac {1}{3}} {\left (x^{5} + x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} + 6 \, {\left (x^{5} + x\right )}^{\frac {2}{3}} {\left (x^{4} + 4 \, x^{2} + 1\right )}}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + \frac {1}{36} \cdot 2^{\frac {2}{3}} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{5} + x\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (x^{4} - 2 \, x^{2} + 1\right )} - 6 \, {\left (x^{5} + x\right )}^{\frac {1}{3}} x}{x^{4} - 2 \, x^{2} + 1}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (x^{5} + x\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{6} \, \log \left (\frac {x^{4} + x^{2} + 3 \, {\left (x^{5} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{4} + x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*sqrt(6)*2^(1/6)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(x^8 + 14*x^6 + 6*x^4 + 14*x^2 + 1)*(x^5 + x)^(2/3
) - sqrt(6)*2^(1/3)*(x^12 - 24*x^10 - 57*x^8 - 56*x^6 - 57*x^4 - 24*x^2 + 1) - 24*sqrt(6)*(x^9 - x^7 - x^3 + x
)*(x^5 + x)^(1/3))/(x^12 + 48*x^10 + 15*x^8 + 88*x^6 + 15*x^4 + 48*x^2 + 1)) - 1/72*2^(2/3)*log((2^(2/3)*(x^8
+ 14*x^6 + 6*x^4 + 14*x^2 + 1) + 12*2^(1/3)*(x^5 + x^3 + x)*(x^5 + x)^(1/3) + 6*(x^5 + x)^(2/3)*(x^4 + 4*x^2 +
1))/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)) + 1/36*2^(2/3)*log((3*2^(2/3)*(x^5 + x)^(2/3) - 2^(1/3)*(x^4 - 2*x^2 +
1) - 6*(x^5 + x)^(1/3)*x)/(x^4 - 2*x^2 + 1)) + 1/3*sqrt(3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(x^5 + x)^(1/3)
)/x) - 1/6*log((x^4 + x^2 + 3*(x^5 + x)^(1/3)*x + 3*(x^5 + x)^(2/3) + 1)/(x^4 + x^2 + 1))

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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^{5} + x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {x^{6}+1}{\left (x^{5}+x \right )^{\frac {1}{3}} \left (x^{6}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)/(x^5+x)^(1/3)/(x^6-1),x)

[Out]

int((x^6+1)/(x^5+x)^(1/3)/(x^6-1),x)

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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^{5} + x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^6 + 1)/((x^6 - 1)*(x^5 + x)^(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^6-1\right )\,{\left (x^5+x\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^6 + 1)/((x^6 - 1)*(x + x^5)^(1/3)), 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 \left (x^{4} + 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**5+x)**(1/3)/(x**6-1),x)

[Out]

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

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