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

Optimal. Leaf size=210 \[ -\frac {2}{3} \log \left (\sqrt [3]{x^4+x^2}+x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^4+x^2}-2 x\right )}{3 \sqrt [3]{2}}+\frac {1}{3} \log \left (x^2-\sqrt [3]{x^4+x^2} x+\left (x^4+x^2\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+x^2} x+\sqrt [3]{2} \left (x^4+x^2\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+x^2}-x}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{\sqrt [3]{2} \sqrt {3}} \]

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Rubi [F]  time = 1.53, 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^3}{\left (-1+x^3\right ) \sqrt [3]{x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*x*(1 + x^2)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (-2*x^2)/(1 - I*Sqrt[3]), -x^2])/(x^2 + x^4)^(1/3) - (2*x*(1
+ x^2)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (-2*x^2)/(1 + I*Sqrt[3]), -x^2])/(x^2 + x^4)^(1/3) - ((I - Sqrt[3])*x^
2*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, (-2*x^2)/(1 - I*Sqrt[3])])/(2*(I + Sqrt[3])*(x^2 + x^4)^(1/
3)) - ((I + Sqrt[3])*x^2*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, (-2*x^2)/(1 + I*Sqrt[3])])/(2*(I - S
qrt[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) +
(2*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-1 + x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(x^2 + x^
4)^(1/3)) - (2*(1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*(1 + x
^6)^(1/3)), x], x, x^(1/3)])/(3*(x^2 + x^4)^(1/3)) - (2*(1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][D
efer[Int][1/((1 + I*Sqrt[3] + 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^3}{\left (-1+x^3\right ) \sqrt [3]{x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1+x^3}{x^{2/3} \sqrt [3]{1+x^2} \left (-1+x^3\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^9}{\sqrt [3]{1+x^6} \left (-1+x^9\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^9\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^9\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 (-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,\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}{(-1+x) \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}{\left (1+x+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^3}{\sqrt [3]{1+x^6} \left (1+x^3+x^6\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}{(-1+x) \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 \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,\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^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,\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}{(-1+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\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}{\left (1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}} \, 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\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}{\left (1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}} \, 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}{(-1+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\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 \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,\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\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 \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,\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}{(-1+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\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 {x^3}{\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\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 {x^3}{\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 (\left (-1-i \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{1+x^6}} \, 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};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^2}{1-i \sqrt {3}},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^2}{1+i \sqrt {3}},-x^2\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}{(-1+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\right ) \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 {x}{\sqrt [3]{1+x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\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\right ) \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 {x}{\sqrt [3]{1+x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^2}{1-i \sqrt {3}},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^2}{1+i \sqrt {3}},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i-\sqrt {3}\right ) x^2 \sqrt [3]{1+x^2} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{2 \left (i+\sqrt {3}\right ) \sqrt [3]{x^2+x^4}}-\frac {\left (i+\sqrt {3}\right ) x^2 \sqrt [3]{1+x^2} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{2 \left (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 (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+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\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}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}\\ \end {align*}

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Mathematica [F]  time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt [3]{x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.78, size = 210, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \log \left (\sqrt [3]{x^4+x^2}+x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^4+x^2}-2 x\right )}{3 \sqrt [3]{2}}+\frac {1}{3} \log \left (x^2-\sqrt [3]{x^4+x^2} x+\left (x^4+x^2\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+x^2} x+\sqrt [3]{2} \left (x^4+x^2\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {2 \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 )}{\sqrt [3]{2} \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 5.95, size = 424, normalized size = 2.02 \begin {gather*} -\frac {1}{18} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{4} + 2 \, x^{3} - 6 \, x^{2} + 2 \, x + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{5} + 14 \, x^{4} + 6 \, x^{3} + 14 \, x^{2} + x\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{7} + 30 \, x^{6} + 51 \, x^{5} + 52 \, x^{4} + 51 \, x^{3} + 30 \, x^{2} + x\right )}}{3 \, {\left (x^{7} - 6 \, x^{6} - 93 \, x^{5} - 20 \, x^{4} - 93 \, x^{3} - 6 \, x^{2} + x\right )}}\right ) - \frac {2}{3} \, \sqrt {3} \arctan \left (-\frac {1078 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (32 \, x^{3} - 605 \, x^{2} + 32 \, x\right )} - 196 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{8 \, x^{3} + 1331 \, x^{2} + 8 \, x}\right ) - \frac {1}{36} \cdot 4^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + 4 \, x + 1\right )} + 4^{\frac {2}{3}} {\left (x^{5} + 14 \, x^{4} + 6 \, x^{3} + 14 \, x^{2} + x\right )} + 24 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} + x\right )}}{x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + x}\right ) + \frac {1}{18} \cdot 4^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} - 6 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{3} \, \log \left (\frac {x^{3} + x^{2} + 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x + 3 \, {\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^3+1)/(x^3-1)/(x^4+x^2)^(1/3),x, algorithm="fricas")

[Out]

-1/18*4^(1/3)*sqrt(3)*arctan(1/3*(3*4^(2/3)*sqrt(3)*(x^4 + 2*x^3 - 6*x^2 + 2*x + 1)*(x^4 + x^2)^(2/3) + 6*4^(1
/3)*sqrt(3)*(x^5 + 14*x^4 + 6*x^3 + 14*x^2 + x)*(x^4 + x^2)^(1/3) + sqrt(3)*(x^7 + 30*x^6 + 51*x^5 + 52*x^4 +
51*x^3 + 30*x^2 + x))/(x^7 - 6*x^6 - 93*x^5 - 20*x^4 - 93*x^3 - 6*x^2 + x)) - 2/3*sqrt(3)*arctan(-(1078*sqrt(3
)*(x^4 + x^2)^(1/3)*x + sqrt(3)*(32*x^3 - 605*x^2 + 32*x) - 196*sqrt(3)*(x^4 + x^2)^(2/3))/(8*x^3 + 1331*x^2 +
8*x)) - 1/36*4^(1/3)*log((6*4^(1/3)*(x^4 + x^2)^(2/3)*(x^2 + 4*x + 1) + 4^(2/3)*(x^5 + 14*x^4 + 6*x^3 + 14*x^
2 + x) + 24*(x^4 + x^2)^(1/3)*(x^3 + x^2 + x))/(x^5 - 4*x^4 + 6*x^3 - 4*x^2 + x)) + 1/18*4^(1/3)*log(-(3*4^(2/
3)*(x^4 + x^2)^(1/3)*x + 4^(1/3)*(x^3 - 2*x^2 + x) - 6*(x^4 + x^2)^(2/3))/(x^3 - 2*x^2 + x)) - 1/3*log((x^3 +
x^2 + 3*(x^4 + x^2)^(1/3)*x + x + 3*(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^{3} + 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 21.35, size = 2327, normalized size = 11.08 \begin {gather*} \text {Expression too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*ln((33000*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^3+37494*RootOf(4*RootO
f(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^3-82500*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^
3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^2-93735*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)
^2*x^2+33000*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x+158472*(x^4+x^2)^(2/3)*R
ootOf(_Z^3-4)^2*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)+37494*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*Roo
tOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x+302664*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x+316944*(x^4+x^2)^(1/3)*Root
Of(_Z^3-4)*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*x+198000*RootOf(_Z^3-4)*x^3+224964*RootOf(4*R
ootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*x^3+176000*RootOf(_Z^3-4)*x^2+199968*RootOf(4*RootOf(_Z^3-4)^2+6*_
Z*RootOf(_Z^3-4)+9*_Z^2)*x^2+198000*RootOf(_Z^3-4)*x+605328*(x^4+x^2)^(2/3)+224964*RootOf(4*RootOf(_Z^3-4)^2+6
*_Z*RootOf(_Z^3-4)+9*_Z^2)*x)/(-1+x)^2/x)*RootOf(_Z^3-4)-1/4*ln((33000*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_
Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^3+37494*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-
4)^2*x^3-82500*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^2-93735*RootOf(4*RootO
f(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+33000*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^
3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x+158472*(x^4+x^2)^(2/3)*RootOf(_Z^3-4)^2*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(
_Z^3-4)+9*_Z^2)+37494*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x+302664*RootOf
(_Z^3-4)^2*(x^4+x^2)^(1/3)*x+316944*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-
4)+9*_Z^2)*x+198000*RootOf(_Z^3-4)*x^3+224964*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*x^3+176000
*RootOf(_Z^3-4)*x^2+199968*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*x^2+198000*RootOf(_Z^3-4)*x+6
05328*(x^4+x^2)^(2/3)+224964*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*x)/(-1+x)^2/x)*RootOf(4*Roo
tOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)+1/4*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*ln(-(8004*
RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^3-37494*RootOf(4*RootOf(_Z^3-4)^2+6*_
Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^3-20010*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*Ro
otOf(_Z^3-4)^3*x^2+93735*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+8004*Roo
tOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x+158472*(x^4+x^2)^(2/3)*RootOf(_Z^3-4)^2*
RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)-37494*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z
^2)^2*RootOf(_Z^3-4)^2*x-91368*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)*x+316944*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*RootOf
(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*x-26680*RootOf(_Z^3-4)*x^3+124980*RootOf(4*RootOf(_Z^3-4)^2+6*
_Z*RootOf(_Z^3-4)+9*_Z^2)*x^3-96048*RootOf(_Z^3-4)*x^2+449928*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*
_Z^2)*x^2-26680*RootOf(_Z^3-4)*x-182736*(x^4+x^2)^(2/3)+124980*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9
*_Z^2)*x)/(-1+x)^2/x)-2/3*ln((3582*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x^
3-8955*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x^2+3582*RootOf(4*RootOf(_Z^3-
4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x+41832*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z
^2)*RootOf(_Z^3-4)^2*x^3+29232*(x^4+x^2)^(2/3)*RootOf(_Z^3-4)^2*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+
9*_Z^2)-38376*(x^4+x^2)^(1/3)*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x-59208*R
ootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2+41832*RootOf(4*RootOf(_Z^3-4)^2+6*_Z
*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x+119680*x^3+180288*(x^4+x^2)^(2/3)+77952*x*(x^4+x^2)^(1/3)-71808*x^2
+119680*x)/x/(x^2+x+1))-1/4*RootOf(_Z^3-4)^2*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*ln(-(3366*R
ootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x^3-8415*RootOf(4*RootOf(_Z^3-4)^2+6*_
Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x^2+3366*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*R
ootOf(_Z^3-4)^4*x-8688*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^3+14616*(x^4+x
^2)^(2/3)*RootOf(_Z^3-4)^2*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)+33804*(x^4+x^2)^(1/3)*RootOf(
4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x-20916*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(
_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2-8688*RootOf(4*RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)
^2*x-19104*x^3-51168*(x^4+x^2)^(2/3)+38976*x*(x^4+x^2)^(1/3)-12736*x^2-19104*x)/x/(x^2+x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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