3.2572 \(\int \frac{1}{\sqrt [3]{1+x} \sqrt [3]{1-x+x^2}} \, dx\)

Optimal. Leaf size=102 \[ \frac{\sqrt [3]{x^3+1} \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}}-\frac{\sqrt [3]{x^3+1} \log \left (\sqrt [3]{x^3+1}-x\right )}{2 \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}} \]

[Out]

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

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Rubi [A]  time = 0.05579, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\sqrt [3]{x^3+1} \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}}-\frac{\sqrt [3]{x^3+1} \log \left (\sqrt [3]{x^3+1}-x\right )}{2 \sqrt [3]{x+1} \sqrt [3]{x^2-x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.1682, size = 148, normalized size = 1.45 \[ - \frac{\left (x + 1\right )^{\frac{2}{3}} \left (x^{2} - x + 1\right )^{\frac{2}{3}} \log{\left (- \frac{x}{\sqrt [3]{x^{3} + 1}} + 1 \right )}}{3 \left (x^{3} + 1\right )^{\frac{2}{3}}} + \frac{\left (x + 1\right )^{\frac{2}{3}} \left (x^{2} - x + 1\right )^{\frac{2}{3}} \log{\left (\frac{x^{2}}{\left (x^{3} + 1\right )^{\frac{2}{3}}} + \frac{x}{\sqrt [3]{x^{3} + 1}} + 1 \right )}}{6 \left (x^{3} + 1\right )^{\frac{2}{3}}} + \frac{\sqrt{3} \left (x + 1\right )^{\frac{2}{3}} \left (x^{2} - x + 1\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{3 \left (x^{3} + 1\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+x)**(1/3)/(x**2-x+1)**(1/3),x)

[Out]

-(x + 1)**(2/3)*(x**2 - x + 1)**(2/3)*log(-x/(x**3 + 1)**(1/3) + 1)/(3*(x**3 + 1
)**(2/3)) + (x + 1)**(2/3)*(x**2 - x + 1)**(2/3)*log(x**2/(x**3 + 1)**(2/3) + x/
(x**3 + 1)**(1/3) + 1)/(6*(x**3 + 1)**(2/3)) + sqrt(3)*(x + 1)**(2/3)*(x**2 - x
+ 1)**(2/3)*atan(sqrt(3)*(2*x/(3*(x**3 + 1)**(1/3)) + 1/3))/(3*(x**3 + 1)**(2/3)
)

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Mathematica [C]  time = 1.11199, size = 281, normalized size = 2.75 \[ \frac{45 \left (2 i x+\sqrt{3}-i\right ) (x+1)^{2/3} \left (2 x+i \sqrt{3}-1\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 i (x+1)}{3 i+\sqrt{3}},-\frac{2 i (x+1)}{-3 i+\sqrt{3}}\right )}{4 \left (x^2-x+1\right )^{4/3} \left (30 i F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 i (x+1)}{3 i+\sqrt{3}},-\frac{2 i (x+1)}{-3 i+\sqrt{3}}\right )+\left (\sqrt{3}+3 i\right ) (x+1) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{2 i (x+1)}{3 i+\sqrt{3}},-\frac{2 i (x+1)}{-3 i+\sqrt{3}}\right )-\left (\sqrt{3}-3 i\right ) (x+1) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{2 i (x+1)}{3 i+\sqrt{3}},-\frac{2 i (x+1)}{-3 i+\sqrt{3}}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(45*(-I + Sqrt[3] + (2*I)*x)*(1 + x)^(2/3)*(-1 + I*Sqrt[3] + 2*x)*AppellF1[2/3,
1/3, 1/3, 5/3, ((2*I)*(1 + x))/(3*I + Sqrt[3]), ((-2*I)*(1 + x))/(-3*I + Sqrt[3]
)])/(4*(1 - x + x^2)^(4/3)*((30*I)*AppellF1[2/3, 1/3, 1/3, 5/3, ((2*I)*(1 + x))/
(3*I + Sqrt[3]), ((-2*I)*(1 + x))/(-3*I + Sqrt[3])] + (3*I + Sqrt[3])*(1 + x)*Ap
pellF1[5/3, 1/3, 4/3, 8/3, ((2*I)*(1 + x))/(3*I + Sqrt[3]), ((-2*I)*(1 + x))/(-3
*I + Sqrt[3])] - (-3*I + Sqrt[3])*(1 + x)*AppellF1[5/3, 4/3, 1/3, 8/3, ((2*I)*(1
+ x))/(3*I + Sqrt[3]), ((-2*I)*(1 + x))/(-3*I + Sqrt[3])]))

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Maple [F]  time = 0.197, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [3]{1+x}}}{\frac{1}{\sqrt [3]{{x}^{2}-x+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)^(1/3)/(x^2-x+1)^(1/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(1/3)*(x + 1)^(1/3)),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(1/3)*(x + 1)^(1/3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{x + 1} \sqrt [3]{x^{2} - x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x)**(1/3)/(x**2-x+1)**(1/3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(1/3)*(x + 1)^(1/3)),x, algorithm="giac")

[Out]

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