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

Optimal. Leaf size=53 \[ -\frac{3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )+\log (x+1) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0963491, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )+\log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.143547, size = 0, normalized size = 0. \[ \int \frac{-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{-1+x}{1+x}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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