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

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

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Rubi [A]  time = 0.0552897, 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, Rules used = {2151} \[ -\frac{3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )+\log (x+1) \]

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

Rule 2151

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*f*ArcTan
[(1 + (2*Rt[b, 3]*(2*c + d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Rt[b, 3]*d), x] + (Simp[(f*Log[c + d*x])/(Rt[
b, 3]*d), x] - Simp[(3*f*Log[Rt[b, 3]*(2*c + d*x) - d*(a + b*x^3)^(1/3)])/(2*Rt[b, 3]*d), x]) /; FreeQ[{a, b,
c, d, e, f}, x] && EqQ[d*e + c*f, 0] && EqQ[2*b*c^3 - a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{-1+x}{(1+x) \sqrt [3]{2+x^3}} \, dx &=\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt{3}}\right )+\log (1+x)-\frac{3}{2} \log \left (2+x-\sqrt [3]{2+x^3}\right )\\ \end{align*}

Mathematica [F]  time = 0.242957, 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]

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{-1+x}{1+x}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \end{align*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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