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

Optimal. Leaf size=145 \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.107369, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2152, 239, 2148} \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]

Int[x/((1 + x)*(1 - x^3)^(1/3)),x]

[Out]

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

Rule 2152

Int[((e_.) + (f_.)*(x_))/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Dist[f/d, Int[1/(a +
b*x^3)^(1/3), x], x] + Dist[(d*e - c*f)/d, Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d,
e, f}, x]

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 2148

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[(Sqrt[3]*ArcTan[(1 - (2^(1/3)*Rt[b,
3]*(c - d*x))/(d*(a + b*x^3)^(1/3)))/Sqrt[3]])/(2^(4/3)*Rt[b, 3]*c), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{x}{(1+x) \sqrt [3]{1-x^3}} \, dx &=\int \frac{1}{\sqrt [3]{1-x^3}} \, dx-\int \frac{1}{(1+x) \sqrt [3]{1-x^3}} \, dx\\ &=\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\log \left ((1-x) (1+x)^2\right )}{4 \sqrt [3]{2}}+\frac{1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )-\frac{3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ \end{align*}

Mathematica [F]  time = 0.0784828, size = 0, normalized size = 0. \[ \int \frac{x}{(1+x) \sqrt [3]{1-x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{1+x}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/((-x^3 + 1)^(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(x/(1+x)/(-x^3+1)^(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}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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