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

Optimal. Leaf size=97 \[ \frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.0438348, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2148} \[ \frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]

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

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{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{\log \left ((1-x) (1+x)^2\right )}{4 \sqrt [3]{2}}+\frac{3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ \end{align*}

Mathematica [F]  time = 0.0762007, size = 0, normalized size = 0. \[ \int \frac{1}{(1+x) \sqrt [3]{1-x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/((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{1}{1+x}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(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{1}{{\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(1/(1+x)/(-x^3+1)^(1/3),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 37.3663, size = 788, normalized size = 8.12 \begin{align*} \frac{1}{12} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{1}{6}}{\left (2^{\frac{5}{6}}{\left (13 \, x^{6} + 2 \, x^{5} + 19 \, x^{4} - 4 \, x^{3} + 19 \, x^{2} + 2 \, x + 13\right )} - 4 \, \sqrt{2}{\left (5 \, x^{5} - 5 \, x^{4} + 6 \, x^{3} - 6 \, x^{2} + 5 \, x - 5\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 16 \cdot 2^{\frac{1}{6}}{\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right )}}{6 \,{\left (3 \, x^{6} - 18 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 3 \, x^{2} - 18 \, x + 3\right )}}\right ) - \frac{1}{24} \cdot 2^{\frac{2}{3}} \log \left (\frac{4 \cdot 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (x^{2} + 1\right )} + 2^{\frac{1}{3}}{\left (5 \, x^{4} + 6 \, x^{2} + 5\right )} - 2 \,{\left (3 \, x^{3} - x^{2} + x - 3\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (\frac{2^{\frac{2}{3}}{\left (x^{2} + 2 \, x + 1\right )} - 2 \cdot 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} - 4 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2} + 2 \, x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(13*x^6 + 2*x^5 + 19*x^4 - 4*x^3 + 19*x^2 + 2*x + 13)
- 4*sqrt(2)*(5*x^5 - 5*x^4 + 6*x^3 - 6*x^2 + 5*x - 5)*(-x^3 + 1)^(1/3) + 16*2^(1/6)*(x^4 + 2*x^3 + 2*x^2 + 2*
x + 1)*(-x^3 + 1)^(2/3))/(3*x^6 - 18*x^5 - 3*x^4 - 28*x^3 - 3*x^2 - 18*x + 3)) - 1/24*2^(2/3)*log((4*2^(2/3)*(
-x^3 + 1)^(2/3)*(x^2 + 1) + 2^(1/3)*(5*x^4 + 6*x^2 + 5) - 2*(3*x^3 - x^2 + x - 3)*(-x^3 + 1)^(1/3))/(x^4 + 4*x
^3 + 6*x^2 + 4*x + 1)) + 1/12*2^(2/3)*log((2^(2/3)*(x^2 + 2*x + 1) - 2*2^(1/3)*(-x^3 + 1)^(1/3)*(x - 1) - 4*(-
x^3 + 1)^(2/3))/(x^2 + 2*x + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(1+x)/(-x**3+1)**(1/3),x)

[Out]

Integral(1/((-(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{1}{{\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(1/(1+x)/(-x^3+1)^(1/3),x, algorithm="giac")

[Out]

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