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

Optimal. Leaf size=232 \[ \frac {1}{2} x F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-\frac {x^3}{8}\right )+\sqrt [3]{1-x^3}-\frac {\log \left (x^3+8\right )}{\sqrt [3]{3}}+\frac {1}{2} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{1-x^3}\right )-\log \left (-\sqrt [3]{1-x^3}-x\right )+\frac {1}{2} 3^{2/3} \log \left (-\sqrt [3]{1-x^3}-\frac {1}{2} 3^{2/3} x\right )-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\sqrt [6]{3} \tan ^{-1}\left (\frac {1-\frac {3^{2/3} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )-\sqrt [6]{3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}}{3 \sqrt [6]{3}}+\frac {1}{\sqrt {3}}\right ) \]

[Out]

(-x^3+1)^(1/3)+1/2*x*AppellF1(1/3,-1/3,1,4/3,x^3,-1/8*x^3)-3^(1/6)*arctan(2/9*(-x^3+1)^(1/3)*3^(5/6)+1/3*3^(1/
2))+3^(1/6)*arctan(1/3*(1-3^(2/3)*x/(-x^3+1)^(1/3))*3^(1/2))-1/3*ln(x^3+8)*3^(2/3)+1/2*3^(2/3)*ln(3^(2/3)-(-x^
3+1)^(1/3))-ln(-x-(-x^3+1)^(1/3))+1/2*3^(2/3)*ln(-1/2*3^(2/3)*x-(-x^3+1)^(1/3))-2/3*arctan(1/3*(1-2*x/(-x^3+1)
^(1/3))*3^(1/2))*3^(1/2)

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Rubi [F]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt [3]{1-x^3}}{2+x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - x^3)^(1/3)/(2 + x),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1-x^3}}{2+x} \, dx &=\int \frac {\sqrt [3]{1-x^3}}{2+x} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{1-x^3}}{2+x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - x^3)^(1/3)/(2 + x),x]

[Out]

Integrate[(1 - x^3)^(1/3)/(2 + x), x]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.85, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x^{3}+1\right )^{\frac {1}{3}}}{x +2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-x^3\right )}^{1/3}}{x+2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^3)^(1/3)/(x + 2),x)

[Out]

int((1 - x^3)^(1/3)/(x + 2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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