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]

1/8*ln((1-x)*(1+x)^2)*2^(2/3)+1/2*ln(x+(-x^3+1)^(1/3))-3/8*ln(-1+x+2^(2/3)*(-x^3+1)^(1/3))*2^(2/3)-1/3*arctan(
1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+1/4*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)*2
^(2/3)

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Rubi [A]  time = 0.11, antiderivative size = 145, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {x}{(1+x) \sqrt [3]{1-x^3}} \, dx \]

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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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/(1+x)/(-x^3+1)^(1/3),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 \[ \int \frac {x}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (x +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (1-x^3\right )}^{1/3}\,\left (x+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((1 - x^3)^(1/3)*(x + 1)),x)

[Out]

int(x/((1 - x^3)^(1/3)*(x + 1)), x)

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

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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