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

Optimal. Leaf size=78 \[ -\frac{\log \left (x^3-1\right )}{6 \sqrt [3]{3}}+\frac{\log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+2}\right )}{2 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{3^{5/6}} \]

[Out]

-(ArcTan[(1 + (2*3^(1/3)*x)/(2 + x^3)^(1/3))/Sqrt[3]]/3^(5/6)) - Log[-1 + x^3]/(6*3^(1/3)) + Log[3^(1/3)*x - (
2 + x^3)^(1/3)]/(2*3^(1/3))

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Rubi [A]  time = 0.0708598, antiderivative size = 107, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {377, 200, 31, 634, 617, 204, 628} \[ \frac{\log \left (1-\frac{\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}\right )}{3 \sqrt [3]{3}}-\frac{\log \left (\frac{3^{2/3} x^2}{\left (x^3+2\right )^{2/3}}+\frac{\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1\right )}{6 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt [6]{3} \sqrt [3]{x^3+2}}+\frac{1}{\sqrt{3}}\right )}{3^{5/6}} \]

Antiderivative was successfully verified.

[In]

Int[1/((-1 + x^3)*(2 + x^3)^(1/3)),x]

[Out]

-(ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(2 + x^3)^(1/3))]/3^(5/6)) + Log[1 - (3^(1/3)*x)/(2 + x^3)^(1/3)]/(3*3^(1/
3)) - Log[1 + (3^(2/3)*x^2)/(2 + x^3)^(2/3) + (3^(1/3)*x)/(2 + x^3)^(1/3)]/(6*3^(1/3))

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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

Rubi steps

\begin{align*} \int \frac{1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{-1+3 x^3} \, dx,x,\frac{x}{\sqrt [3]{2+x^3}}\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt [3]{3} x} \, dx,x,\frac{x}{\sqrt [3]{2+x^3}}\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{-2-\sqrt [3]{3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{2+x^3}}\right )\\ &=\frac{\log \left (1-\frac{\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{3 \sqrt [3]{3}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{2+x^3}}\right )-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{3}+2\ 3^{2/3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{2+x^3}}\right )}{6 \sqrt [3]{3}}\\ &=\frac{\log \left (1-\frac{\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{3 \sqrt [3]{3}}-\frac{\log \left (1+\frac{3^{2/3} x^2}{\left (2+x^3\right )^{2/3}}+\frac{\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{6 \sqrt [3]{3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{\sqrt [3]{3}}\\ &=-\frac{\tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{3} x}{\sqrt [3]{2+x^3}}}{\sqrt{3}}\right )}{3^{5/6}}+\frac{\log \left (1-\frac{\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{3 \sqrt [3]{3}}-\frac{\log \left (1+\frac{3^{2/3} x^2}{\left (2+x^3\right )^{2/3}}+\frac{\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{6 \sqrt [3]{3}}\\ \end{align*}

Mathematica [A]  time = 0.0766959, size = 104, normalized size = 1.33 \[ \frac{\sqrt{3} \left (2 \log \left (1-\frac{\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}\right )-\log \left (\frac{3^{2/3} x^2}{\left (x^3+2\right )^{2/3}}+\frac{\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1\right )\right )-6 \tan ^{-1}\left (\frac{2 x}{\sqrt [6]{3} \sqrt [3]{x^3+2}}+\frac{1}{\sqrt{3}}\right )}{6\ 3^{5/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((-1 + x^3)*(2 + x^3)^(1/3)),x]

[Out]

(-6*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(2 + x^3)^(1/3))] + Sqrt[3]*(2*Log[1 - (3^(1/3)*x)/(2 + x^3)^(1/3)] - Lo
g[1 + (3^(2/3)*x^2)/(2 + x^3)^(2/3) + (3^(1/3)*x)/(2 + x^3)^(1/3)]))/(6*3^(5/6))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}-1}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 27.4718, size = 624, normalized size = 8. \begin{align*} \frac{1}{27} \cdot 3^{\frac{2}{3}} \log \left (\frac{9 \cdot 3^{\frac{1}{3}}{\left (x^{3} + 2\right )}^{\frac{1}{3}} x^{2} - 2 \cdot 3^{\frac{2}{3}}{\left (x^{3} - 1\right )} - 9 \,{\left (x^{3} + 2\right )}^{\frac{2}{3}} x}{x^{3} - 1}\right ) - \frac{1}{54} \cdot 3^{\frac{2}{3}} \log \left (\frac{3 \cdot 3^{\frac{2}{3}}{\left (7 \, x^{4} + 2 \, x\right )}{\left (x^{3} + 2\right )}^{\frac{2}{3}} + 3^{\frac{1}{3}}{\left (31 \, x^{6} + 46 \, x^{3} + 4\right )} + 9 \,{\left (5 \, x^{5} + 4 \, x^{2}\right )}{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) - \frac{1}{9} \cdot 3^{\frac{1}{6}} \arctan \left (\frac{3^{\frac{1}{6}}{\left (12 \cdot 3^{\frac{2}{3}}{\left (7 \, x^{7} - 5 \, x^{4} - 2 \, x\right )}{\left (x^{3} + 2\right )}^{\frac{2}{3}} - 3^{\frac{1}{3}}{\left (127 \, x^{9} + 402 \, x^{6} + 192 \, x^{3} + 8\right )} - 18 \,{\left (31 \, x^{8} + 46 \, x^{5} + 4 \, x^{2}\right )}{\left (x^{3} + 2\right )}^{\frac{1}{3}}\right )}}{3 \,{\left (251 \, x^{9} + 462 \, x^{6} + 24 \, x^{3} - 8\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*3^(2/3)*log((9*3^(1/3)*(x^3 + 2)^(1/3)*x^2 - 2*3^(2/3)*(x^3 - 1) - 9*(x^3 + 2)^(2/3)*x)/(x^3 - 1)) - 1/54
*3^(2/3)*log((3*3^(2/3)*(7*x^4 + 2*x)*(x^3 + 2)^(2/3) + 3^(1/3)*(31*x^6 + 46*x^3 + 4) + 9*(5*x^5 + 4*x^2)*(x^3
+ 2)^(1/3))/(x^6 - 2*x^3 + 1)) - 1/9*3^(1/6)*arctan(1/3*3^(1/6)*(12*3^(2/3)*(7*x^7 - 5*x^4 - 2*x)*(x^3 + 2)^(
2/3) - 3^(1/3)*(127*x^9 + 402*x^6 + 192*x^3 + 8) - 18*(31*x^8 + 46*x^5 + 4*x^2)*(x^3 + 2)^(1/3))/(251*x^9 + 46
2*x^6 + 24*x^3 - 8))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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