3.4.14 \(\int \frac {1}{(-1+x^3) \sqrt [3]{2+x^3}} \, dx\) [314]

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

[Out]

-1/3*arctan(1/3*(1+2*3^(1/3)*x/(x^3+2)^(1/3))*3^(1/2))*3^(1/6)-1/18*ln(x^3-1)*3^(2/3)+1/6*ln(3^(1/3)*x-(x^3+2)
^(1/3))*3^(2/3)

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Rubi [A]
time = 0.01, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 = {384} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{3^{5/6}}-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-1 + x^3)*(2 + x^3)^(1/3)),x]

[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))

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx &=\text {Subst}\left (\int \frac {1}{-1+3 x^3} \, dx,x,\frac {x}{\sqrt [3]{2+x^3}}\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{2+x^3}}\right )+\frac {1}{3} \text {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} \text {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 {\text {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 {\text {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*}

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Mathematica [A]
time = 0.22, size = 110, normalized size = 1.41 \begin {gather*} \frac {-6 \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2+x^3}}\right )+\sqrt {3} \left (2 \log \left (-3 x+3^{2/3} \sqrt [3]{2+x^3}\right )-\log \left (3 x^2+3^{2/3} x \sqrt [3]{2+x^3}+\sqrt [3]{3} \left (2+x^3\right )^{2/3}\right )\right )}{6\ 3^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.70, size = 904, normalized size = 11.59

method	result	size
trager	\(\text {Expression too large to display}\)	\(904\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*ln(-(27*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-9*x^3*RootOf(_Z^3-9)^
3*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)-15*(x^3+2)^(2/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)
^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x-45*(x^3+2)^(1/3)*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf
(_Z^3-9)*x^2+2*(x^3+2)^(1/3)*RootOf(_Z^3-9)^2*x^2-9*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3+3
*RootOf(_Z^3-9)*x^3+6*x*(x^3+2)^(2/3)-18*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)+6*RootOf(_Z^3-9)
)/(-1+x)/(x^2+x+1))*RootOf(_Z^3-9)-ln(-(27*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-
9)^2*x^3-9*x^3*RootOf(_Z^3-9)^3*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)-15*(x^3+2)^(2/3)*RootOf(_
Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x-45*(x^3+2)^(1/3)*RootOf(RootOf(_Z^3-9)^2+9*_Z*
RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*x^2+2*(x^3+2)^(1/3)*RootOf(_Z^3-9)^2*x^2-9*RootOf(RootOf(_Z^3-9)^2+9*_Z
*RootOf(_Z^3-9)+81*_Z^2)*x^3+3*RootOf(_Z^3-9)*x^3+6*x*(x^3+2)^(2/3)-18*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^
3-9)+81*_Z^2)+6*RootOf(_Z^3-9))/(-1+x)/(x^2+x+1))*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)+RootOf(
RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*ln(-(27*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*R
ootOf(_Z^3-9)^2*x^3+12*x^3*RootOf(_Z^3-9)^3*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)+15*(x^3+2)^(2
/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x+45*(x^3+2)^(1/3)*RootOf(RootOf(_Z^
3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*x^2+7*(x^3+2)^(1/3)*RootOf(_Z^3-9)^2*x^2+36*RootOf(RootOf(_
Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3+16*RootOf(_Z^3-9)*x^3+21*x*(x^3+2)^(2/3)+18*RootOf(RootOf(_Z^3-9)^2+
9*_Z*RootOf(_Z^3-9)+81*_Z^2)+8*RootOf(_Z^3-9))/(-1+x)/(x^2+x+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (59) = 118\).
time = 3.64, size = 232, normalized size = 2.97 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x - 1\right ) \sqrt [3]{x^{3} + 2} \left (x^{2} + x + 1\right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x^3-1\right )\,{\left (x^3+2\right )}^{1/3}} \,d x \end {gather*}

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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