3.12.36 \(\int \frac {x}{(-1+x^3) (-1+2 x^3)^{2/3}} \, dx\) [1136]

Optimal. Leaf size=85 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+2 x^3}+\left (-1+2 x^3\right )^{2/3}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 60, normalized size of antiderivative = 0.71, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {503} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{2 x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^3-1\right )+\frac {1}{2} \log \left (x-\sqrt [3]{2 x^3-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 503

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

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 85, normalized size = 1.00 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+2 x^3}+\left (-1+2 x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.60, size = 447, normalized size = 5.26

method	result	size
trager	\(\frac {\ln \left (-\frac {-27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-30 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+4 \left (2 x^{3}-1\right )^{\frac {2}{3}} x -5 \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-\frac {\ln \left (\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {2}{3}} x -12 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+\left (2 x^{3}-1\right )^{\frac {2}{3}} x -5 \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-\ln \left (\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {2}{3}} x -12 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+\left (2 x^{3}-1\right )^{\frac {2}{3}} x -5 \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(447\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*ln(-(-27*RootOf(9*_Z^2+3*_Z+1)^2*x^3+15*RootOf(9*_Z^2+3*_Z+1)*(2*x^3-1)^(2/3)*x-3*RootOf(9*_Z^2+3*_Z+1)*(2
*x^3-1)^(1/3)*x^2-30*RootOf(9*_Z^2+3*_Z+1)*x^3+4*(2*x^3-1)^(2/3)*x-5*(2*x^3-1)^(1/3)*x^2-3*x^3+9*RootOf(9*_Z^2
+3*_Z+1)+1)/(-1+x)/(x^2+x+1))-1/3*ln((-9*RootOf(9*_Z^2+3*_Z+1)^2*x^3+15*RootOf(9*_Z^2+3*_Z+1)*(2*x^3-1)^(2/3)*
x-12*RootOf(9*_Z^2+3*_Z+1)*(2*x^3-1)^(1/3)*x^2-3*RootOf(9*_Z^2+3*_Z+1)*x^3+(2*x^3-1)^(2/3)*x-5*(2*x^3-1)^(1/3)
*x^2+6*x^3-3*RootOf(9*_Z^2+3*_Z+1)-3)/(-1+x)/(x^2+x+1))-ln((-9*RootOf(9*_Z^2+3*_Z+1)^2*x^3+15*RootOf(9*_Z^2+3*
_Z+1)*(2*x^3-1)^(2/3)*x-12*RootOf(9*_Z^2+3*_Z+1)*(2*x^3-1)^(1/3)*x^2-3*RootOf(9*_Z^2+3*_Z+1)*x^3+(2*x^3-1)^(2/
3)*x-5*(2*x^3-1)^(1/3)*x^2+6*x^3-3*RootOf(9*_Z^2+3*_Z+1)-3)/(-1+x)/(x^2+x+1))*RootOf(9*_Z^2+3*_Z+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(x/(x^3-1)/(2*x^3-1)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.62, size = 104, normalized size = 1.22 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2 \, x^{3} - 1\right )}}{10 \, x^{3} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*arctan(-(4*sqrt(3)*(2*x^3 - 1)^(1/3)*x^2 - 2*sqrt(3)*(2*x^3 - 1)^(2/3)*x + sqrt(3)*(2*x^3 - 1))/(1
0*x^3 - 1)) + 1/6*log((x^3 + 3*(2*x^3 - 1)^(1/3)*x^2 - 3*(2*x^3 - 1)^(2/3)*x - 1)/(x^3 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/((x - 1)*(2*x**3 - 1)**(2/3)*(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(x/(x^3-1)/(2*x^3-1)^(2/3),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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