3.33 \(\int \frac{1}{-1+2 x^3} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0440628, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (2^{2/3} x^2+\sqrt [3]{2} x+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (1-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{2} x+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x^3)^(-1),x]

[Out]

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

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

Mathematica [A]  time = 0.0221478, size = 66, normalized size = 0.85 \[ -\frac{\log \left (2^{2/3} x^2+\sqrt [3]{2} x+1\right )-2 \log \left (1-\sqrt [3]{2} x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{2} x+1}{\sqrt{3}}\right )}{6 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x^3)^(-1),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 58, normalized size = 0.7 \begin{align*}{\frac{{2}^{{\frac{2}{3}}}}{6}\ln \left ( x-{\frac{{2}^{{\frac{2}{3}}}}{2}} \right ) }-{\frac{{2}^{{\frac{2}{3}}}}{12}\ln \left ({x}^{2}+{\frac{{2}^{{\frac{2}{3}}}x}{2}}+{\frac{\sqrt [3]{2}}{2}} \right ) }-{\frac{{2}^{{\frac{2}{3}}}\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,\sqrt [3]{2}x \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*2^(2/3)*ln(x-1/2*2^(2/3))-1/12*2^(2/3)*ln(x^2+1/2*2^(2/3)*x+1/2*2^(1/3))-1/6*arctan(1/3*(1+2*2^(1/3)*x)*3^
(1/2))*2^(2/3)*3^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.45058, size = 89, normalized size = 1.14 \begin{align*} -\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2 \cdot 2^{\frac{2}{3}} x + 2^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} x^{2} + 2^{\frac{1}{3}} x + 1\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (\frac{1}{2} \cdot 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} x - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2*2^(2/3)*x + 2^(1/3))) - 1/12*2^(2/3)*log(2^(2/3)*x^2 + 2^(1
/3)*x + 1) + 1/6*2^(2/3)*log(1/2*2^(2/3)*(2^(1/3)*x - 1))

________________________________________________________________________________________

Fricas [A]  time = 1.93149, size = 209, normalized size = 2.68 \begin{align*} -\frac{1}{6} \, \sqrt{6} 2^{\frac{1}{6}} \arctan \left (\frac{1}{6} \, \sqrt{6} 2^{\frac{1}{6}}{\left (2 \cdot 2^{\frac{2}{3}} x + 2^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2 \, x^{2} + 2^{\frac{2}{3}} x + 2^{\frac{1}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (2 \, x - 2^{\frac{2}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(6)*2^(1/6)*arctan(1/6*sqrt(6)*2^(1/6)*(2*2^(2/3)*x + 2^(1/3))) - 1/12*2^(2/3)*log(2*x^2 + 2^(2/3)*x
+ 2^(1/3)) + 1/6*2^(2/3)*log(2*x - 2^(2/3))

________________________________________________________________________________________

Sympy [A]  time = 0.308794, size = 78, normalized size = 1. \begin{align*} \frac{2^{\frac{2}{3}} \log{\left (x - \frac{2^{\frac{2}{3}}}{2} \right )}}{6} - \frac{2^{\frac{2}{3}} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x}{2} + \frac{\sqrt [3]{2}}{2} \right )}}{12} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt [3]{2} \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2**(2/3)*log(x - 2**(2/3)/2)/6 - 2**(2/3)*log(x**2 + 2**(2/3)*x/2 + 2**(1/3)/2)/12 - 2**(2/3)*sqrt(3)*atan(2*2
**(1/3)*sqrt(3)*x/3 + sqrt(3)/3)/6

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError