3.40 \(\int \frac{1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{3}{4} \log \left (\sqrt [3]{x^3-3 x^2+7 x-5}-x+1\right )+\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 (x-1)}{\sqrt{3} \sqrt [3]{x^3-3 x^2+7 x-5}}+\frac{1}{\sqrt{3}}\right )+\frac{1}{4} \log (1-x) \]

[Out]

(Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(-1 + x))/(Sqrt[3]*(-5 + 7*x - 3*x^2 + x^3)^(1/3))])/2 + Log[1 - x]/4 - (3*Log[
1 - x + (-5 + 7*x - 3*x^2 + x^3)^(1/3)])/4

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Rubi [A]  time = 0.0728112, antiderivative size = 131, normalized size of antiderivative = 1.62, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {2067, 2011, 329, 275, 239} \[ \frac{\sqrt{3} \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \tan ^{-1}\left (\frac{\frac{2 (x-1)^{2/3}}{\sqrt [3]{(x-1)^2+4}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{(x-1)^3+4 (x-1)}}-\frac{3 \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \log \left ((x-1)^{2/3}-\sqrt [3]{(x-1)^2+4}\right )}{4 \sqrt [3]{(x-1)^3+4 (x-1)}} \]

Antiderivative was successfully verified.

[In]

Int[(-5 + 7*x - 3*x^2 + x^3)^(-1/3),x]

[Out]

(Sqrt[3]*(4 + (-1 + x)^2)^(1/3)*(-1 + x)^(1/3)*ArcTan[(1 + (2*(-1 + x)^(2/3))/(4 + (-1 + x)^2)^(1/3))/Sqrt[3]]
)/(2*(4*(-1 + x) + (-1 + x)^3)^(1/3)) - (3*(4 + (-1 + x)^2)^(1/3)*(-1 + x)^(1/3)*Log[-(4 + (-1 + x)^2)^(1/3) +
(-1 + x)^(2/3)])/(4*(4*(-1 + x) + (-1 + x)^3)^(1/3))

Rule 2067

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x,
x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rule 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{-5+7 x-3 x^2+x^3}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{4 x+x^3}} \, dx,x,-1+x\right )\\ &=\frac{\left (\sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} \sqrt [3]{4+x^2}} \, dx,x,-1+x\right )}{\sqrt [3]{4 (-1+x)+(-1+x)^3}}\\ &=\frac{\left (3 \sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt [3]{4+x^6}} \, dx,x,\sqrt [3]{-1+x}\right )}{\sqrt [3]{4 (-1+x)+(-1+x)^3}}\\ &=\frac{\left (3 \sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{4+x^3}} \, dx,x,(-1+x)^{2/3}\right )}{2 \sqrt [3]{4 (-1+x)+(-1+x)^3}}\\ &=\frac{\sqrt{3} \sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x} \tan ^{-1}\left (\frac{1+\frac{2 (-1+x)^{2/3}}{\sqrt [3]{4+(-1+x)^2}}}{\sqrt{3}}\right )}{2 \sqrt [3]{-4 (1-x)+(-1+x)^3}}-\frac{3 \sqrt [3]{4+(-1+x)^2} \sqrt [3]{-1+x} \log \left (\sqrt [3]{4+(-1+x)^2}-(-1+x)^{2/3}\right )}{4 \sqrt [3]{-4 (1-x)+(-1+x)^3}}\\ \end{align*}

Mathematica [C]  time = 0.0139155, size = 85, normalized size = 1.05 \[ \frac{3 \sqrt [3]{i x+(2-i)} \sqrt [3]{i (x-1)} (x-(1-2 i)) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{1}{4} i (x-(1-2 i)),-\frac{1}{2} i (x-(1-2 i))\right )}{4 \sqrt [3]{x^3-3 x^2+7 x-5}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(-5 + 7*x - 3*x^2 + x^3)^(-1/3),x]

[Out]

(3*((2 - I) + I*x)^(1/3)*(I*(-1 + x))^(1/3)*((-1 + 2*I) + x)*AppellF1[2/3, 1/3, 1/3, 5/3, (-I/4)*((-1 + 2*I) +
x), (-I/2)*((-1 + 2*I) + x)])/(4*(-5 + 7*x - 3*x^2 + x^3)^(1/3))

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Maple [F]  time = 0.011, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{{x}^{3}-3\,{x}^{2}+7\,x-5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3-3*x^2+7*x-5)^(1/3),x)

[Out]

int(1/(x^3-3*x^2+7*x-5)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-3*x^2+7*x-5)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^3 - 3*x^2 + 7*x - 5)^(-1/3), x)

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Fricas [A]  time = 4.68439, size = 410, normalized size = 5.06 \begin{align*} -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{22791076 \, \sqrt{3}{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac{1}{3}}{\left (x - 1\right )} + \sqrt{3}{\left (20389537 \, x^{2} - 40779074 \, x + 53222437\right )} + 17987998 \, \sqrt{3}{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac{2}{3}}}{7204617 \, x^{2} - 14409234 \, x - 20666867}\right ) - \frac{1}{4} \, \log \left (3 \,{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac{1}{3}}{\left (x - 1\right )} - 3 \,{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac{2}{3}} + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-3*x^2+7*x-5)^(1/3),x, algorithm="fricas")

[Out]

-1/2*sqrt(3)*arctan((22791076*sqrt(3)*(x^3 - 3*x^2 + 7*x - 5)^(1/3)*(x - 1) + sqrt(3)*(20389537*x^2 - 40779074
*x + 53222437) + 17987998*sqrt(3)*(x^3 - 3*x^2 + 7*x - 5)^(2/3))/(7204617*x^2 - 14409234*x - 20666867)) - 1/4*
log(3*(x^3 - 3*x^2 + 7*x - 5)^(1/3)*(x - 1) - 3*(x^3 - 3*x^2 + 7*x - 5)^(2/3) + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{x^{3} - 3 x^{2} + 7 x - 5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**3-3*x**2+7*x-5)**(1/3),x)

[Out]

Integral((x**3 - 3*x**2 + 7*x - 5)**(-1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-3*x^2+7*x-5)^(1/3),x, algorithm="giac")

[Out]

integrate((x^3 - 3*x^2 + 7*x - 5)^(-1/3), x)