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 = 0.139111, 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 \[ \frac{\sqrt{3} \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \tan ^{-1}\left (\frac{\frac{2 (x-1)^{2/3}}{\sqrt [3]{(x-1)^2+4}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{(x-1)^3-4 (1-x)}}-\frac{3 \sqrt [3]{(x-1)^2+4} \sqrt [3]{x-1} \log \left ((x-1)^{2/3}-\sqrt [3]{(x-1)^2+4}\right )}{4 \sqrt [3]{(x-1)^3-4 (1-x)}} \]

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))

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Rubi in Sympy [A]  time = 7.39252, size = 204, normalized size = 2.52 \[ - \frac{\sqrt [3]{x - 1} \sqrt [3]{x^{2} - 2 x + 5} \log{\left (- \frac{\left (x - 1\right )^{\frac{2}{3}}}{\sqrt [3]{\left (x - 1\right )^{2} + 4}} + 1 \right )}}{2 \sqrt [3]{x^{3} - 3 x^{2} + 7 x - 5}} + \frac{\sqrt [3]{x - 1} \sqrt [3]{x^{2} - 2 x + 5} \log{\left (\frac{\left (x - 1\right )^{\frac{4}{3}}}{\left (\left (x - 1\right )^{2} + 4\right )^{\frac{2}{3}}} + \frac{\left (x - 1\right )^{\frac{2}{3}}}{\sqrt [3]{\left (x - 1\right )^{2} + 4}} + 1 \right )}}{4 \sqrt [3]{x^{3} - 3 x^{2} + 7 x - 5}} + \frac{\sqrt{3} \sqrt [3]{x - 1} \sqrt [3]{x^{2} - 2 x + 5} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \left (x - 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{\left (x - 1\right )^{2} + 4}} + \frac{1}{3}\right ) \right )}}{2 \sqrt [3]{x^{3} - 3 x^{2} + 7 x - 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-3*x**2+7*x-5)**(1/3),x)

[Out]

-(x - 1)**(1/3)*(x**2 - 2*x + 5)**(1/3)*log(-(x - 1)**(2/3)/((x - 1)**2 + 4)**(1
/3) + 1)/(2*(x**3 - 3*x**2 + 7*x - 5)**(1/3)) + (x - 1)**(1/3)*(x**2 - 2*x + 5)*
*(1/3)*log((x - 1)**(4/3)/((x - 1)**2 + 4)**(2/3) + (x - 1)**(2/3)/((x - 1)**2 +
4)**(1/3) + 1)/(4*(x**3 - 3*x**2 + 7*x - 5)**(1/3)) + sqrt(3)*(x - 1)**(1/3)*(x
**2 - 2*x + 5)**(1/3)*atan(sqrt(3)*(2*(x - 1)**(2/3)/(3*((x - 1)**2 + 4)**(1/3))
+ 1/3))/(2*(x**3 - 3*x**2 + 7*x - 5)**(1/3))

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Mathematica [C]  time = 0.0224462, 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 = 0.019, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [3]{{x}^{3}-3\,{x}^{2}+7\,x-5}}}\, dx \]

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. \[ \int \frac{1}{{\left (x^{3} - 3 \, x^{2} + 7 \, x - 5\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)