3.577 \(\int \frac{1}{\sqrt [3]{a-b x^3}} \, dx\)

Optimal. Leaf size=72 \[ \frac{\log \left (\sqrt [3]{a-b x^3}+\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.0303382, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\log \left (\sqrt [3]{a-b x^3}+\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^3)^(-1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.488, size = 114, normalized size = 1.58 \[ \frac{\log{\left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a - b x^{3}}} + 1 \right )}}{3 \sqrt [3]{b}} - \frac{\log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a - b x^{3}\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{b} x}{\sqrt [3]{a - b x^{3}}} + 1 \right )}}{6 \sqrt [3]{b}} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a - b x^{3}}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-b*x**3+a)**(1/3),x)

[Out]

log(b**(1/3)*x/(a - b*x**3)**(1/3) + 1)/(3*b**(1/3)) - log(b**(2/3)*x**2/(a - b*
x**3)**(2/3) - b**(1/3)*x/(a - b*x**3)**(1/3) + 1)/(6*b**(1/3)) - sqrt(3)*atan(s
qrt(3)*(-2*b**(1/3)*x/(3*(a - b*x**3)**(1/3)) + 1/3))/(3*b**(1/3))

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Mathematica [A]  time = 0.132911, size = 116, normalized size = 1.61 \[ \frac{-\log \left (\frac{b^{2/3} x^2}{\left (a-b x^3\right )^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}+1\right )+2 \log \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}-1}{\sqrt{3}}\right )}{6 \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^3)^(-1/3),x]

[Out]

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

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [3]{-b{x}^{3}+a}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^3+a)^(1/3),x)

[Out]

int(1/(-b*x^3+a)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^3 + a)^(-1/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242721, size = 161, normalized size = 2.24 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} \log \left (\frac{b x +{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}}}{x}\right ) - \sqrt{3} \log \left (\frac{b x^{2} -{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{1}{3}}}{x^{2}}\right ) - 6 \, \arctan \left (-\frac{\sqrt{3} b x - 2 \, \sqrt{3}{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}}}{3 \, b x}\right )\right )}}{18 \, b^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^3 + a)^(-1/3),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*(2*sqrt(3)*log((b*x + (-b*x^3 + a)^(1/3)*b^(2/3))/x) - sqrt(3)*log(
(b*x^2 - (-b*x^3 + a)^(1/3)*b^(2/3)*x + (-b*x^3 + a)^(2/3)*b^(1/3))/x^2) - 6*arc
tan(-1/3*(sqrt(3)*b*x - 2*sqrt(3)*(-b*x^3 + a)^(1/3)*b^(2/3))/(b*x)))/b^(1/3)

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Sympy [A]  time = 3.65963, size = 37, normalized size = 0.51 \[ \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x**3+a)**(1/3),x)

[Out]

x*gamma(1/3)*hyper((1/3, 1/3), (4/3,), b*x**3*exp_polar(2*I*pi)/a)/(3*a**(1/3)*g
amma(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^3 + a)^(-1/3),x, algorithm="giac")

[Out]

integrate((-b*x^3 + a)^(-1/3), x)