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

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 = 0.0100193, 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, Rules used = {239} \[ \frac{\log \left (\sqrt [3]{a-b x^3}+\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]

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))

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]{a-b x^3}} \, dx &=-\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}}+\frac{\log \left (\sqrt [3]{b} x+\sqrt [3]{a-b x^3}\right )}{2 \sqrt [3]{b}}\\ \end{align*}

Mathematica [A]  time = 0.0695784, 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 = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{-b{x}^{3}+a}}}\, dx \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.56207, size = 772, normalized size = 10.72 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (-3 \, b x^{3} - 3 \,{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}} x^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (b^{\frac{4}{3}} x^{3} -{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b x^{2} - 2 \,{\left (-b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{2}{3}} x\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} + 2 \, a\right ) + 2 \, b^{\frac{2}{3}} \log \left (\frac{b^{\frac{1}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - b^{\frac{2}{3}} \log \left (\frac{b^{\frac{2}{3}} x^{2} -{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{1}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right )}{6 \, b}, \frac{6 \, \sqrt{\frac{1}{3}} b^{\frac{2}{3}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (b^{\frac{1}{3}} x - 2 \,{\left (-b x^{3} + a\right )}^{\frac{1}{3}}\right )}}{b^{\frac{1}{3}} x}\right ) + 2 \, b^{\frac{2}{3}} \log \left (\frac{b^{\frac{1}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - b^{\frac{2}{3}} \log \left (\frac{b^{\frac{2}{3}} x^{2} -{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{1}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right )}{6 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*sqrt(1/3)*b*sqrt(-1/b^(2/3))*log(-3*b*x^3 - 3*(-b*x^3 + a)^(1/3)*b^(2/3)*x^2 + 3*sqrt(1/3)*(b^(4/3)*x^
3 - (-b*x^3 + a)^(1/3)*b*x^2 - 2*(-b*x^3 + a)^(2/3)*b^(2/3)*x)*sqrt(-1/b^(2/3)) + 2*a) + 2*b^(2/3)*log((b^(1/3
)*x + (-b*x^3 + a)^(1/3))/x) - b^(2/3)*log((b^(2/3)*x^2 - (-b*x^3 + a)^(1/3)*b^(1/3)*x + (-b*x^3 + a)^(2/3))/x
^2))/b, 1/6*(6*sqrt(1/3)*b^(2/3)*arctan(sqrt(1/3)*(b^(1/3)*x - 2*(-b*x^3 + a)^(1/3))/(b^(1/3)*x)) + 2*b^(2/3)*
log((b^(1/3)*x + (-b*x^3 + a)^(1/3))/x) - b^(2/3)*log((b^(2/3)*x^2 - (-b*x^3 + a)^(1/3)*b^(1/3)*x + (-b*x^3 +
a)^(2/3))/x^2))/b]

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Sympy [C]  time = 1.37568, size = 37, normalized size = 0.51 \begin{align*} \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 )} \end{align*}

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)*gamma(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-b*x^3 + a)^(-1/3), x)