3.6.79 \(\int \frac {1}{\sqrt [3]{a-b x^3}} \, dx\) [579]

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

[Out]

1/2*ln(b^(1/3)*x+(-b*x^3+a)^(1/3))/b^(1/3)-1/3*arctan(1/3*(1-2*b^(1/3)*x/(-b*x^3+a)^(1/3))*3^(1/2))/b^(1/3)*3^
(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 = {245} \begin {gather*} \frac {\log \left (\sqrt [3]{a-b x^3}+\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}-\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}} \end {gather*}

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 245

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*}

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Mathematica [A]
time = 0.06, size = 116, normalized size = 1.61 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {-1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}}{\sqrt {3}}\right )-\log \left (1+\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}}\right )+2 \log \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}\right )}{6 \sqrt [3]{b}} \end {gather*}

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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx\]

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 [A]
time = 0.53, size = 103, normalized size = 1.43 \begin {gather*} -\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (b^{\frac {1}{3}} - \frac {2 \, {\left (-b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{3 \, b^{\frac {1}{3}}} - \frac {\log \left (b^{\frac {2}{3}} - \frac {{\left (-b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (-b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, b^{\frac {1}{3}}} + \frac {\log \left (b^{\frac {1}{3}} + \frac {{\left (-b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{3 \, b^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (55) = 110\).
time = 0.37, size = 284, normalized size = 3.94 \begin {gather*} \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 {gather*}

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] Result contains complex when optimal does not.
time = 0.44, size = 37, normalized size = 0.51 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 1.07, size = 38, normalized size = 0.53 \begin {gather*} \frac {x\,{\left (1-\frac {b\,x^3}{a}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ \frac {b\,x^3}{a}\right )}{{\left (a-b\,x^3\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*x^3)^(1/3),x)

[Out]

(x*(1 - (b*x^3)/a)^(1/3)*hypergeom([1/3, 1/3], 4/3, (b*x^3)/a))/(a - b*x^3)^(1/3)

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