3.1968 \(\int \frac{1}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=119 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3}}+\frac{x}{a} \]

[Out]

x/a + (b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(4/
3)) - (b^(1/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(4/3)) + (b^(1/3)*Log[b^(2/3) - a^
(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(4/3))

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Rubi [A]  time = 0.155705, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{4/3}}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^3)^(-1),x]

[Out]

x/a + (b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(4/
3)) - (b^(1/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(4/3)) + (b^(1/3)*Log[b^(2/3) - a^
(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(4/3))

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Rubi in Sympy [A]  time = 29.5204, size = 112, normalized size = 0.94 \[ \frac{x}{a} - \frac{\sqrt [3]{b} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 a^{\frac{4}{3}}} + \frac{\sqrt [3]{b} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 a^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**3),x)

[Out]

x/a - b**(1/3)*log(a**(1/3)*x + b**(1/3))/(3*a**(4/3)) + b**(1/3)*log(a**(2/3)*x
**2 - a**(1/3)*b**(1/3)*x + b**(2/3))/(6*a**(4/3)) + sqrt(3)*b**(1/3)*atan(sqrt(
3)*(-2*a**(1/3)*x/3 + b**(1/3)/3)/b**(1/3))/(3*a**(4/3))

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Mathematica [A]  time = 0.0291277, size = 108, normalized size = 0.91 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )+6 \sqrt [3]{a} x}{6 a^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^3)^(-1),x]

[Out]

(6*a^(1/3)*x + 2*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] - 2
*b^(1/3)*Log[b^(1/3) + a^(1/3)*x] + b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^
(2/3)*x^2])/(6*a^(4/3))

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Maple [A]  time = 0.003, size = 99, normalized size = 0.8 \[{\frac{x}{a}}-{\frac{b}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{6\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b\sqrt{3}}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3),x)

[Out]

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

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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(1/(a + b/x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232219, size = 157, normalized size = 1.32 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 6 \, \sqrt{3} x + 6 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{b}{a}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )\right )}}{18 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.27161, size = 22, normalized size = 0.18 \[ \operatorname{RootSum}{\left (27 t^{3} a^{4} + b, \left ( t \mapsto t \log{\left (- 3 t a + x \right )} \right )\right )} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3),x)

[Out]

RootSum(27*_t**3*a**4 + b, Lambda(_t, _t*log(-3*_t*a + x))) + x/a

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GIAC/XCAS [A]  time = 0.223646, size = 150, normalized size = 1.26 \[ \frac{\left (-\frac{b}{a}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a} + \frac{x}{a} - \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2}} - \frac{\left (-a^{2} b\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(-b/a)^(1/3)*ln(abs(x - (-b/a)^(1/3)))/a + x/a - 1/3*sqrt(3)*(-a^2*b)^(1/3)*
arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/a^2 - 1/6*(-a^2*b)^(1/3)*l
n(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^2