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

Optimal. Leaf size=115 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.105793, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.9216, size = 109, normalized size = 0.95 \[ \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**3+a),x)

[Out]

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

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Mathematica [A]  time = 0.0217544, size = 89, normalized size = 0.77 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(-1),x]

[Out]

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

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Maple [A]  time = 0.002, size = 91, normalized size = 0.8 \[{\frac{1}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a),x)

[Out]

1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)
^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(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/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246363, size = 120, normalized size = 1.04 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right )\right )}}{18 \, \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.347146, size = 20, normalized size = 0.17 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log{\left (3 t a + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*a**2*b - 1, Lambda(_t, _t*log(3*_t*a + x)))

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GIAC/XCAS [A]  time = 0.21968, size = 151, normalized size = 1.31 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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