3.1323 \(\int \frac{x}{a+b x^6} \, dx\)

Optimal. Leaf size=123 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{2/3} \sqrt [3]{b}} \]

[Out]

-ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(2*Sqrt[3]*a^(2/3)*b^(1/3))
+ Log[a^(1/3) + b^(1/3)*x^2]/(6*a^(2/3)*b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3
)*x^2 + b^(2/3)*x^4]/(12*a^(2/3)*b^(1/3))

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Rubi [A]  time = 0.172227, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^6),x]

[Out]

-ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))]/(2*Sqrt[3]*a^(2/3)*b^(1/3))
+ Log[a^(1/3) + b^(1/3)*x^2]/(6*a^(2/3)*b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3
)*x^2 + b^(2/3)*x^4]/(12*a^(2/3)*b^(1/3))

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Rubi in Sympy [A]  time = 28.4407, size = 114, normalized size = 0.93 \[ \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} x^{2} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{2} + b^{\frac{2}{3}} x^{4} \right )}}{12 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x^{2}}{3}\right )}{\sqrt [3]{a}} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x**6+a),x)

[Out]

log(a**(1/3) + b**(1/3)*x**2)/(6*a**(2/3)*b**(1/3)) - log(a**(2/3) - a**(1/3)*b*
*(1/3)*x**2 + b**(2/3)*x**4)/(12*a**(2/3)*b**(1/3)) - sqrt(3)*atan(sqrt(3)*(a**(
1/3)/3 - 2*b**(1/3)*x**2/3)/a**(1/3))/(6*a**(2/3)*b**(1/3))

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Mathematica [A]  time = 0.0510894, size = 154, normalized size = 1.25 \[ -\frac{-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{12 a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^6),x]

[Out]

-(2*Sqrt[3]*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*Sqrt[3]*ArcTan[Sqrt[3] +
(2*b^(1/6)*x)/a^(1/6)] - 2*Log[a^(1/3) + b^(1/3)*x^2] + Log[a^(1/3) - Sqrt[3]*a
^(1/6)*b^(1/6)*x + b^(1/3)*x^2] + Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1
/3)*x^2])/(12*a^(2/3)*b^(1/3))

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Maple [A]  time = 0.002, size = 97, normalized size = 0.8 \[{\frac{1}{6\,b}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{12\,b}\ln \left ({x}^{4}-{x}^{2}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{6\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^6+a),x)

[Out]

1/6/b/(a/b)^(2/3)*ln(x^2+(a/b)^(1/3))-1/12/b/(a/b)^(2/3)*ln(x^4-x^2*(a/b)^(1/3)+
(a/b)^(2/3))+1/6/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^2-1))

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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(x/(b*x^6 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224558, size = 128, normalized size = 1.04 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{4} - \left (a^{2} b\right )^{\frac{1}{3}} a x^{2} + a^{2}\right ) - 2 \, \sqrt{3} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x^{2} + a\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x^{2} - \sqrt{3} a}{3 \, a}\right )\right )}}{36 \, \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^6 + a),x, algorithm="fricas")

[Out]

-1/36*sqrt(3)*(sqrt(3)*log((a^2*b)^(2/3)*x^4 - (a^2*b)^(1/3)*a*x^2 + a^2) - 2*sq
rt(3)*log((a^2*b)^(1/3)*x^2 + a) - 6*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x^2 - s
qrt(3)*a)/a))/(a^2*b)^(1/3)

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Sympy [A]  time = 0.496687, size = 22, normalized size = 0.18 \[ \operatorname{RootSum}{\left (216 t^{3} a^{2} b - 1, \left ( t \mapsto t \log{\left (6 t a + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**6+a),x)

[Out]

RootSum(216*_t**3*a**2*b - 1, Lambda(_t, _t*log(6*_t*a + x**2)))

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GIAC/XCAS [A]  time = 0.229002, size = 159, normalized size = 1.29 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{6 \, a} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{6 \, a b} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{12 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^6 + a),x, algorithm="giac")

[Out]

-1/6*(-a/b)^(1/3)*ln(abs(x^2 - (-a/b)^(1/3)))/a + 1/6*sqrt(3)*(-a*b^2)^(1/3)*arc
tan(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b) + 1/12*(-a*b^2)^(1/3)
*ln(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b)