3.1324 \(\int \frac{1}{a+b x^6} \, dx\)

Optimal. Leaf size=215 \[ -\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{5/6} \sqrt [6]{b}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{5/6} \sqrt [6]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}} \]

[Out]

ArcTan[(b^(1/6)*x)/a^(1/6)]/(3*a^(5/6)*b^(1/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^
(1/6)*x)/a^(1/6)]/(6*a^(5/6)*b^(1/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a
^(1/6)]/(6*a^(5/6)*b^(1/6)) - Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*
x^2]/(4*Sqrt[3]*a^(5/6)*b^(1/6)) + Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(
1/3)*x^2]/(4*Sqrt[3]*a^(5/6)*b^(1/6))

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Rubi [A]  time = 0.806408, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{5/6} \sqrt [6]{b}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{5/6} \sqrt [6]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^6)^(-1),x]

[Out]

ArcTan[(b^(1/6)*x)/a^(1/6)]/(3*a^(5/6)*b^(1/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^
(1/6)*x)/a^(1/6)]/(6*a^(5/6)*b^(1/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a
^(1/6)]/(6*a^(5/6)*b^(1/6)) - Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*
x^2]/(4*Sqrt[3]*a^(5/6)*b^(1/6)) + Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(
1/3)*x^2]/(4*Sqrt[3]*a^(5/6)*b^(1/6))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**6+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0347063, size = 154, normalized size = 0.72 \[ \frac{-\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{12 a^{5/6} \sqrt [6]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^6)^(-1),x]

[Out]

(4*ArcTan[(b^(1/6)*x)/a^(1/6)] - 2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*A
rcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] - Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b
^(1/6)*x + b^(1/3)*x^2] + Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1
/3)*x^2])/(12*a^(5/6)*b^(1/6))

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Maple [A]  time = 0.041, size = 162, normalized size = 0.8 \[{\frac{1}{3\,a}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{\sqrt{3}}{12\,a}\sqrt [6]{{\frac{a}{b}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{a}{b}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{6\,a}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{\sqrt{3}}{12\,a}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{6\,a}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^6+a),x)

[Out]

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

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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^6 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232715, size = 425, normalized size = 1.98 \[ -\frac{1}{3} \, \sqrt{3} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}}}{a \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + 2 \, x + 2 \, \sqrt{a^{2} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{3}} + a x \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x^{2}}}\right ) - \frac{1}{3} \, \sqrt{3} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}}}{a \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} - 2 \, x - 2 \, \sqrt{a^{2} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{3}} - a x \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x^{2}}}\right ) + \frac{1}{12} \, \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{3}} + a x \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x^{2}\right ) - \frac{1}{12} \, \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{3}} - a x \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x^{2}\right ) + \frac{1}{6} \, \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \log \left (a \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x\right ) - \frac{1}{6} \, \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \log \left (-a \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^6 + a),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*(-1/(a^5*b))^(1/6)*arctan(sqrt(3)*a*(-1/(a^5*b))^(1/6)/(a*(-1/(a^5*
b))^(1/6) + 2*x + 2*sqrt(a^2*(-1/(a^5*b))^(1/3) + a*x*(-1/(a^5*b))^(1/6) + x^2))
) - 1/3*sqrt(3)*(-1/(a^5*b))^(1/6)*arctan(-sqrt(3)*a*(-1/(a^5*b))^(1/6)/(a*(-1/(
a^5*b))^(1/6) - 2*x - 2*sqrt(a^2*(-1/(a^5*b))^(1/3) - a*x*(-1/(a^5*b))^(1/6) + x
^2))) + 1/12*(-1/(a^5*b))^(1/6)*log(a^2*(-1/(a^5*b))^(1/3) + a*x*(-1/(a^5*b))^(1
/6) + x^2) - 1/12*(-1/(a^5*b))^(1/6)*log(a^2*(-1/(a^5*b))^(1/3) - a*x*(-1/(a^5*b
))^(1/6) + x^2) + 1/6*(-1/(a^5*b))^(1/6)*log(a*(-1/(a^5*b))^(1/6) + x) - 1/6*(-1
/(a^5*b))^(1/6)*log(-a*(-1/(a^5*b))^(1/6) + x)

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Sympy [A]  time = 0.430496, size = 20, normalized size = 0.09 \[ \operatorname{RootSum}{\left (46656 t^{6} a^{5} b + 1, \left ( t \mapsto t \log{\left (6 t a + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**6+a),x)

[Out]

RootSum(46656*_t**6*a**5*b + 1, Lambda(_t, _t*log(6*_t*a + x)))

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GIAC/XCAS [A]  time = 0.222479, size = 257, normalized size = 1.2 \[ \frac{\sqrt{3} \left (a b^{5}\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} + \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{6}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{12 \, a b} - \frac{\sqrt{3} \left (a b^{5}\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} - \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{6}} + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{12 \, a b} + \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{6 \, a b} + \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{6 \, a b} + \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{x}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{3 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^6 + a),x, algorithm="giac")

[Out]

1/12*sqrt(3)*(a*b^5)^(1/6)*ln(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a*b) -
1/12*sqrt(3)*(a*b^5)^(1/6)*ln(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a*b)
+ 1/6*(a*b^5)^(1/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a*b) + 1/6*
(a*b^5)^(1/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a*b) + 1/3*(a*b^5
)^(1/6)*arctan(x/(a/b)^(1/6))/(a*b)