3.2107 \(\int \frac{1}{a+\frac{b}{x^6}} \, dx\)

Optimal. Leaf size=220 \[ \frac{\sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{a} x+\sqrt{3} \sqrt [6]{b}}{\sqrt [6]{b}}\right )}{6 a^{7/6}}+\frac{x}{a} \]

[Out]

x/a - (b^(1/6)*ArcTan[(a^(1/6)*x)/b^(1/6)])/(3*a^(7/6)) + (b^(1/6)*ArcTan[(Sqrt[
3]*b^(1/6) - 2*a^(1/6)*x)/b^(1/6)])/(6*a^(7/6)) - (b^(1/6)*ArcTan[(Sqrt[3]*b^(1/
6) + 2*a^(1/6)*x)/b^(1/6)])/(6*a^(7/6)) + (b^(1/6)*Log[b^(1/3) - Sqrt[3]*a^(1/6)
*b^(1/6)*x + a^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6)) - (b^(1/6)*Log[b^(1/3) + Sqrt[3]*
a^(1/6)*b^(1/6)*x + a^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6))

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Rubi [A]  time = 0.988402, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889 \[ \frac{\sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{a} x+\sqrt{3} \sqrt [6]{b}}{\sqrt [6]{b}}\right )}{6 a^{7/6}}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^6)^(-1),x]

[Out]

x/a - (b^(1/6)*ArcTan[(a^(1/6)*x)/b^(1/6)])/(3*a^(7/6)) + (b^(1/6)*ArcTan[(Sqrt[
3]*b^(1/6) - 2*a^(1/6)*x)/b^(1/6)])/(6*a^(7/6)) - (b^(1/6)*ArcTan[(Sqrt[3]*b^(1/
6) + 2*a^(1/6)*x)/b^(1/6)])/(6*a^(7/6)) + (b^(1/6)*Log[b^(1/3) - Sqrt[3]*a^(1/6)
*b^(1/6)*x + a^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6)) - (b^(1/6)*Log[b^(1/3) + Sqrt[3]*
a^(1/6)*b^(1/6)*x + a^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6))

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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/(a+b/x**6),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0924812, size = 182, normalized size = 0.83 \[ \frac{\sqrt{3} \sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )-\sqrt{3} \sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )-4 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )+2 \sqrt [6]{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )-2 \sqrt [6]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{a} x}{\sqrt [6]{b}}+\sqrt{3}\right )+12 \sqrt [6]{a} x}{12 a^{7/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^6)^(-1),x]

[Out]

(12*a^(1/6)*x - 4*b^(1/6)*ArcTan[(a^(1/6)*x)/b^(1/6)] + 2*b^(1/6)*ArcTan[Sqrt[3]
- (2*a^(1/6)*x)/b^(1/6)] - 2*b^(1/6)*ArcTan[Sqrt[3] + (2*a^(1/6)*x)/b^(1/6)] +
Sqrt[3]*b^(1/6)*Log[b^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + a^(1/3)*x^2] - Sqrt[3]
*b^(1/6)*Log[b^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + a^(1/3)*x^2])/(12*a^(7/6))

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Maple [A]  time = 0.089, size = 167, normalized size = 0.8 \[{\frac{x}{a}}-{\frac{\sqrt{3}}{12\,a}\sqrt [6]{{\frac{b}{a}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{b}{a}}}x+\sqrt [3]{{\frac{b}{a}}} \right ) }-{\frac{1}{6\,a}\sqrt [6]{{\frac{b}{a}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{b}{a}}}}}}+\sqrt{3} \right ) }+{\frac{\sqrt{3}}{12\,a}\sqrt [6]{{\frac{b}{a}}}\ln \left ( \sqrt{3}\sqrt [6]{{\frac{b}{a}}}x-{x}^{2}-\sqrt [3]{{\frac{b}{a}}} \right ) }-{\frac{1}{6\,a}\sqrt [6]{{\frac{b}{a}}}\arctan \left ( -\sqrt{3}+2\,{x{\frac{1}{\sqrt [6]{{\frac{b}{a}}}}}} \right ) }-{\frac{1}{3\,a}\sqrt [6]{{\frac{b}{a}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{b}{a}}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^6),x)

[Out]

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

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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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253097, size = 389, normalized size = 1.77 \[ \frac{4 \, \sqrt{3} a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \arctan \left (\frac{\sqrt{3} a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}}}{a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + 2 \, x + 2 \, \sqrt{a^{2} \left (-\frac{b}{a^{7}}\right )^{\frac{1}{3}} + a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x^{2}}}\right ) + 4 \, \sqrt{3} a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\sqrt{3} a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}}}{a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} - 2 \, x - 2 \, \sqrt{a^{2} \left (-\frac{b}{a^{7}}\right )^{\frac{1}{3}} - a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x^{2}}}\right ) - a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{b}{a^{7}}\right )^{\frac{1}{3}} + a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) + a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{b}{a^{7}}\right )^{\frac{1}{3}} - a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) - 2 \, a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x\right ) + 2 \, a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (-a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x\right ) + 12 \, x}{12 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(4*sqrt(3)*a*(-b/a^7)^(1/6)*arctan(sqrt(3)*a*(-b/a^7)^(1/6)/(a*(-b/a^7)^(1/
6) + 2*x + 2*sqrt(a^2*(-b/a^7)^(1/3) + a*x*(-b/a^7)^(1/6) + x^2))) + 4*sqrt(3)*a
*(-b/a^7)^(1/6)*arctan(-sqrt(3)*a*(-b/a^7)^(1/6)/(a*(-b/a^7)^(1/6) - 2*x - 2*sqr
t(a^2*(-b/a^7)^(1/3) - a*x*(-b/a^7)^(1/6) + x^2))) - a*(-b/a^7)^(1/6)*log(a^2*(-
b/a^7)^(1/3) + a*x*(-b/a^7)^(1/6) + x^2) + a*(-b/a^7)^(1/6)*log(a^2*(-b/a^7)^(1/
3) - a*x*(-b/a^7)^(1/6) + x^2) - 2*a*(-b/a^7)^(1/6)*log(a*(-b/a^7)^(1/6) + x) +
2*a*(-b/a^7)^(1/6)*log(-a*(-b/a^7)^(1/6) + x) + 12*x)/a

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Sympy [A]  time = 1.34074, size = 22, normalized size = 0.1 \[ \operatorname{RootSum}{\left (46656 t^{6} a^{7} + b, \left ( t \mapsto t \log{\left (- 6 t a + x \right )} \right )\right )} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**6),x)

[Out]

RootSum(46656*_t**6*a**7 + b, Lambda(_t, _t*log(-6*_t*a + x))) + x/a

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GIAC/XCAS [A]  time = 0.217526, size = 243, normalized size = 1.1 \[ \frac{x}{a} - \frac{\sqrt{3} \left (a^{5} b\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} + \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{6}} + \left (\frac{b}{a}\right )^{\frac{1}{3}}\right )}{12 \, a^{2}} + \frac{\sqrt{3} \left (a^{5} b\right )^{\frac{1}{6}}{\rm ln}\left (x^{2} - \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{6}} + \left (\frac{b}{a}\right )^{\frac{1}{3}}\right )}{12 \, a^{2}} - \frac{\left (a^{5} b\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{6}}}{\left (\frac{b}{a}\right )^{\frac{1}{6}}}\right )}{6 \, a^{2}} - \frac{\left (a^{5} b\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{6}}}{\left (\frac{b}{a}\right )^{\frac{1}{6}}}\right )}{6 \, a^{2}} - \frac{\left (a^{5} b\right )^{\frac{1}{6}} \arctan \left (\frac{x}{\left (\frac{b}{a}\right )^{\frac{1}{6}}}\right )}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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