3.2108 \(\int \frac{1}{a+\frac{b}{x^8}} \, dx\)

Optimal. Leaf size=272 \[ -\frac{\sqrt [8]{b} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac{x}{a} \]

[Out]

x/a + (b^(1/8)*ArcTan[((-a)^(1/8)*x)/b^(1/8)])/(4*(-a)^(9/8)) - (b^(1/8)*ArcTan[
1 - (Sqrt[2]*(-a)^(1/8)*x)/b^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*ArcTan[1
+ (Sqrt[2]*(-a)^(1/8)*x)/b^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*ArcTanh[((-
a)^(1/8)*x)/b^(1/8)])/(4*(-a)^(9/8)) - (b^(1/8)*Log[b^(1/4) - Sqrt[2]*(-a)^(1/8)
*b^(1/8)*x + (-a)^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*Log[b^(1/4) + Sq
rt[2]*(-a)^(1/8)*b^(1/8)*x + (-a)^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8))

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Rubi [A]  time = 0.609334, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.333 \[ -\frac{\sqrt [8]{b} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^8)^(-1),x]

[Out]

x/a + (b^(1/8)*ArcTan[((-a)^(1/8)*x)/b^(1/8)])/(4*(-a)^(9/8)) - (b^(1/8)*ArcTan[
1 - (Sqrt[2]*(-a)^(1/8)*x)/b^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*ArcTan[1
+ (Sqrt[2]*(-a)^(1/8)*x)/b^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*ArcTanh[((-
a)^(1/8)*x)/b^(1/8)])/(4*(-a)^(9/8)) - (b^(1/8)*Log[b^(1/4) - Sqrt[2]*(-a)^(1/8)
*b^(1/8)*x + (-a)^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*Log[b^(1/4) + Sq
rt[2]*(-a)^(1/8)*b^(1/8)*x + (-a)^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8))

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Rubi in Sympy [A]  time = 102.358, size = 250, normalized size = 0.92 \[ - \frac{\sqrt{2} \sqrt [8]{b} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} + x^{2} \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt{2} \sqrt [8]{b} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} + x^{2} \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt [8]{b} \operatorname{atan}{\left (\frac{x \sqrt [8]{- a}}{\sqrt [8]{b}} \right )}}{4 \left (- a\right )^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{b} \operatorname{atan}{\left (1 - \frac{\sqrt{2} x \sqrt [8]{- a}}{\sqrt [8]{b}} \right )}}{8 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt{2} \sqrt [8]{b} \operatorname{atan}{\left (1 + \frac{\sqrt{2} x \sqrt [8]{- a}}{\sqrt [8]{b}} \right )}}{8 \left (- a\right )^{\frac{9}{8}}} + \frac{\sqrt [8]{b} \operatorname{atanh}{\left (\frac{x \sqrt [8]{- a}}{\sqrt [8]{b}} \right )}}{4 \left (- a\right )^{\frac{9}{8}}} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**8),x)

[Out]

-sqrt(2)*b**(1/8)*log(-sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b**(1/4) + x**2*(-a)**(1
/4))/(16*(-a)**(9/8)) + sqrt(2)*b**(1/8)*log(sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b*
*(1/4) + x**2*(-a)**(1/4))/(16*(-a)**(9/8)) + b**(1/8)*atan(x*(-a)**(1/8)/b**(1/
8))/(4*(-a)**(9/8)) - sqrt(2)*b**(1/8)*atan(1 - sqrt(2)*x*(-a)**(1/8)/b**(1/8))/
(8*(-a)**(9/8)) + sqrt(2)*b**(1/8)*atan(1 + sqrt(2)*x*(-a)**(1/8)/b**(1/8))/(8*(
-a)**(9/8)) + b**(1/8)*atanh(x*(-a)**(1/8)/b**(1/8))/(4*(-a)**(9/8)) + x/a

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Mathematica [A]  time = 0.407333, size = 367, normalized size = 1.35 \[ \frac{\sqrt [8]{b} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-\sqrt [8]{b} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )+\sqrt [8]{b} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-\sqrt [8]{b} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-2 \sqrt [8]{b} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{a} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{b}}-\tan \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{b} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{a} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{b}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt [8]{b} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{a} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{b}}\right )-2 \sqrt [8]{b} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{a} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{b}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [8]{a} x}{8 a^{9/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^8)^(-1),x]

[Out]

(8*a^(1/8)*x - 2*b^(1/8)*ArcTan[(a^(1/8)*x*Sec[Pi/8])/b^(1/8) - Tan[Pi/8]]*Cos[P
i/8] - 2*b^(1/8)*ArcTan[(a^(1/8)*x*Sec[Pi/8])/b^(1/8) + Tan[Pi/8]]*Cos[Pi/8] + b
^(1/8)*Cos[Pi/8]*Log[b^(1/4) + a^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - b^
(1/8)*Cos[Pi/8]*Log[b^(1/4) + a^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + 2*b
^(1/8)*ArcTan[Cot[Pi/8] - (a^(1/8)*x*Csc[Pi/8])/b^(1/8)]*Sin[Pi/8] - 2*b^(1/8)*A
rcTan[Cot[Pi/8] + (a^(1/8)*x*Csc[Pi/8])/b^(1/8)]*Sin[Pi/8] + b^(1/8)*Log[b^(1/4)
+ a^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8] - b^(1/8)*Log[b^(1/4)
+ a^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8])/(8*a^(9/8))

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Maple [C]  time = 0.023, size = 34, normalized size = 0.1 \[{\frac{x}{a}}-{\frac{b}{8\,{a}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}+b \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^8),x)

[Out]

x/a-1/8*b/a^2*sum(1/_R^7*ln(x-_R),_R=RootOf(_Z^8*a+b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{b \int \frac{1}{a x^{8} + b}\,{d x}}{a} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^8),x, algorithm="maxima")

[Out]

-b*integrate(1/(a*x^8 + b), x)/a + x/a

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Fricas [A]  time = 0.252853, size = 478, normalized size = 1.76 \[ \frac{\sqrt{2}{\left (4 \, \sqrt{2} a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}}}{x + \sqrt{a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}}}\right ) - \sqrt{2} a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + x\right ) + \sqrt{2} a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (-a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + x\right ) + 4 \, a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}}}{\sqrt{2} x + a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}}}\right ) + 4 \, a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}}}{\sqrt{2} x - a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{-\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}}}\right ) - a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}\right ) + a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}\right ) + 8 \, \sqrt{2} x\right )}}{16 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^8),x, algorithm="fricas")

[Out]

1/16*sqrt(2)*(4*sqrt(2)*a*(-b/a^9)^(1/8)*arctan(a*(-b/a^9)^(1/8)/(x + sqrt(a^2*(
-b/a^9)^(1/4) + x^2))) - sqrt(2)*a*(-b/a^9)^(1/8)*log(a*(-b/a^9)^(1/8) + x) + sq
rt(2)*a*(-b/a^9)^(1/8)*log(-a*(-b/a^9)^(1/8) + x) + 4*a*(-b/a^9)^(1/8)*arctan(a*
(-b/a^9)^(1/8)/(sqrt(2)*x + a*(-b/a^9)^(1/8) + sqrt(2)*sqrt(sqrt(2)*a*x*(-b/a^9)
^(1/8) + a^2*(-b/a^9)^(1/4) + x^2))) + 4*a*(-b/a^9)^(1/8)*arctan(a*(-b/a^9)^(1/8
)/(sqrt(2)*x - a*(-b/a^9)^(1/8) + sqrt(2)*sqrt(-sqrt(2)*a*x*(-b/a^9)^(1/8) + a^2
*(-b/a^9)^(1/4) + x^2))) - a*(-b/a^9)^(1/8)*log(sqrt(2)*a*x*(-b/a^9)^(1/8) + a^2
*(-b/a^9)^(1/4) + x^2) + a*(-b/a^9)^(1/8)*log(-sqrt(2)*a*x*(-b/a^9)^(1/8) + a^2*
(-b/a^9)^(1/4) + x^2) + 8*sqrt(2)*x)/a

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Sympy [A]  time = 1.4569, size = 22, normalized size = 0.08 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{9} + b, \left ( t \mapsto t \log{\left (- 8 t a + x \right )} \right )\right )} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**8),x)

[Out]

RootSum(16777216*_t**8*a**9 + b, Lambda(_t, _t*log(-8*_t*a + x))) + x/a

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GIAC/XCAS [A]  time = 0.226716, size = 586, normalized size = 2.15 \[ -\frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{16 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^8),x, algorithm="giac")

[Out]

-1/8*sqrt(sqrt(2) + 2)*(b/a)^(1/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(b/a)^(1/8))
/(sqrt(sqrt(2) + 2)*(b/a)^(1/8)))/a - 1/8*sqrt(sqrt(2) + 2)*(b/a)^(1/8)*arctan((
2*x - sqrt(-sqrt(2) + 2)*(b/a)^(1/8))/(sqrt(sqrt(2) + 2)*(b/a)^(1/8)))/a - 1/8*s
qrt(-sqrt(2) + 2)*(b/a)^(1/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(b/a)^(1/8))/(sqrt
(-sqrt(2) + 2)*(b/a)^(1/8)))/a - 1/8*sqrt(-sqrt(2) + 2)*(b/a)^(1/8)*arctan((2*x
- sqrt(sqrt(2) + 2)*(b/a)^(1/8))/(sqrt(-sqrt(2) + 2)*(b/a)^(1/8)))/a - 1/16*sqrt
(sqrt(2) + 2)*(b/a)^(1/8)*ln(x^2 + x*sqrt(sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4)
)/a + 1/16*sqrt(sqrt(2) + 2)*(b/a)^(1/8)*ln(x^2 - x*sqrt(sqrt(2) + 2)*(b/a)^(1/8
) + (b/a)^(1/4))/a - 1/16*sqrt(-sqrt(2) + 2)*(b/a)^(1/8)*ln(x^2 + x*sqrt(-sqrt(2
) + 2)*(b/a)^(1/8) + (b/a)^(1/4))/a + 1/16*sqrt(-sqrt(2) + 2)*(b/a)^(1/8)*ln(x^2
- x*sqrt(-sqrt(2) + 2)*(b/a)^(1/8) + (b/a)^(1/4))/a + x/a