3.1467 \(\int \frac{1}{x^8 \left (a+b x^8\right )} \, dx\)

Optimal. Leaf size=277 \[ \frac{b^{7/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{15/8}}-\frac{b^{7/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{15/8}}-\frac{b^{7/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac{b^{7/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{15/8}}-\frac{b^{7/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{15/8}}-\frac{b^{7/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}-\frac{1}{7 a x^7} \]

[Out]

-1/(7*a*x^7) - (b^(7/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(15/8)) + (b^(7/
8)*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(15/8)) - (b^(7/8
)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(15/8)) - (b^(7/8)
*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(15/8)) + (b^(7/8)*Log[(-a)^(1/4) - Sq
rt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(15/8)) - (b^(7/8)*Lo
g[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(15/
8))

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Rubi [A]  time = 0.514474, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846 \[ \frac{b^{7/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{15/8}}-\frac{b^{7/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{15/8}}-\frac{b^{7/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac{b^{7/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{15/8}}-\frac{b^{7/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{15/8}}-\frac{b^{7/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}-\frac{1}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(a + b*x^8)),x]

[Out]

-1/(7*a*x^7) - (b^(7/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(15/8)) + (b^(7/
8)*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(15/8)) - (b^(7/8
)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(15/8)) - (b^(7/8)
*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(15/8)) + (b^(7/8)*Log[(-a)^(1/4) - Sq
rt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(15/8)) - (b^(7/8)*Lo
g[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(15/
8))

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Rubi in Sympy [A]  time = 115.804, size = 255, normalized size = 0.92 \[ \frac{\sqrt{2} b^{\frac{7}{8}} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{15}{8}}} - \frac{\sqrt{2} b^{\frac{7}{8}} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{15}{8}}} - \frac{b^{\frac{7}{8}} \operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{15}{8}}} - \frac{\sqrt{2} b^{\frac{7}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 \left (- a\right )^{\frac{15}{8}}} - \frac{\sqrt{2} b^{\frac{7}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 \left (- a\right )^{\frac{15}{8}}} - \frac{b^{\frac{7}{8}} \operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{15}{8}}} - \frac{1}{7 a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(b*x**8+a),x)

[Out]

sqrt(2)*b**(7/8)*log(-sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b**(1/4)*x**2 + (-a)**(1/
4))/(16*(-a)**(15/8)) - sqrt(2)*b**(7/8)*log(sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b*
*(1/4)*x**2 + (-a)**(1/4))/(16*(-a)**(15/8)) - b**(7/8)*atan(b**(1/8)*x/(-a)**(1
/8))/(4*(-a)**(15/8)) - sqrt(2)*b**(7/8)*atan(sqrt(2)*b**(1/8)*x/(-a)**(1/8) - 1
)/(8*(-a)**(15/8)) - sqrt(2)*b**(7/8)*atan(sqrt(2)*b**(1/8)*x/(-a)**(1/8) + 1)/(
8*(-a)**(15/8)) - b**(7/8)*atanh(b**(1/8)*x/(-a)**(1/8))/(4*(-a)**(15/8)) - 1/(7
*a*x**7)

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Mathematica [A]  time = 0.563118, size = 395, normalized size = 1.43 \[ -\frac{8 a^{7/8}+14 b^{7/8} x^7 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+14 b^{7/8} x^7 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )-14 b^{7/8} x^7 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+14 b^{7/8} x^7 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )-7 b^{7/8} x^7 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+7 b^{7/8} x^7 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-7 b^{7/8} x^7 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+7 b^{7/8} x^7 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{56 a^{15/8} x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(a + b*x^8)),x]

[Out]

-(8*a^(7/8) + 14*b^(7/8)*x^7*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*C
os[Pi/8] + 14*b^(7/8)*x^7*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[
Pi/8] - 7*b^(7/8)*x^7*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*
Cos[Pi/8]] + 7*b^(7/8)*x^7*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/
8)*x*Cos[Pi/8]] - 14*b^(7/8)*x^7*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8
)]*Sin[Pi/8] + 14*b^(7/8)*x^7*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*
Sin[Pi/8] - 7*b^(7/8)*x^7*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi
/8]]*Sin[Pi/8] + 7*b^(7/8)*x^7*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*S
in[Pi/8]]*Sin[Pi/8])/(56*a^(15/8)*x^7)

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Maple [C]  time = 0.007, size = 36, normalized size = 0.1 \[ -{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}}-{\frac{1}{7\,a{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(b*x^8+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{b \int \frac{1}{b x^{8} + a}\,{d x}}{a} - \frac{1}{7 \, a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*integrate(1/(b*x^8 + a), x)/a - 1/7/(a*x^7)

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Fricas [A]  time = 0.242872, size = 671, normalized size = 2.42 \[ \frac{\sqrt{2}{\left (28 \, \sqrt{2} a x^{7} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}}}{b x + b \sqrt{\frac{a^{4} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{4}} + b^{2} x^{2}}{b^{2}}}}\right ) - 7 \, \sqrt{2} a x^{7} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} \log \left (a^{2} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} + b x\right ) + 7 \, \sqrt{2} a x^{7} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} \log \left (-a^{2} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} + b x\right ) + 28 \, a x^{7} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}}}{\sqrt{2} b x + a^{2} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} + \sqrt{2} b \sqrt{\frac{\sqrt{2} a^{2} b x \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} + a^{4} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{4}} + b^{2} x^{2}}{b^{2}}}}\right ) + 28 \, a x^{7} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}}}{\sqrt{2} b x - a^{2} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} + \sqrt{2} b \sqrt{-\frac{\sqrt{2} a^{2} b x \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} - a^{4} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{4}} - b^{2} x^{2}}{b^{2}}}}\right ) - 7 \, a x^{7} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{2} b x \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} + a^{4} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{4}} + b^{2} x^{2}\right ) + 7 \, a x^{7} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{2} b x \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{8}} + a^{4} \left (-\frac{b^{7}}{a^{15}}\right )^{\frac{1}{4}} + b^{2} x^{2}\right ) - 8 \, \sqrt{2}\right )}}{112 \, a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/112*sqrt(2)*(28*sqrt(2)*a*x^7*(-b^7/a^15)^(1/8)*arctan(a^2*(-b^7/a^15)^(1/8)/(
b*x + b*sqrt((a^4*(-b^7/a^15)^(1/4) + b^2*x^2)/b^2))) - 7*sqrt(2)*a*x^7*(-b^7/a^
15)^(1/8)*log(a^2*(-b^7/a^15)^(1/8) + b*x) + 7*sqrt(2)*a*x^7*(-b^7/a^15)^(1/8)*l
og(-a^2*(-b^7/a^15)^(1/8) + b*x) + 28*a*x^7*(-b^7/a^15)^(1/8)*arctan(a^2*(-b^7/a
^15)^(1/8)/(sqrt(2)*b*x + a^2*(-b^7/a^15)^(1/8) + sqrt(2)*b*sqrt((sqrt(2)*a^2*b*
x*(-b^7/a^15)^(1/8) + a^4*(-b^7/a^15)^(1/4) + b^2*x^2)/b^2))) + 28*a*x^7*(-b^7/a
^15)^(1/8)*arctan(a^2*(-b^7/a^15)^(1/8)/(sqrt(2)*b*x - a^2*(-b^7/a^15)^(1/8) + s
qrt(2)*b*sqrt(-(sqrt(2)*a^2*b*x*(-b^7/a^15)^(1/8) - a^4*(-b^7/a^15)^(1/4) - b^2*
x^2)/b^2))) - 7*a*x^7*(-b^7/a^15)^(1/8)*log(sqrt(2)*a^2*b*x*(-b^7/a^15)^(1/8) +
a^4*(-b^7/a^15)^(1/4) + b^2*x^2) + 7*a*x^7*(-b^7/a^15)^(1/8)*log(-sqrt(2)*a^2*b*
x*(-b^7/a^15)^(1/8) + a^4*(-b^7/a^15)^(1/4) + b^2*x^2) - 8*sqrt(2))/(a*x^7)

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Sympy [A]  time = 15.228, size = 32, normalized size = 0.12 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{15} + b^{7}, \left ( t \mapsto t \log{\left (- \frac{8 t a^{2}}{b} + x \right )} \right )\right )} - \frac{1}{7 a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(b*x**8+a),x)

[Out]

RootSum(16777216*_t**8*a**15 + b**7, Lambda(_t, _t*log(-8*_t*a**2/b + x))) - 1/(
7*a*x**7)

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GIAC/XCAS [A]  time = 0.236106, size = 601, normalized size = 2.17 \[ -\frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} + \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{1}{7 \, a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*b*sqrt(sqrt(2) + 2)*(a/b)^(1/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(a/b)^(1/8
))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)))/a^2 - 1/8*b*sqrt(sqrt(2) + 2)*(a/b)^(1/8)*ar
ctan((2*x - sqrt(-sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(a/b)^(1/8)))/a^2
- 1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(1/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(a/b)^(1
/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))/a^2 - 1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(1/8
)*arctan((2*x - sqrt(sqrt(2) + 2)*(a/b)^(1/8))/(sqrt(-sqrt(2) + 2)*(a/b)^(1/8)))
/a^2 - 1/16*b*sqrt(sqrt(2) + 2)*(a/b)^(1/8)*ln(x^2 + x*sqrt(sqrt(2) + 2)*(a/b)^(
1/8) + (a/b)^(1/4))/a^2 + 1/16*b*sqrt(sqrt(2) + 2)*(a/b)^(1/8)*ln(x^2 - x*sqrt(s
qrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/a^2 - 1/16*b*sqrt(-sqrt(2) + 2)*(a/b)^(1/
8)*ln(x^2 + x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/a^2 + 1/16*b*sqrt(-s
qrt(2) + 2)*(a/b)^(1/8)*ln(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))
/a^2 - 1/7/(a*x^7)