3.1465 \(\int \frac{1}{x^4 \left (a+b x^8\right )} \, dx\)

Optimal. Leaf size=277 \[ -\frac{b^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{1}{3 a x^3} \]

[Out]

-1/(3*a*x^3) - (b^(3/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/
8)*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(11/8)) + (b^(3/8
)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(11/8)) - (b^(3/8)
*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/8)*Log[(-a)^(1/4) - Sq
rt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/8)) + (b^(3/8)*Lo
g[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/
8))

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Rubi [A]  time = 0.510975, 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 \[ -\frac{b^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^8)),x]

[Out]

-1/(3*a*x^3) - (b^(3/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/
8)*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(11/8)) + (b^(3/8
)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(11/8)) - (b^(3/8)
*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/8)*Log[(-a)^(1/4) - Sq
rt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/8)) + (b^(3/8)*Lo
g[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/
8))

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Rubi in Sympy [A]  time = 109.79, size = 255, normalized size = 0.92 \[ - \frac{\sqrt{2} b^{\frac{3}{8}} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} b^{\frac{3}{8}} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{11}{8}}} - \frac{b^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} b^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} b^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 \left (- a\right )^{\frac{11}{8}}} - \frac{b^{\frac{3}{8}} \operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{11}{8}}} - \frac{1}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**8+a),x)

[Out]

-sqrt(2)*b**(3/8)*log(-sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b**(1/4)*x**2 + (-a)**(1
/4))/(16*(-a)**(11/8)) + sqrt(2)*b**(3/8)*log(sqrt(2)*b**(1/8)*x*(-a)**(1/8) + b
**(1/4)*x**2 + (-a)**(1/4))/(16*(-a)**(11/8)) - b**(3/8)*atan(b**(1/8)*x/(-a)**(
1/8))/(4*(-a)**(11/8)) + sqrt(2)*b**(3/8)*atan(sqrt(2)*b**(1/8)*x/(-a)**(1/8) -
1)/(8*(-a)**(11/8)) + sqrt(2)*b**(3/8)*atan(sqrt(2)*b**(1/8)*x/(-a)**(1/8) + 1)/
(8*(-a)**(11/8)) - b**(3/8)*atanh(b**(1/8)*x/(-a)**(1/8))/(4*(-a)**(11/8)) - 1/(
3*a*x**3)

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Mathematica [A]  time = 0.365373, size = 395, normalized size = 1.43 \[ \frac{-8 a^{3/8}+6 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+6 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+6 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-6 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+3 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-3 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-3 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+3 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{24 a^{11/8} x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^8)),x]

[Out]

(-8*a^(3/8) + 6*b^(3/8)*x^3*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Co
s[Pi/8] - 6*b^(3/8)*x^3*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Cos[Pi
/8] + 3*b^(3/8)*x^3*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Si
n[Pi/8]] - 3*b^(3/8)*x^3*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)
*x*Sin[Pi/8]] + 6*b^(3/8)*x^3*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*
Sin[Pi/8] + 6*b^(3/8)*x^3*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Sin[
Pi/8] - 3*b^(3/8)*x^3*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]]
*Sin[Pi/8] + 3*b^(3/8)*x^3*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[P
i/8]]*Sin[Pi/8])/(24*a^(11/8)*x^3)

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Maple [C]  time = 0.007, size = 36, normalized size = 0.1 \[ -{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}}}}-{\frac{1}{3\,a{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^8+a),x)

[Out]

-1/8/a*sum(1/_R^3*ln(x-_R),_R=RootOf(_Z^8*b+a))-1/3/a/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{b \int \frac{x^{4}}{b x^{8} + a}\,{d x}}{a} - \frac{1}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^4),x, algorithm="maxima")

[Out]

-b*integrate(x^4/(b*x^8 + a), x)/a - 1/3/(a*x^3)

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Fricas [A]  time = 0.241778, size = 698, normalized size = 2.52 \[ -\frac{\sqrt{2}{\left (12 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}}}{b^{2} x + b^{2} \sqrt{-\frac{a^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} - b x^{2}}{b}}}\right ) - 3 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + b^{2} x\right ) + 3 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (-a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + b^{2} x\right ) - 12 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}}}{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + \sqrt{2} b^{2} x + \sqrt{2} b^{2} \sqrt{\frac{\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}}{b^{2}}}}\right ) - 12 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}}}{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - \sqrt{2} b^{2} x - \sqrt{2} b^{2} \sqrt{-\frac{\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} - b^{2} x^{2}}{b^{2}}}}\right ) + 3 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}\right ) - 3 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}\right ) + 8 \, \sqrt{2}\right )}}{48 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^4),x, algorithm="fricas")

[Out]

-1/48*sqrt(2)*(12*sqrt(2)*a*x^3*(-b^3/a^11)^(1/8)*arctan(a^7*(-b^3/a^11)^(5/8)/(
b^2*x + b^2*sqrt(-(a^3*(-b^3/a^11)^(1/4) - b*x^2)/b))) - 3*sqrt(2)*a*x^3*(-b^3/a
^11)^(1/8)*log(a^7*(-b^3/a^11)^(5/8) + b^2*x) + 3*sqrt(2)*a*x^3*(-b^3/a^11)^(1/8
)*log(-a^7*(-b^3/a^11)^(5/8) + b^2*x) - 12*a*x^3*(-b^3/a^11)^(1/8)*arctan(a^7*(-
b^3/a^11)^(5/8)/(a^7*(-b^3/a^11)^(5/8) + sqrt(2)*b^2*x + sqrt(2)*b^2*sqrt((sqrt(
2)*a^7*x*(-b^3/a^11)^(5/8) - a^3*b*(-b^3/a^11)^(1/4) + b^2*x^2)/b^2))) - 12*a*x^
3*(-b^3/a^11)^(1/8)*arctan(-a^7*(-b^3/a^11)^(5/8)/(a^7*(-b^3/a^11)^(5/8) - sqrt(
2)*b^2*x - sqrt(2)*b^2*sqrt(-(sqrt(2)*a^7*x*(-b^3/a^11)^(5/8) + a^3*b*(-b^3/a^11
)^(1/4) - b^2*x^2)/b^2))) + 3*a*x^3*(-b^3/a^11)^(1/8)*log(sqrt(2)*a^7*x*(-b^3/a^
11)^(5/8) - a^3*b*(-b^3/a^11)^(1/4) + b^2*x^2) - 3*a*x^3*(-b^3/a^11)^(1/8)*log(-
sqrt(2)*a^7*x*(-b^3/a^11)^(5/8) - a^3*b*(-b^3/a^11)^(1/4) + b^2*x^2) + 8*sqrt(2)
)/(a*x^3)

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Sympy [A]  time = 2.50367, size = 36, normalized size = 0.13 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{11} + b^{3}, \left ( t \mapsto t \log{\left (\frac{32768 t^{5} a^{7}}{b^{2}} + x \right )} \right )\right )} - \frac{1}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**8+a),x)

[Out]

RootSum(16777216*_t**8*a**11 + b**3, Lambda(_t, _t*log(32768*_t**5*a**7/b**2 + x
))) - 1/(3*a*x**3)

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GIAC/XCAS [A]  time = 0.257668, size = 601, normalized size = 2.17 \[ \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} + \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{1}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^4),x, algorithm="giac")

[Out]

1/8*b*sqrt(-sqrt(2) + 2)*(a/b)^(5/8)*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)^(5/8)*a
rctan((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)^(5/8)*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)^(5/8)
*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)^(5/8)*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)^(5/8)*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)^(5/
8)*ln(x^2 + x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/a^2 + 1/16*b*sqrt(sq
rt(2) + 2)*(a/b)^(5/8)*ln(x^2 - x*sqrt(-sqrt(2) + 2)*(a/b)^(1/8) + (a/b)^(1/4))/
a^2 - 1/3/(a*x^3)