3.1459 \(\int \frac{x^8}{a+b x^8} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.615694, antiderivative size = 272, 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 \[ \frac{\sqrt [8]{-a} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{\sqrt [8]{-a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} b^{9/8}}-\frac{\sqrt [8]{-a} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 b^{9/8}}+\frac{x}{b} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 111.698, size = 250, normalized size = 0.92 \[ \frac{x}{b} + \frac{\sqrt{2} \sqrt [8]{- a} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 b^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{- a} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 b^{\frac{9}{8}}} - \frac{\sqrt [8]{- a} \operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 b^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{- a} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 b^{\frac{9}{8}}} - \frac{\sqrt{2} \sqrt [8]{- a} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 b^{\frac{9}{8}}} - \frac{\sqrt [8]{- a} \operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 b^{\frac{9}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.351058, size = 367, normalized size = 1.35 \[ \frac{\sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{a} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \sqrt [8]{a} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [8]{b} x}{8 b^{9/8}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^8),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^8+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.242761, size = 478, normalized size = 1.76 \[ \frac{\sqrt{2}{\left (4 \, \sqrt{2} b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}}}{x + \sqrt{b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}}}\right ) - \sqrt{2} b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \log \left (b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + x\right ) + \sqrt{2} b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \log \left (-b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + x\right ) + 4 \, b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}}}{\sqrt{2} x + b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{\sqrt{2} b x \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}}}\right ) + 4 \, b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \arctan \left (\frac{b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}}}{\sqrt{2} x - b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{-\sqrt{2} b x \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}}}\right ) - b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} b x \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}\right ) + b \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} b x \left (-\frac{a}{b^{9}}\right )^{\frac{1}{8}} + b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + x^{2}\right ) + 8 \, \sqrt{2} x\right )}}{16 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.4112, size = 22, normalized size = 0.08 \[ \operatorname{RootSum}{\left (16777216 t^{8} b^{9} + a, \left ( t \mapsto t \log{\left (- 8 t b + x \right )} \right )\right )} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**8+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.237656, size = 586, normalized size = 2.15 \[ -\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, b} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, b} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, b} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, b} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, b} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, b} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, b} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, b} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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