3.1463 \(\int \frac{1}{a+b x^8} \, dx\)

Optimal. Leaf size=267 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{7/8} \sqrt [8]{b}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}} \]

[Out]

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

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Rubi [A]  time = 0.403607, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.111 \[ \frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{7/8} \sqrt [8]{b}}-\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{7/8} \sqrt [8]{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^8)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 99.9133, size = 246, normalized size = 0.92 \[ \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \sqrt [8]{b} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \sqrt [8]{b} \left (- a\right )^{\frac{7}{8}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \sqrt [8]{b} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 \sqrt [8]{b} \left (- a\right )^{\frac{7}{8}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 \sqrt [8]{b} \left (- a\right )^{\frac{7}{8}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \sqrt [8]{b} \left (- a\right )^{\frac{7}{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**8+a),x)

[Out]

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

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Mathematica [A]  time = 0.20543, size = 324, normalized size = 1.21 \[ \frac{-\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )-2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{8 a^{7/8} \sqrt [8]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^8)^(-1),x]

[Out]

(2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8] + 2*ArcTan[(b^(1/
8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8] - Cos[Pi/8]*Log[a^(1/4) + b^(1/4)
*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/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*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8
)]*Sin[Pi/8] + 2*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] - L
og[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/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*a^(7/8)*b^(1/8))

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Maple [C]  time = 0.002, size = 27, normalized size = 0.1 \[{\frac{1}{8\,b}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^8+a),x)

[Out]

1/8/b*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. \[ \int \frac{1}{b x^{8} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(b*x^8 + a), x)

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Fricas [A]  time = 0.235425, size = 518, normalized size = 1.94 \[ -\frac{1}{16} \, \sqrt{2}{\left (4 \, \sqrt{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}}}{x + \sqrt{a^{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}}}\right ) - \sqrt{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (a \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + x\right ) + \sqrt{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (-a \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + x\right ) + 4 \, \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}}}{\sqrt{2} x + a \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{\sqrt{2} a x \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}}}\right ) + 4 \, \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}}}{\sqrt{2} x - a \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{-\sqrt{2} a x \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}}}\right ) - \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a x \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}\right ) + \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a x \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*sqrt(2)*(4*sqrt(2)*(-1/(a^7*b))^(1/8)*arctan(a*(-1/(a^7*b))^(1/8)/(x + sqr
t(a^2*(-1/(a^7*b))^(1/4) + x^2))) - sqrt(2)*(-1/(a^7*b))^(1/8)*log(a*(-1/(a^7*b)
)^(1/8) + x) + sqrt(2)*(-1/(a^7*b))^(1/8)*log(-a*(-1/(a^7*b))^(1/8) + x) + 4*(-1
/(a^7*b))^(1/8)*arctan(a*(-1/(a^7*b))^(1/8)/(sqrt(2)*x + a*(-1/(a^7*b))^(1/8) +
sqrt(2)*sqrt(sqrt(2)*a*x*(-1/(a^7*b))^(1/8) + a^2*(-1/(a^7*b))^(1/4) + x^2))) +
4*(-1/(a^7*b))^(1/8)*arctan(a*(-1/(a^7*b))^(1/8)/(sqrt(2)*x - a*(-1/(a^7*b))^(1/
8) + sqrt(2)*sqrt(-sqrt(2)*a*x*(-1/(a^7*b))^(1/8) + a^2*(-1/(a^7*b))^(1/4) + x^2
))) - (-1/(a^7*b))^(1/8)*log(sqrt(2)*a*x*(-1/(a^7*b))^(1/8) + a^2*(-1/(a^7*b))^(
1/4) + x^2) + (-1/(a^7*b))^(1/8)*log(-sqrt(2)*a*x*(-1/(a^7*b))^(1/8) + a^2*(-1/(
a^7*b))^(1/4) + x^2))

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Sympy [A]  time = 0.464769, size = 20, normalized size = 0.07 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{7} b + 1, \left ( t \mapsto t \log{\left (8 t a + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**8+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.231627, size = 579, normalized size = 2.17 \[ \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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)))/a + 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)))/a + 1/8*sq
rt(-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 + 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)))/a + 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))
/a - 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))/a + 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))/a - 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))/a