3.1468 \(\int \frac{1}{a-b x^8} \, dx\)

Optimal. Leaf size=239 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}} \]

[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 = 0.377881, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1. \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}} \]

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 = 73.1661, size = 223, normalized size = 0.93 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x + \sqrt [4]{a} + \sqrt [4]{b} x^{2} \right )}}{16 a^{\frac{7}{8}} \sqrt [8]{b}} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x + \sqrt [4]{a} + \sqrt [4]{b} x^{2} \right )}}{16 a^{\frac{7}{8}} \sqrt [8]{b}} + \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}} \right )}}{4 a^{\frac{7}{8}} \sqrt [8]{b}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}} \right )}}{8 a^{\frac{7}{8}} \sqrt [8]{b}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}} \right )}}{8 a^{\frac{7}{8}} \sqrt [8]{b}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}} \right )}}{4 a^{\frac{7}{8}} \sqrt [8]{b}} \]

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

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Mathematica [A]  time = 0.101395, size = 198, normalized size = 0.83 \[ \frac{-\sqrt{2} \log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt{2} \log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \log \left (\sqrt [8]{a}-\sqrt [8]{b} x\right )+2 \log \left (\sqrt [8]{a}+\sqrt [8]{b} x\right )+4 \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )-2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )+2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{16 a^{7/8} \sqrt [8]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^8)^(-1),x]

[Out]

(4*ArcTan[(b^(1/8)*x)/a^(1/8)] - 2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/a^(1/8
)] + 2*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/a^(1/8)] - 2*Log[a^(1/8) - b^(1/8)
*x] + 2*Log[a^(1/8) + b^(1/8)*x] - Sqrt[2]*Log[a^(1/4) - Sqrt[2]*a^(1/8)*b^(1/8)
*x + b^(1/4)*x^2] + Sqrt[2]*Log[a^(1/4) + Sqrt[2]*a^(1/8)*b^(1/8)*x + b^(1/4)*x^
2])/(16*a^(7/8)*b^(1/8))

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Maple [C]  time = 0.019, size = 29, normalized size = 0.1 \[ -{\frac{1}{8\,b}\sum _{{\it \_R}={\it RootOf} \left ( b{{\it \_Z}}^{8}-a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(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 = 0.236282, size = 487, normalized size = 2.04 \[ -\frac{1}{16} \, \sqrt{2}{\left (4 \, \sqrt{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}}}{x + \sqrt{a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}}}\right ) - \sqrt{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + x\right ) + \sqrt{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (-a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + x\right ) + 4 \, \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}}}{\sqrt{2} x + a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{\sqrt{2} a x \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}}}\right ) + 4 \, \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (\frac{a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}}}{\sqrt{2} x - a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + \sqrt{2} \sqrt{-\sqrt{2} a x \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}}}\right ) - \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a x \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}\right ) + \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a x \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}\right )\right )} \]

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 + sqrt(
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)*sq
rt(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)*sq
rt(-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 = 0.498425, size = 22, normalized size = 0.09 \[ - \operatorname{RootSum}{\left (16777216 t^{8} a^{7} b - 1, \left ( t \mapsto t \log{\left (- 8 t a + x \right )} \right )\right )} \]

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 = 0.227837, size = 612, normalized size = 2.56 \[ \frac{\sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}} + \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}} + \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}} + \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{8}} + \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} \]

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)*arcta
n((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*sqrt(-sqrt(2) + 2)*(-a/b)^(1/8)*arc
tan((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