3.1453 \(\int \frac{x}{a+b x^8} \, dx\)

Optimal. Leaf size=193 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b}}-\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{3/4} \sqrt [4]{b}} \]

[Out]

-ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)]/(4*Sqrt[2]*a^(3/4)*b^(1/4)) + ArcTan[
1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)]/(4*Sqrt[2]*a^(3/4)*b^(1/4)) - Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4]/(8*Sqrt[2]*a^(3/4)*b^(1/4)) + Log[Sqrt
[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4]/(8*Sqrt[2]*a^(3/4)*b^(1/4))

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Rubi [A]  time = 0.33071, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b}}-\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{3/4} \sqrt [4]{b}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^8),x]

[Out]

-ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)]/(4*Sqrt[2]*a^(3/4)*b^(1/4)) + ArcTan[
1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)]/(4*Sqrt[2]*a^(3/4)*b^(1/4)) - Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4]/(8*Sqrt[2]*a^(3/4)*b^(1/4)) + Log[Sqrt
[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4]/(8*Sqrt[2]*a^(3/4)*b^(1/4))

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Rubi in Sympy [A]  time = 53.6773, size = 178, normalized size = 0.92 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 a^{\frac{3}{4}} \sqrt [4]{b}} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 a^{\frac{3}{4}} \sqrt [4]{b}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x**8+a),x)

[Out]

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

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Mathematica [A]  time = 0.334635, size = 279, normalized size = 1.45 \[ -\frac{-\log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{8 \sqrt{2} a^{3/4} \sqrt [4]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^8),x]

[Out]

-(2*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] + 2*ArcTan[Cot[Pi/8] + (b^
(1/8)*x*Csc[Pi/8])/a^(1/8)] - 2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]
] + 2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]] + Log[a^(1/4) + b^(1/4)*
x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(
1/8)*x*Cos[Pi/8]] - Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]] -
Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]])/(8*Sqrt[2]*a^(3/4)*
b^(1/4))

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Maple [A]  time = 0.002, size = 136, normalized size = 0.7 \[{\frac{\sqrt{2}}{16\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{4}+\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{4}-\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{8\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{8\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^8+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230295, size = 150, normalized size = 0.78 \[ -\frac{1}{2} \, \left (-\frac{1}{a^{3} b}\right )^{\frac{1}{4}} \arctan \left (\frac{a \left (-\frac{1}{a^{3} b}\right )^{\frac{1}{4}}}{x^{2} + \sqrt{x^{4} + a^{2} \sqrt{-\frac{1}{a^{3} b}}}}\right ) + \frac{1}{8} \, \left (-\frac{1}{a^{3} b}\right )^{\frac{1}{4}} \log \left (x^{2} + a \left (-\frac{1}{a^{3} b}\right )^{\frac{1}{4}}\right ) - \frac{1}{8} \, \left (-\frac{1}{a^{3} b}\right )^{\frac{1}{4}} \log \left (x^{2} - a \left (-\frac{1}{a^{3} b}\right )^{\frac{1}{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-1/(a^3*b))^(1/4)*arctan(a*(-1/(a^3*b))^(1/4)/(x^2 + sqrt(x^4 + a^2*sqrt(-
1/(a^3*b))))) + 1/8*(-1/(a^3*b))^(1/4)*log(x^2 + a*(-1/(a^3*b))^(1/4)) - 1/8*(-1
/(a^3*b))^(1/4)*log(x^2 - a*(-1/(a^3*b))^(1/4))

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Sympy [A]  time = 0.531322, size = 22, normalized size = 0.11 \[ \operatorname{RootSum}{\left (4096 t^{4} a^{3} b + 1, \left ( t \mapsto t \log{\left (8 t a + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**8+a),x)

[Out]

RootSum(4096*_t**4*a**3*b + 1, Lambda(_t, _t*log(8*_t*a + x**2)))

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GIAC/XCAS [A]  time = 0.235663, size = 252, normalized size = 1.31 \[ \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a b} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a b} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{4} + \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a b} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{4} - \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2)*(a/b)^(1/4))/(a/b)
^(1/4))/(a*b) + 1/8*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x^2 - sqrt(2)*(a
/b)^(1/4))/(a/b)^(1/4))/(a*b) + 1/16*sqrt(2)*(a*b^3)^(1/4)*ln(x^4 + sqrt(2)*x^2*
(a/b)^(1/4) + sqrt(a/b))/(a*b) - 1/16*sqrt(2)*(a*b^3)^(1/4)*ln(x^4 - sqrt(2)*x^2
*(a/b)^(1/4) + sqrt(a/b))/(a*b)