3.137 \(\int \frac{x}{a^5+x^5} \, dx\)

Optimal. Leaf size=201 \[ -\frac{\log (a+x)}{5 a^3}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) a-4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )}{5 a^3}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (\left (1+\sqrt{5}\right ) a-4 x\right )}{2 a}\right )}{5 a^3}+\frac{\left (1+\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2\right )}{20 a^3}+\frac{\left (1-\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )}{20 a^3} \]

[Out]

(Sqrt[(5 - Sqrt[5])/2]*ArcTan[((1 - Sqrt[5])*a - 4*x)/(Sqrt[2*(5 + Sqrt[5])]*a)]
)/(5*a^3) - (Sqrt[(5 + Sqrt[5])/2]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*((1 + Sqrt[5])
*a - 4*x))/(2*a)])/(5*a^3) - Log[a + x]/(5*a^3) + ((1 + Sqrt[5])*Log[a^2 - ((1 -
Sqrt[5])*a*x)/2 + x^2])/(20*a^3) + ((1 - Sqrt[5])*Log[a^2 - ((1 + Sqrt[5])*a*x)
/2 + x^2])/(20*a^3)

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Rubi [A]  time = 0.572741, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ -\frac{\log (a+x)}{5 a^3}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) a-4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )}{5 a^3}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (\left (1+\sqrt{5}\right ) a-4 x\right )}{2 a}\right )}{5 a^3}+\frac{\left (1+\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2\right )}{20 a^3}+\frac{\left (1-\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )}{20 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x/(a^5 + x^5),x]

[Out]

(Sqrt[(5 - Sqrt[5])/2]*ArcTan[((1 - Sqrt[5])*a - 4*x)/(Sqrt[2*(5 + Sqrt[5])]*a)]
)/(5*a^3) - (Sqrt[(5 + Sqrt[5])/2]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*((1 + Sqrt[5])
*a - 4*x))/(2*a)])/(5*a^3) - Log[a + x]/(5*a^3) + ((1 + Sqrt[5])*Log[a^2 - ((1 -
Sqrt[5])*a*x)/2 + x^2])/(20*a^3) + ((1 - Sqrt[5])*Log[a^2 - ((1 + Sqrt[5])*a*x)
/2 + x^2])/(20*a^3)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a**5+x**5),x)

[Out]

Timed out

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Mathematica [A]  time = 0.132639, size = 204, normalized size = 1.01 \[ \frac{\sqrt{5} \log \left (a^2+\frac{1}{2} \left (\sqrt{5}-1\right ) a x+x^2\right )+\log \left (a^2+\frac{1}{2} \left (\sqrt{5}-1\right ) a x+x^2\right )-\sqrt{5} \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )+\log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )-4 \log (a+x)-2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{\left (\sqrt{5}-1\right ) a+4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )+2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 x-\left (1+\sqrt{5}\right ) a}{\sqrt{10-2 \sqrt{5}} a}\right )}{20 a^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/(a^5 + x^5),x]

[Out]

(-2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[((-1 + Sqrt[5])*a + 4*x)/(Sqrt[2*(5 + Sqrt[5])]*
a)] + 2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(-((1 + Sqrt[5])*a) + 4*x)/(Sqrt[10 - 2*Sqr
t[5]]*a)] - 4*Log[a + x] + Log[a^2 + ((-1 + Sqrt[5])*a*x)/2 + x^2] + Sqrt[5]*Log
[a^2 + ((-1 + Sqrt[5])*a*x)/2 + x^2] + Log[a^2 - ((1 + Sqrt[5])*a*x)/2 + x^2] -
Sqrt[5]*Log[a^2 - ((1 + Sqrt[5])*a*x)/2 + x^2])/(20*a^3)

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Maple [C]  time = 0.011, size = 97, normalized size = 0.5 \[ -{\frac{\ln \left ( a+x \right ) }{5\,{a}^{3}}}+{\frac{1}{5\,{a}^{3}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-a{{\it \_Z}}^{3}+{a}^{2}{{\it \_Z}}^{2}-{a}^{3}{\it \_Z}+{a}^{4} \right ) }{\frac{ \left ({{\it \_R}}^{3}-2\,{{\it \_R}}^{2}a+3\,{\it \_R}\,{a}^{2}+{a}^{3} \right ) \ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}a+2\,{\it \_R}\,{a}^{2}-{a}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a^5+x^5),x)

[Out]

-1/5*ln(a+x)/a^3+1/5/a^3*sum((_R^3-2*_R^2*a+3*_R*a^2+a^3)/(4*_R^3-3*_R^2*a+2*_R*
a^2-a^3)*ln(x-_R),_R=RootOf(_Z^4-_Z^3*a+_Z^2*a^2-_Z*a^3+a^4))

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Maxima [A]  time = 1.54449, size = 358, normalized size = 1.78 \[ -\frac{\log \left (x +{\left (a^{5}\right )}^{\frac{1}{5}}\right )}{5 \,{\left (a^{5}\right )}^{\frac{3}{5}}} + \frac{\sqrt{5} \log \left (\frac{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - 4 \, x +{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - 4 \, x -{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}\right )}{5 \,{\left (a^{5}\right )}^{\frac{3}{5}} \sqrt{2 \, \sqrt{5} - 10}} - \frac{\sqrt{5} \log \left (\frac{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + 4 \, x -{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + 4 \, x +{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}\right )}{5 \,{\left (a^{5}\right )}^{\frac{3}{5}} \sqrt{-2 \, \sqrt{5} - 10}} - \frac{\log \left (-{\left (a^{5}\right )}^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, x^{2} + 2 \,{\left (a^{5}\right )}^{\frac{2}{5}}\right )}{5 \,{\left (a^{5}\right )}^{\frac{3}{5}}{\left (\sqrt{5} + 1\right )}} + \frac{\log \left ({\left (a^{5}\right )}^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, x^{2} + 2 \,{\left (a^{5}\right )}^{\frac{2}{5}}\right )}{5 \,{\left (a^{5}\right )}^{\frac{3}{5}}{\left (\sqrt{5} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a^5 + x^5),x, algorithm="maxima")

[Out]

-1/5*log(x + (a^5)^(1/5))/(a^5)^(3/5) + 1/5*sqrt(5)*log(((a^5)^(1/5)*(sqrt(5) +
1) - 4*x + (a^5)^(1/5)*sqrt(2*sqrt(5) - 10))/((a^5)^(1/5)*(sqrt(5) + 1) - 4*x -
(a^5)^(1/5)*sqrt(2*sqrt(5) - 10)))/((a^5)^(3/5)*sqrt(2*sqrt(5) - 10)) - 1/5*sqrt
(5)*log(((a^5)^(1/5)*(sqrt(5) - 1) + 4*x - (a^5)^(1/5)*sqrt(-2*sqrt(5) - 10))/((
a^5)^(1/5)*(sqrt(5) - 1) + 4*x + (a^5)^(1/5)*sqrt(-2*sqrt(5) - 10)))/((a^5)^(3/5
)*sqrt(-2*sqrt(5) - 10)) - log(-(a^5)^(1/5)*x*(sqrt(5) + 1) + 2*x^2 + 2*(a^5)^(2
/5))/((a^5)^(3/5)*((5*sqrt(5)) + 5)) + 1/5*log((a^5)^(1/5)*x*(sqrt(5) - 1) + 2*x
^2 + 2*(a^5)^(2/5))/((a^5)^(3/5)*(sqrt(5) - 1))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a^5 + x^5),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.156183, size = 41, normalized size = 0.2 \[ \frac{- \frac{\log{\left (a + x \right )}}{5} + \operatorname{RootSum}{\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left ( t \mapsto t \log{\left (- 125 t^{3} a + x \right )} \right )\right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a**5+x**5),x)

[Out]

(-log(a + x)/5 + RootSum(625*_t**4 - 125*_t**3 + 25*_t**2 - 5*_t + 1, Lambda(_t,
_t*log(-125*_t**3*a + x))))/a**3

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GIAC/XCAS [A]  time = 0.222489, size = 239, normalized size = 1.19 \[ -\frac{\sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{a{\left (\sqrt{5} - 1\right )} + 4 \, x}{a \sqrt{2 \, \sqrt{5} + 10}}\right )}{10 \, a^{3}} + \frac{\sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{a{\left (\sqrt{5} + 1\right )} - 4 \, x}{a \sqrt{-2 \, \sqrt{5} + 10}}\right )}{10 \, a^{3}} - \frac{\sqrt{5}{\rm ln}\left (a^{2} - \frac{1}{2} \,{\left (\sqrt{5} a + a\right )} x + x^{2}\right )}{20 \, a^{3}} + \frac{\sqrt{5}{\rm ln}\left (a^{2} + \frac{1}{2} \,{\left (\sqrt{5} a - a\right )} x + x^{2}\right )}{20 \, a^{3}} + \frac{{\rm ln}\left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a^{3}} - \frac{{\rm ln}\left ({\left | a + x \right |}\right )}{5 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a^5 + x^5),x, algorithm="giac")

[Out]

-1/10*sqrt(-2*sqrt(5) + 10)*arctan((a*(sqrt(5) - 1) + 4*x)/(a*sqrt(2*sqrt(5) + 1
0)))/a^3 + 1/10*sqrt(2*sqrt(5) + 10)*arctan(-(a*(sqrt(5) + 1) - 4*x)/(a*sqrt(-2*
sqrt(5) + 10)))/a^3 - 1/20*sqrt(5)*ln(a^2 - 1/2*(sqrt(5)*a + a)*x + x^2)/a^3 + 1
/20*sqrt(5)*ln(a^2 + 1/2*(sqrt(5)*a - a)*x + x^2)/a^3 + 1/20*ln(abs(a^4 - a^3*x
+ a^2*x^2 - a*x^3 + x^4))/a^3 - 1/5*ln(abs(a + x))/a^3