3.136 \(\int \frac{1}{a^5+x^5} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.650836, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ \frac{\log (a+x)}{5 a^4}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) a-4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )}{5 a^4}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (\left (1+\sqrt{5}\right ) a-4 x\right )}{2 a}\right )}{5 a^4}-\frac{\left (1-\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2\right )}{20 a^4}-\frac{\left (1+\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )}{20 a^4} \]

Antiderivative was successfully verified.

[In]  Int[(a^5 + x^5)^(-1),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 144.858, size = 262, normalized size = 1.3 \[ \frac{\log{\left (a + x \right )}}{5 a^{4}} - \frac{\left (- \frac{\sqrt{5}}{20} + \frac{1}{20}\right ) \log{\left (a^{2} + a x \left (- \frac{1}{2} + \frac{\sqrt{5}}{2}\right ) + x^{2} \right )}}{a^{4}} - \frac{\left (\frac{1}{20} + \frac{\sqrt{5}}{20}\right ) \log{\left (a^{2} + a x \left (- \frac{\sqrt{5}}{2} - \frac{1}{2}\right ) + x^{2} \right )}}{a^{4}} + \frac{2 \left (- \left (\frac{1}{4} + \frac{\sqrt{5}}{4}\right )^{2} + 1\right ) \operatorname{atan}{\left (\frac{- \frac{a \left (1 + \sqrt{5}\right )}{4} + x}{a \sqrt{- \frac{\sqrt{5}}{4} + \frac{3}{4}} \sqrt{\frac{\sqrt{5}}{4} + \frac{5}{4}}} \right )}}{5 a^{4} \sqrt{- \frac{\sqrt{5}}{4} + \frac{3}{4}} \sqrt{\frac{\sqrt{5}}{4} + \frac{5}{4}}} + \frac{2 \left (- \left (- \frac{\sqrt{5}}{4} + \frac{1}{4}\right )^{2} + 1\right ) \operatorname{atan}{\left (\frac{\frac{a \left (-1 + \sqrt{5}\right )}{4} + x}{a \sqrt{- \frac{\sqrt{5}}{4} + \frac{5}{4}} \sqrt{\frac{\sqrt{5}}{4} + \frac{3}{4}}} \right )}}{5 a^{4} \sqrt{- \frac{\sqrt{5}}{4} + \frac{5}{4}} \sqrt{\frac{\sqrt{5}}{4} + \frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a**5+x**5),x)

[Out]

log(a + x)/(5*a**4) - (-sqrt(5)/20 + 1/20)*log(a**2 + a*x*(-1/2 + sqrt(5)/2) + x
**2)/a**4 - (1/20 + sqrt(5)/20)*log(a**2 + a*x*(-sqrt(5)/2 - 1/2) + x**2)/a**4 +
2*(-(1/4 + sqrt(5)/4)**2 + 1)*atan((-a*(1 + sqrt(5))/4 + x)/(a*sqrt(-sqrt(5)/4
+ 3/4)*sqrt(sqrt(5)/4 + 5/4)))/(5*a**4*sqrt(-sqrt(5)/4 + 3/4)*sqrt(sqrt(5)/4 + 5
/4)) + 2*(-(-sqrt(5)/4 + 1/4)**2 + 1)*atan((a*(-1 + sqrt(5))/4 + x)/(a*sqrt(-sqr
t(5)/4 + 5/4)*sqrt(sqrt(5)/4 + 3/4)))/(5*a**4*sqrt(-sqrt(5)/4 + 5/4)*sqrt(sqrt(5
)/4 + 3/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.29973, size = 204, normalized size = 1.01 \[ -\frac{-\sqrt{5} \log \left (a^2+\frac{1}{2} \left (\sqrt{5}-1\right ) a x+x^2\right )+\log \left (a^2+\frac{1}{2} \left (\sqrt{5}-1\right ) a x+x^2\right )+\sqrt{5} \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )+\log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )-4 \log (a+x)-2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (\sqrt{5}-1\right ) a+4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )-2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 x-\left (1+\sqrt{5}\right ) a}{\sqrt{10-2 \sqrt{5}} a}\right )}{20 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^5 + x^5)^(-1),x]

[Out]

-(-2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[((-1 + Sqrt[5])*a + 4*x)/(Sqrt[2*(5 + Sqrt[5])
]*a)] - 2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(-((1 + Sqrt[5])*a) + 4*x)/(Sqrt[10 - 2*Sq
rt[5]]*a)] - 4*Log[a + x] + Log[a^2 + ((-1 + Sqrt[5])*a*x)/2 + x^2] - Sqrt[5]*Lo
g[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^4)

_______________________________________________________________________________________

Maple [C]  time = 0.046, size = 101, normalized size = 0.5 \[{\frac{\ln \left ( a+x \right ) }{5\,{a}^{4}}}+{\frac{1}{5\,{a}^{4}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-a{{\it \_Z}}^{3}+{a}^{2}{{\it \_Z}}^{2}-{a}^{3}{\it \_Z}+{a}^{4} \right ) }{\frac{ \left ( -{{\it \_R}}^{3}+2\,{{\it \_R}}^{2}a-3\,{\it \_R}\,{a}^{2}+4\,{a}^{3} \right ) \ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}a+2\,{\it \_R}\,{a}^{2}-{a}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a^5+x^5),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.4789, size = 385, normalized size = 1.92 \[ \frac{\sqrt{5}{\left (\sqrt{5} - 1\right )} \log \left (\frac{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - 4 \, x +{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - 4 \, x -{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}\right )}{10 \,{\left (a^{5}\right )}^{\frac{4}{5}} \sqrt{2 \, \sqrt{5} - 10}} + \frac{\sqrt{5}{\left (\sqrt{5} + 1\right )} \log \left (\frac{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + 4 \, x -{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + 4 \, x +{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}\right )}{10 \,{\left (a^{5}\right )}^{\frac{4}{5}} \sqrt{-2 \, \sqrt{5} - 10}} - \frac{{\left (\sqrt{5} + 3\right )} \log \left (-{\left (a^{5}\right )}^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, x^{2} + 2 \,{\left (a^{5}\right )}^{\frac{2}{5}}\right )}{10 \,{\left (a^{5}\right )}^{\frac{4}{5}}{\left (\sqrt{5} + 1\right )}} - \frac{{\left (\sqrt{5} - 3\right )} \log \left ({\left (a^{5}\right )}^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, x^{2} + 2 \,{\left (a^{5}\right )}^{\frac{2}{5}}\right )}{10 \,{\left (a^{5}\right )}^{\frac{4}{5}}{\left (\sqrt{5} - 1\right )}} + \frac{\log \left (x +{\left (a^{5}\right )}^{\frac{1}{5}}\right )}{5 \,{\left (a^{5}\right )}^{\frac{4}{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a^5 + x^5),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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(1/(a^5 + x^5),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 0.158108, size = 39, normalized size = 0.19 \[ \frac{\frac{\log{\left (a + x \right )}}{5} + \operatorname{RootSum}{\left (625 t^{4} + 125 t^{3} + 25 t^{2} + 5 t + 1, \left ( t \mapsto t \log{\left (5 t a + x \right )} \right )\right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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(5*_t*a + x))))/a**4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216174, size = 239, normalized size = 1.19 \[ \frac{\sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{a{\left (\sqrt{5} - 1\right )} + 4 \, x}{a \sqrt{2 \, \sqrt{5} + 10}}\right )}{10 \, a^{4}} + \frac{\sqrt{-2 \, \sqrt{5} + 10} \arctan \left (-\frac{a{\left (\sqrt{5} + 1\right )} - 4 \, x}{a \sqrt{-2 \, \sqrt{5} + 10}}\right )}{10 \, a^{4}} - \frac{\sqrt{5}{\rm ln}\left (a^{2} - \frac{1}{2} \,{\left (\sqrt{5} a + a\right )} x + x^{2}\right )}{20 \, a^{4}} + \frac{\sqrt{5}{\rm ln}\left (a^{2} + \frac{1}{2} \,{\left (\sqrt{5} a - a\right )} x + x^{2}\right )}{20 \, a^{4}} - \frac{{\rm ln}\left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a^{4}} + \frac{{\rm ln}\left ({\left | a + x \right |}\right )}{5 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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) + 10)
))/a^4 + 1/10*sqrt(-2*sqrt(5) + 10)*arctan(-(a*(sqrt(5) + 1) - 4*x)/(a*sqrt(-2*s
qrt(5) + 10)))/a^4 - 1/20*sqrt(5)*ln(a^2 - 1/2*(sqrt(5)*a + a)*x + x^2)/a^4 + 1/
20*sqrt(5)*ln(a^2 + 1/2*(sqrt(5)*a - a)*x + x^2)/a^4 - 1/20*ln(abs(a^4 - a^3*x +
a^2*x^2 - a*x^3 + x^4))/a^4 + 1/5*ln(abs(a + x))/a^4