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

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

[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 = 0.271211, 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, Rules used = {293, 634, 618, 204, 628, 31} \[ \frac{\left (1+\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2\right )}{20 a^3}+\frac{\left (1-\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )}{20 a^3}-\frac{\log (a+x)}{5 a^3}+\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) a-4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )}{5 a^3}-\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (\left (1+\sqrt{5}\right ) a-4 x\right )}{2 a}\right )}{5 a^3} \]

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)

Rule 293

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x]; -(((-r)^(m + 1)*Int[1/(r + s*x), x])/(a*n*s^m)) + Dist[(2*r^(m + 1))/(a*n*s
^m), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n -
1] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x}{a^5+x^5} \, dx &=\frac{2 \int \frac{\frac{1}{4} \left (1-\sqrt{5}\right ) a-\frac{1}{4} \left (-1-\sqrt{5}\right ) x}{a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2} \, dx}{5 a^3}+\frac{2 \int \frac{\frac{1}{4} \left (1+\sqrt{5}\right ) a-\frac{1}{4} \left (-1+\sqrt{5}\right ) x}{a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2} \, dx}{5 a^3}-\frac{\int \frac{1}{a+x} \, dx}{5 a^3}\\ &=-\frac{\log (a+x)}{5 a^3}+\frac{\left (1-\sqrt{5}\right ) \int \frac{-\frac{1}{2} \left (1+\sqrt{5}\right ) a+2 x}{a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2} \, dx}{20 a^3}+\frac{\left (1+\sqrt{5}\right ) \int \frac{-\frac{1}{2} \left (1-\sqrt{5}\right ) a+2 x}{a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2} \, dx}{20 a^3}-\frac{\int \frac{1}{a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2} \, dx}{2 \sqrt{5} a^2}+\frac{\int \frac{1}{a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2} \, dx}{2 \sqrt{5} a^2}\\ &=-\frac{\log (a+x)}{5 a^3}+\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^2-a x-\sqrt{5} a x+2 x^2\right )}{20 a^3}+\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^2-a x+\sqrt{5} a x+2 x^2\right )}{20 a^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5-\sqrt{5}\right ) a^2-x^2} \, dx,x,-\frac{1}{2} \left (1+\sqrt{5}\right ) a+2 x\right )}{\sqrt{5} a^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5+\sqrt{5}\right ) a^2-x^2} \, dx,x,-\frac{1}{2} \left (1-\sqrt{5}\right ) a+2 x\right )}{\sqrt{5} a^2}\\ &=\frac{\sqrt{\frac{2}{5 \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 )}{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{\log (a+x)}{5 a^3}+\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^2-a x-\sqrt{5} a x+2 x^2\right )}{20 a^3}+\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^2-a x+\sqrt{5} a x+2 x^2\right )}{20 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0565078, 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*Sqrt[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]*Lo
g[a^2 - ((1 + Sqrt[5])*a*x)/2 + x^2])/(20*a^3)

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Maple [C]  time = 0.005, size = 97, normalized size = 0.5 \begin{align*}{\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}}}}-{\frac{\ln \left ( a+x \right ) }{5\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a^5+x^5),x)

[Out]

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))-1/5*ln(a+x)/a^3

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Maxima [A]  time = 1.42026, size = 358, normalized size = 1.78 \begin{align*} -\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 )}} \end{align*}

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)) - 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)) + 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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [A]  time = 0.126691, size = 41, normalized size = 0.2 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.08677, size = 239, normalized size = 1.19 \begin{align*} -\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} \log \left (a^{2} - \frac{1}{2} \,{\left (\sqrt{5} a + a\right )} x + x^{2}\right )}{20 \, a^{3}} + \frac{\sqrt{5} \log \left (a^{2} + \frac{1}{2} \,{\left (\sqrt{5} a - a\right )} x + x^{2}\right )}{20 \, a^{3}} + \frac{\log \left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a^{3}} - \frac{\log \left ({\left | a + x \right |}\right )}{5 \, a^{3}} \end{align*}

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