3.2.37 \(\int \frac {x}{a^5+x^5} \, dx\) [137]

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

[Out]

-1/5*ln(a+x)/a^3+1/20*ln(a^2+x^2-1/2*a*x*(5^(1/2)+1))*(-5^(1/2)+1)/a^3+1/20*ln(a^2+x^2-1/2*a*x*(-5^(1/2)+1))*(
5^(1/2)+1)/a^3+1/10*arctan((-4*x+a*(-5^(1/2)+1))/a/(10+2*5^(1/2))^(1/2))*(10-2*5^(1/2))^(1/2)/a^3-1/10*arctan(
1/20*(-4*x+a*(5^(1/2)+1))*(50+10*5^(1/2))^(1/2)/a)*(10+2*5^(1/2))^(1/2)/a^3

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Rubi [A]
time = 0.19, antiderivative size = 201, normalized size of antiderivative = 1.00, 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 = {299, 648, 632, 210, 642, 31} \begin {gather*} \frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {ArcTan}\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 )} \text {ArcTan}\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 (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} \end {gather*}

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 31

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

Rule 210

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

Rule 299

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)/(a*n*s^m))*Int[1/(r + s*x), x] + 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 632

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 642

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 648

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]

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 {\text {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 {\text {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*}

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Mathematica [A]
time = 0.03, size = 204, normalized size = 1.01 \begin {gather*} \frac {-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {\left (-1+\sqrt {5}\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 {-\left (\left (1+\sqrt {5}\right ) a\right )+4 x}{\sqrt {10-2 \sqrt {5}} a}\right )-4 \log (a+x)+\log \left (a^2+\frac {1}{2} \left (-1+\sqrt {5}\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 )-\sqrt {5} \log \left (a^2-\frac {1}{2} \left (1+\sqrt {5}\right ) a x+x^2\right )}{20 a^3} \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.05, size = 97, normalized size = 0.48

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{12} \textit {\_Z}^{4}-a^{9} \textit {\_Z}^{3}+a^{6} \textit {\_Z}^{2}-a^{3} \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (-a^{10} \textit {\_R}^{3}+x \right )\right )}{5}-\frac {\ln \left (a +x \right )}{5 a^{3}}\)	\(60\)
default	\(-\frac {\ln \left (a +x \right )}{5 a^{3}}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}+\textit {\_Z}^{2} a^{2}-a^{3} \textit {\_Z} +a^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{3}-2 \textit {\_R}^{2} a +3 a^{2} \textit {\_R} +a^{3}\right ) \ln \left (-\textit {\_R} +x \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a +2 a^{2} \textit {\_R} -a^{3}}}{5 a^{3}}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a^5+x^5),x,method=_RETURNVERBOSE)

[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(-_R+x),_R=RootOf(_
Z^4-_Z^3*a+_Z^2*a^2-_Z*a^3+a^4))

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Maxima [A]
time = 3.01, size = 160, normalized size = 0.80 \begin {gather*} -\frac {2 \, \sqrt {5} \arctan \left (\frac {a {\left (\sqrt {5} - 1\right )} + 4 \, x}{a \sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, a^{3} \sqrt {2 \, \sqrt {5} + 10}} + \frac {2 \, \sqrt {5} \arctan \left (-\frac {a {\left (\sqrt {5} + 1\right )} - 4 \, x}{a \sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, a^{3} \sqrt {-2 \, \sqrt {5} + 10}} - \frac {\log \left (a + x\right )}{5 \, a^{3}} - \frac {\log \left (-a x {\left (\sqrt {5} + 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{5 \, a^{3} {\left (\sqrt {5} + 1\right )}} + \frac {\log \left (a x {\left (\sqrt {5} - 1\right )} + 2 \, a^{2} + 2 \, x^{2}\right )}{5 \, a^{3} {\left (\sqrt {5} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 1.29, size = 18781, normalized size = 93.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6000*(300*a^3*(10*sqrt(-1/50*sqrt(5)/a^6 - 1/10/a^6) + sqrt(5)/a^3 - 1/a^3)*log(1/64*a^10*(10*sqrt(-1/50*sq
rt(5)/a^6 - 1/10/a^6) + sqrt(5)/a^3 - 1/a^3)^3 + x) + 2*(3125*(1/25)^(2/3)*(-I*sqrt(3) + 1)*((2*sqrt(5)*sqrt(1
/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)^2/a^6 - 12*sqrt(5)*(sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt
(5)*(sqrt(5) + 1)/a^6) + 5*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 2*sqrt(5))/a^6)/(15625*sqrt(5)*(2*
sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)^3/a^9 - 1406250*(2*sqrt(5)*sqrt(1/2)*a
^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)*(sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) +
5*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 2*sqrt(5))/a^9 + 4218750*sqrt(5)*(sqrt(5)*sqrt(1/2)*a^9*(-
sqrt(5)*(sqrt(5) + 1)/a^6)^(3/2) + 13*sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 5*sqrt(1/2)*a^3
*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 4*sqrt(5) - 20)/a^9 + 18*sqrt(-610351562500*sqrt(5)*sqrt(1/2)*a^15*(-sqrt(
5)*(sqrt(5) + 1)/a^6)^(5/2) - 6103515625000/3*sqrt(1/2)*a^9*(9*sqrt(5) - 5)*(-sqrt(5)*(sqrt(5) + 1)/a^6)^(3/2)
- 30517578125000/3*sqrt(1/2)*a^3*(11*sqrt(5) - 7)*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) - 23651123046875/6*sqrt(5)
*(sqrt(5) + 1)^3 + 762939453125/3*(9*sqrt(5) + 305)*(sqrt(5) + 1)^2 - 5340576171875/3*sqrt(5)*(sqrt(5) + 51)*(
sqrt(5) + 1) - 115966796875000/3*sqrt(5) + 396728515625000/3)/a^9)^(1/3) + (1/25)^(1/3)*(I*sqrt(3) + 1)*(15625
*sqrt(5)*(2*sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)^3/a^9 - 1406250*(2*sqrt(5)
*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)*(sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5)
+ 1)/a^6) + 5*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 2*sqrt(5))/a^9 + 4218750*sqrt(5)*(sqrt(5)*sqrt
(1/2)*a^9*(-sqrt(5)*(sqrt(5) + 1)/a^6)^(3/2) + 13*sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 5*s
qrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 4*sqrt(5) - 20)/a^9 + 18*sqrt(-610351562500*sqrt(5)*sqrt(1/2)*
a^15*(-sqrt(5)*(sqrt(5) + 1)/a^6)^(5/2) - 6103515625000/3*sqrt(1/2)*a^9*(9*sqrt(5) - 5)*(-sqrt(5)*(sqrt(5) + 1
)/a^6)^(3/2) - 30517578125000/3*sqrt(1/2)*a^3*(11*sqrt(5) - 7)*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) - 236511230468
75/6*sqrt(5)*(sqrt(5) + 1)^3 + 762939453125/3*(9*sqrt(5) + 305)*(sqrt(5) + 1)^2 - 5340576171875/3*sqrt(5)*(sqr
t(5) + 51)*(sqrt(5) + 1) - 115966796875000/3*sqrt(5) + 396728515625000/3)/a^9)^(1/3) - 10*sqrt(5)*(2*sqrt(5)*s
qrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)/a^3)*a^3*log(-1/64*a^10*(10*sqrt(-1/50*sqrt(5)/
a^6 - 1/10/a^6) + sqrt(5)/a^3 - 1/a^3)^3 - 1/16*a^7*(10*sqrt(-1/50*sqrt(5)/a^6 - 1/10/a^6) + sqrt(5)/a^3 - 1/a
^3)^2 - 1/4*a^4*(10*sqrt(-1/50*sqrt(5)/a^6 - 1/10/a^6) + sqrt(5)/a^3 - 1/a^3) - 1/1440000*(a^10*(10*sqrt(-1/50
*sqrt(5)/a^6 - 1/10/a^6) + sqrt(5)/a^3 - 1/a^3) + 4*a^7)*(3125*(1/25)^(2/3)*(-I*sqrt(3) + 1)*((2*sqrt(5)*sqrt(
1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)^2/a^6 - 12*sqrt(5)*(sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqr
t(5)*(sqrt(5) + 1)/a^6) + 5*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 2*sqrt(5))/a^6)/(15625*sqrt(5)*(2
*sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)^3/a^9 - 1406250*(2*sqrt(5)*sqrt(1/2)*
a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)*(sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6)
+ 5*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 2*sqrt(5))/a^9 + 4218750*sqrt(5)*(sqrt(5)*sqrt(1/2)*a^9*(
-sqrt(5)*(sqrt(5) + 1)/a^6)^(3/2) + 13*sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 5*sqrt(1/2)*a^
3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 4*sqrt(5) - 20)/a^9 + 18*sqrt(-610351562500*sqrt(5)*sqrt(1/2)*a^15*(-sqrt
(5)*(sqrt(5) + 1)/a^6)^(5/2) - 6103515625000/3*sqrt(1/2)*a^9*(9*sqrt(5) - 5)*(-sqrt(5)*(sqrt(5) + 1)/a^6)^(3/2
) - 30517578125000/3*sqrt(1/2)*a^3*(11*sqrt(5) - 7)*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) - 23651123046875/6*sqrt(5
)*(sqrt(5) + 1)^3 + 762939453125/3*(9*sqrt(5) + 305)*(sqrt(5) + 1)^2 - 5340576171875/3*sqrt(5)*(sqrt(5) + 51)*
(sqrt(5) + 1) - 115966796875000/3*sqrt(5) + 396728515625000/3)/a^9)^(1/3) + (1/25)^(1/3)*(I*sqrt(3) + 1)*(1562
5*sqrt(5)*(2*sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)^3/a^9 - 1406250*(2*sqrt(5
)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 3*sqrt(5) + 5)*(sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5
) + 1)/a^6) + 5*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 2*sqrt(5))/a^9 + 4218750*sqrt(5)*(sqrt(5)*sqr
t(1/2)*a^9*(-sqrt(5)*(sqrt(5) + 1)/a^6)^(3/2) + 13*sqrt(5)*sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 5*
sqrt(1/2)*a^3*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) + 4*sqrt(5) - 20)/a^9 + 18*sqrt(-610351562500*sqrt(5)*sqrt(1/2)
*a^15*(-sqrt(5)*(sqrt(5) + 1)/a^6)^(5/2) - 6103515625000/3*sqrt(1/2)*a^9*(9*sqrt(5) - 5)*(-sqrt(5)*(sqrt(5) +
1)/a^6)^(3/2) - 30517578125000/3*sqrt(1/2)*a^3*(11*sqrt(5) - 7)*sqrt(-sqrt(5)*(sqrt(5) + 1)/a^6) - 23651123046
875/6*sqrt(5)*(sqrt(5) + 1)^3 + 762939453125/3*(9*sqrt(5) + 305)*(sqrt(5) + 1)^2 - 5340576171875/3*sqrt(5)*(sq
rt(5) + 51)*(sqrt(5) + 1) - 115966796875000/3*s...

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Sympy [A]
time = 0.05, size = 41, normalized size = 0.20 \begin {gather*} \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 {gather*}

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 = 0.52, size = 177, normalized size = 0.88 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.59, size = 182, normalized size = 0.91 \begin {gather*} \frac {\ln \left (x-\frac {a\,{\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}^3}{64}\right )\,\left (\sqrt {5}-\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^3}-\frac {\ln \left (a+x\right )}{5\,a^3}+\frac {\ln \left (x-\frac {a\,{\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}^3}{64}\right )\,\left (\sqrt {-2\,\sqrt {5}-10}-\sqrt {5}+1\right )}{20\,a^3}-\frac {\ln \left (x+\frac {a\,{\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}^3}{64}\right )\,\left (\sqrt {5}+\sqrt {-2\,\sqrt {5}-10}-1\right )}{20\,a^3}+\frac {\ln \left (x-\frac {a\,{\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}^3}{64}\right )\,\left (\sqrt {5}+\sqrt {2\,\sqrt {5}-10}+1\right )}{20\,a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a^5 + x^5),x)

[Out]

(log(x - (a*(5^(1/2) - (2*5^(1/2) - 10)^(1/2) + 1)^3)/64)*(5^(1/2) - (2*5^(1/2) - 10)^(1/2) + 1))/(20*a^3) - l
og(a + x)/(5*a^3) + (log(x - (a*((- 2*5^(1/2) - 10)^(1/2) - 5^(1/2) + 1)^3)/64)*((- 2*5^(1/2) - 10)^(1/2) - 5^
(1/2) + 1))/(20*a^3) - (log(x + (a*(5^(1/2) + (- 2*5^(1/2) - 10)^(1/2) - 1)^3)/64)*(5^(1/2) + (- 2*5^(1/2) - 1
0)^(1/2) - 1))/(20*a^3) + (log(x - (a*(5^(1/2) + (2*5^(1/2) - 10)^(1/2) + 1)^3)/64)*(5^(1/2) + (2*5^(1/2) - 10
)^(1/2) + 1))/(20*a^3)

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