3.1300 \(\int \frac{x^5}{1+x^5} \, dx\)

Optimal. Leaf size=186 \[ \frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+x-\frac{1}{5} \log (x+1)-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]

[Out]

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

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Rubi [A]  time = 0.228627, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {321, 201, 634, 618, 204, 628, 31} \[ \frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+x-\frac{1}{5} \log (x+1)-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(1 + x^5),x]

[Out]

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (r*
Int[1/(r + s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[
(n - 3)/2, 0] && 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^5}{1+x^5} \, dx &=x-\int \frac{1}{1+x^5} \, dx\\ &=x-\frac{2}{5} \int \frac{1-\frac{1}{4} \left (1-\sqrt{5}\right ) x}{1-\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx-\frac{2}{5} \int \frac{1-\frac{1}{4} \left (1+\sqrt{5}\right ) x}{1-\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx-\frac{1}{5} \int \frac{1}{1+x} \, dx\\ &=x-\frac{1}{5} \log (1+x)-\frac{1}{20} \left (-1-\sqrt{5}\right ) \int \frac{\frac{1}{2} \left (-1-\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (-1-\sqrt{5}\right ) x+x^2} \, dx-\frac{1}{20} \left (5-\sqrt{5}\right ) \int \frac{1}{1+\frac{1}{2} \left (-1-\sqrt{5}\right ) x+x^2} \, dx-\frac{1}{20} \left (-1+\sqrt{5}\right ) \int \frac{\frac{1}{2} \left (-1+\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (-1+\sqrt{5}\right ) x+x^2} \, dx-\frac{1}{20} \left (5+\sqrt{5}\right ) \int \frac{1}{1+\frac{1}{2} \left (-1+\sqrt{5}\right ) x+x^2} \, dx\\ &=x-\frac{1}{5} \log (1+x)+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (2-x-\sqrt{5} x+2 x^2\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (2-x+\sqrt{5} x+2 x^2\right )-\frac{1}{10} \left (-5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5+\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (-1-\sqrt{5}\right )+2 x\right )+\frac{1}{10} \left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5-\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (-1+\sqrt{5}\right )+2 x\right )\\ &=x+\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1-\sqrt{5}-4 x}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (1+\sqrt{5}-4 x\right )\right )-\frac{1}{5} \log (1+x)+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (2-x-\sqrt{5} x+2 x^2\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (2-x+\sqrt{5} x+2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0981023, size = 147, normalized size = 0.79 \[ \frac{1}{20} \left (-\left (\sqrt{5}-1\right ) \log \left (x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) x+1\right )+\left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+20 x-4 \log (x+1)+2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{-4 x+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )-2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 x+\sqrt{5}-1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 + x^5),x]

[Out]

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

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Maple [A]  time = 0.028, size = 219, normalized size = 1.2 \begin{align*} x-{\frac{\ln \left ( 1+x \right ) }{5}}+{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{20}}+{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{20}}-{\frac{1}{\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }-{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{20}}+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{20}}-{\frac{1}{\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^5+1),x)

[Out]

x-1/5*ln(1+x)+1/20*ln(-x*5^(1/2)+2*x^2-x+2)*5^(1/2)+1/20*ln(-x*5^(1/2)+2*x^2-x+2)-1/(10-2*5^(1/2))^(1/2)*arcta
n((-5^(1/2)+4*x-1)/(10-2*5^(1/2))^(1/2))+1/5/(10-2*5^(1/2))^(1/2)*arctan((-5^(1/2)+4*x-1)/(10-2*5^(1/2))^(1/2)
)*5^(1/2)-1/20*ln(x*5^(1/2)+2*x^2-x+2)*5^(1/2)+1/20*ln(x*5^(1/2)+2*x^2-x+2)-1/(10+2*5^(1/2))^(1/2)*arctan((5^(
1/2)+4*x-1)/(10+2*5^(1/2))^(1/2))-1/5/(10+2*5^(1/2))^(1/2)*arctan((5^(1/2)+4*x-1)/(10+2*5^(1/2))^(1/2))*5^(1/2
)

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Maxima [A]  time = 1.46357, size = 196, normalized size = 1.05 \begin{align*} -\frac{\sqrt{5}{\left (\sqrt{5} + 1\right )} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right )}{5 \, \sqrt{2 \, \sqrt{5} + 10}} - \frac{\sqrt{5}{\left (\sqrt{5} - 1\right )} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right )}{5 \, \sqrt{-2 \, \sqrt{5} + 10}} + x + \frac{{\left (\sqrt{5} + 3\right )} \log \left (2 \, x^{2} - x{\left (\sqrt{5} + 1\right )} + 2\right )}{10 \,{\left (\sqrt{5} + 1\right )}} + \frac{{\left (\sqrt{5} - 3\right )} \log \left (2 \, x^{2} + x{\left (\sqrt{5} - 1\right )} + 2\right )}{10 \,{\left (\sqrt{5} - 1\right )}} - \frac{1}{5} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^5+1),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 11.7575, size = 1909, normalized size = 10.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^5+1),x, algorithm="fricas")

[Out]

1/20*(sqrt(5) - 2*sqrt(-3/16*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2 - 1/8*(sqrt(5) + 10*sqrt(-1/50*sq
rt(5) - 1/10) + 3)*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10) - 1) - 3/16*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/1
0) - 1)^2 - 1/2*sqrt(5) - 5*sqrt(-1/50*sqrt(5) - 1/10) - 5/2) + 1)*log(2*x - 1/2*sqrt(5) + sqrt(-3/16*(sqrt(5)
+ 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2 - 1/8*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) + 3)*(sqrt(5) - 10*sqrt
(-1/50*sqrt(5) - 1/10) - 1) - 3/16*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2 - 1/2*sqrt(5) - 5*sqrt(-1/5
0*sqrt(5) - 1/10) - 5/2) - 1/2) + 1/20*(sqrt(5) + 2*sqrt(-3/16*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2
- 1/8*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) + 3)*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10) - 1) - 3/16*(sqr
t(5) - 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2 - 1/2*sqrt(5) - 5*sqrt(-1/50*sqrt(5) - 1/10) - 5/2) + 1)*log(2*x -
1/2*sqrt(5) - sqrt(-3/16*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2 - 1/8*(sqrt(5) + 10*sqrt(-1/50*sqrt(
5) - 1/10) + 3)*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10) - 1) - 3/16*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10)
- 1)^2 - 1/2*sqrt(5) - 5*sqrt(-1/50*sqrt(5) - 1/10) - 5/2) - 1/2) - 1/20*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/
10) - 1)*log(x + 1/4*sqrt(5) + 5/2*sqrt(-1/50*sqrt(5) - 1/10) - 1/4) - 1/20*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) -
1/10) - 1)*log(x + 1/4*sqrt(5) - 5/2*sqrt(-1/50*sqrt(5) - 1/10) - 1/4) + x - 1/5*log(x + 1)

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Sympy [A]  time = 0.916771, size = 36, normalized size = 0.19 \begin{align*} x - \frac{\log{\left (x + 1 \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 + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**5+1),x)

[Out]

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

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Giac [A]  time = 1.13675, size = 173, normalized size = 0.93 \begin{align*} -\frac{1}{10} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) - \frac{1}{10} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) + \frac{1}{20} \, \sqrt{5} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) - \frac{1}{20} \, \sqrt{5} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) + x + \frac{1}{20} \, \log \left (x^{4} - x^{3} + x^{2} - x + 1\right ) - \frac{1}{5} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^5+1),x, algorithm="giac")

[Out]

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