3.1306 \(\int \frac{1}{x^3 (1+x^5)} \, dx\)

Optimal. Leaf size=192 \[ -\frac{1}{2 x^2}+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-\frac{1}{5} \log (x+1)+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]

[Out]

-1/(2*x^2) + (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 = 0.276318, antiderivative size = 192, 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 = {325, 293, 634, 618, 204, 628, 31} \[ -\frac{1}{2 x^2}+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-\frac{1}{5} \log (x+1)+\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )+\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(1 + x^5)),x]

[Out]

-1/(2*x^2) + (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 325

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

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{1}{x^3 \left (1+x^5\right )} \, dx &=-\frac{1}{2 x^2}-\int \frac{x^2}{1+x^5} \, dx\\ &=-\frac{1}{2 x^2}-\frac{2}{5} \int \frac{\frac{1}{4} \left (-1-\sqrt{5}\right )-\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{\frac{1}{4} \left (-1+\sqrt{5}\right )-\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\\ &=-\frac{1}{2 x^2}-\frac{1}{5} \log (1+x)-\frac{\int \frac{1}{1+\frac{1}{2} \left (-1-\sqrt{5}\right ) x+x^2} \, dx}{2 \sqrt{5}}+\frac{\int \frac{1}{1+\frac{1}{2} \left (-1+\sqrt{5}\right ) x+x^2} \, dx}{2 \sqrt{5}}-\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 (-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}{2 x^2}-\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{\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 )}{\sqrt{5}}+\frac{\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 )}{\sqrt{5}}\\ &=-\frac{1}{2 x^2}-\sqrt{\frac{2}{5 \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 = 0.111824, size = 149, normalized size = 0.78 \[ \frac{1}{20} \left (-\frac{10}{x^2}+\left (1+\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) x+1\right )-\left (\sqrt{5}-1\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-4 \log (x+1)+2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{-4 x+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )+2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 x+\sqrt{5}-1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^3*(1 + x^5)),x]

[Out]

(-10/x^2 + 2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(1 + Sqrt[5] - 4*x)/Sqrt[10 - 2*Sqrt[5]]] + 2*Sqrt[10 - 2*Sqrt[5]]*A
rcTan[(-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 = 0.015, size = 161, normalized size = 0.8 \begin{align*} -{\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{2\,\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{2\,\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{1}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(x^5+1),x)

[Out]

-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)-2/5/(10-2*5^(1/2))^(1/2)*arct
an((-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)+2/5/(10+2*5^(1/2))^(1/2)*arctan((5^(1/2)+4*x-1)/(10+2*5^(1/2))^(1/2))*5^(1/2)-1/2/x^2

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Maxima [A]  time = 1.49875, size = 174, normalized size = 0.91 \begin{align*} \frac{2 \, \sqrt{5} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right )}{5 \, \sqrt{2 \, \sqrt{5} + 10}} - \frac{2 \, \sqrt{5} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right )}{5 \, \sqrt{-2 \, \sqrt{5} + 10}} - \frac{\log \left (2 \, x^{2} - x{\left (\sqrt{5} + 1\right )} + 2\right )}{5 \,{\left (\sqrt{5} + 1\right )}} + \frac{\log \left (2 \, x^{2} + x{\left (\sqrt{5} - 1\right )} + 2\right )}{5 \,{\left (\sqrt{5} - 1\right )}} - \frac{1}{2 \, x^{2}} - \frac{1}{5} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^5+1),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 11.8622, size = 2504, normalized size = 13.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^5+1),x, algorithm="fricas")

[Out]

-1/40*(2*x^2*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)*log(1/16*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) -
1)^2 + x) + 2*x^2*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)*log(1/16*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1
/10) - 1)^2 + x) + 8*x^2*log(x + 1) - (x^2*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) - 1) + x^2*(sqrt(5) - 10*s
qrt(-1/50*sqrt(5) - 1/10) - 1) + 4*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)*x^2 + 4*x^2)*log(-1/16*(sqrt(5)
+ 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2 - 1/16*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2 + 1/2*sqrt(-3/1
6*(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)*(sqrt(5) + 1) + 2*x - 1) - (x^2*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) -
1) + x^2*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10) - 1) - 4*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)*x^2 + 4
*x^2)*log(-1/16*(sqrt(5) + 10*sqrt(-1/50*sqrt(5) - 1/10) - 1)^2 - 1/16*(sqrt(5) - 10*sqrt(-1/50*sqrt(5) - 1/10
) - 1)^2 - 1/2*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*(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)*(sqrt(5) + 1) + 2*x - 1) + 20)/x^2

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Sympy [A]  time = 0.937331, size = 42, normalized size = 0.22 \begin{align*} - \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 (25 t^{2} + x \right )} \right )\right )} - \frac{1}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(x**5+1),x)

[Out]

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

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Giac [A]  time = 1.55118, size = 178, 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 ) - \frac{1}{2 \, x^{2}} + \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(1/x^3/(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)*arctan
((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)*l
og(x^2 + 1/2*x*(sqrt(5) - 1) + 1) - 1/2/x^2 + 1/20*log(x^4 - x^3 + x^2 - x + 1) - 1/5*log(abs(x + 1))