3.347 \(\int \frac{1}{\frac{1}{x^2}+x^3} \, dx\)

Optimal. Leaf size=185 \[ -\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+\frac{1}{5} \log (x+1)-\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]

[Out]

-(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

________________________________________________________________________________________

Rubi [A]  time = 0.345543, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {1593, 293, 634, 618, 204, 628, 31} \[ -\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+\frac{1}{5} \log (x+1)-\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} x+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x^(-2) + x^3)^(-1),x]

[Out]

-(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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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}{\frac{1}{x^2}+x^3} \, dx &=\int \frac{x^2}{1+x^5} \, 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{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}{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}{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}}\\ &=\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 = 0.125276, size = 144, normalized size = 0.78 \[ \frac{1}{20} \left (-\left (1+\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) x+1\right )+\left (\sqrt{5}-1\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+4 \log (x+1)-2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{-4 x+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )-2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 x+\sqrt{5}-1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^(-2) + x^3)^(-1),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 156, 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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/x^2+x^3),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)*arcta
n((-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)

________________________________________________________________________________________

Maxima [A]  time = 1.54593, size = 167, normalized size = 0.9 \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}{5} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/x^2+x^3),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) + 1/5*log(x + 1)

________________________________________________________________________________________

Fricas [B]  time = 5.63503, size = 2210, normalized size = 11.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/x^2+x^3),x, algorithm="fricas")

[Out]

-1/20*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*log(1/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 +
x) + 1/20*(sqrt(5) + 2*sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqr
t(5) - 5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) -
sqrt(5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2) - 1)*log(-1/16*(2*sqrt(1/2)*sqrt(sqrt(5) -
5) + sqrt(5) + 1)^2 - 1/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + 1/2*sqrt(-3/16*(2*sqrt(1/2)*sqrt(
sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) -
5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*s
qrt(5) - 5/2)*(sqrt(5) - 1) + 2*x - 1) + 1/20*(sqrt(5) - 2*sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5)
+ 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/
16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2) - 1)*log
(-1/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 - 1/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2
- 1/2*sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5
) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 +
sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2)*(sqrt(5) - 1) + 2*x - 1) + 1/20*(2*sqrt(1/2)*sqrt(sqrt(5) - 5
) - sqrt(5) - 1)*log(1/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + x) + 1/5*log(x + 1)

________________________________________________________________________________________

Sympy [A]  time = 1.6551, size = 36, normalized size = 0.19 \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1/x**2+x**3),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)))

________________________________________________________________________________________

Giac [A]  time = 1.15905, size = 151, normalized size = 0.82 \begin{align*} \frac{1}{20} \,{\left (\sqrt{5} - 1\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) - \frac{1}{20} \,{\left (\sqrt{5} + 1\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\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}{10} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\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/(1/x^2+x^3),x, algorithm="giac")

[Out]

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