3.22 \(\int \frac{1}{-1+4 x-4 x^2+16 x^3} \, dx\)

Optimal. Leaf size=31 \[ -\frac{1}{10} \log \left (4 x^2+1\right )+\frac{1}{5} \log (1-4 x)-\frac{1}{10} \tan ^{-1}(2 x) \]

[Out]

-ArcTan[2*x]/10 + Log[1 - 4*x]/5 - Log[1 + 4*x^2]/10

________________________________________________________________________________________

Rubi [A]  time = 0.0198343, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2058, 635, 203, 260} \[ -\frac{1}{10} \log \left (4 x^2+1\right )+\frac{1}{5} \log (1-4 x)-\frac{1}{10} \tan ^{-1}(2 x) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 4*x - 4*x^2 + 16*x^3)^(-1),x]

[Out]

-ArcTan[2*x]/10 + Log[1 - 4*x]/5 - Log[1 + 4*x^2]/10

Rule 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rule 635

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

Rule 203

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

Rule 260

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

Rubi steps

\begin{align*} \int \frac{1}{-1+4 x-4 x^2+16 x^3} \, dx &=\int \left (\frac{4}{5 (-1+4 x)}+\frac{-1-4 x}{5 \left (1+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{5} \log (1-4 x)+\frac{1}{5} \int \frac{-1-4 x}{1+4 x^2} \, dx\\ &=\frac{1}{5} \log (1-4 x)-\frac{1}{5} \int \frac{1}{1+4 x^2} \, dx-\frac{4}{5} \int \frac{x}{1+4 x^2} \, dx\\ &=-\frac{1}{10} \tan ^{-1}(2 x)+\frac{1}{5} \log (1-4 x)-\frac{1}{10} \log \left (1+4 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0075836, size = 31, normalized size = 1. \[ -\frac{1}{10} \log \left (4 x^2+1\right )+\frac{1}{5} \log (1-4 x)-\frac{1}{10} \tan ^{-1}(2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 4*x - 4*x^2 + 16*x^3)^(-1),x]

[Out]

-ArcTan[2*x]/10 + Log[1 - 4*x]/5 - Log[1 + 4*x^2]/10

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 26, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( -1+4\,x \right ) }{5}}-{\frac{\ln \left ( 4\,{x}^{2}+1 \right ) }{10}}-{\frac{\arctan \left ( 2\,x \right ) }{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(16*x^3-4*x^2+4*x-1),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.55369, size = 34, normalized size = 1.1 \begin{align*} -\frac{1}{10} \, \arctan \left (2 \, x\right ) - \frac{1}{10} \, \log \left (4 \, x^{2} + 1\right ) + \frac{1}{5} \, \log \left (4 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(16*x^3-4*x^2+4*x-1),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.83125, size = 81, normalized size = 2.61 \begin{align*} -\frac{1}{10} \, \arctan \left (2 \, x\right ) - \frac{1}{10} \, \log \left (4 \, x^{2} + 1\right ) + \frac{1}{5} \, \log \left (4 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(16*x^3-4*x^2+4*x-1),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.131789, size = 24, normalized size = 0.77 \begin{align*} \frac{\log{\left (x - \frac{1}{4} \right )}}{5} - \frac{\log{\left (x^{2} + \frac{1}{4} \right )}}{10} - \frac{\operatorname{atan}{\left (2 x \right )}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(16*x**3-4*x**2+4*x-1),x)

[Out]

log(x - 1/4)/5 - log(x**2 + 1/4)/10 - atan(2*x)/10

________________________________________________________________________________________

Giac [A]  time = 1.23862, size = 35, normalized size = 1.13 \begin{align*} -\frac{1}{10} \, \arctan \left (2 \, x\right ) - \frac{1}{10} \, \log \left (4 \, x^{2} + 1\right ) + \frac{1}{5} \, \log \left ({\left | 4 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(16*x^3-4*x^2+4*x-1),x, algorithm="giac")

[Out]

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