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

Optimal. Leaf size=27 \[ \frac{x+1}{2 \left (1-(x+1)^2\right )}+\frac{1}{2} \tanh ^{-1}(x+1) \]

[Out]

(1 + x)/(2*(1 - (1 + x)^2)) + ArcTanh[1 + x]/2

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Rubi [A]  time = 0.0107282, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1106, 199, 206} \[ \frac{x+1}{2 \left (1-(x+1)^2\right )}+\frac{1}{2} \tanh ^{-1}(x+1) \]

Antiderivative was successfully verified.

[In]

Int[(4*x^2 + 4*x^3 + x^4)^(-1),x]

[Out]

(1 + x)/(2*(1 - (1 + x)^2)) + ArcTanh[1 + x]/2

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{4 x^2+4 x^3+x^4} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,1+x\right )\\ &=\frac{1+x}{2 \left (1-(1+x)^2\right )}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,1+x\right )\\ &=\frac{1+x}{2 \left (1-(1+x)^2\right )}+\frac{1}{2} \tanh ^{-1}(1+x)\\ \end{align*}

Mathematica [A]  time = 0.0142848, size = 26, normalized size = 0.96 \[ \frac{1}{4} \left (-\frac{2 (x+1)}{x (x+2)}-\log (x)+\log (x+2)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*(1 + x))/(x*(2 + x)) - Log[x] + Log[2 + x])/4

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Maple [A]  time = 0.009, size = 24, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,x}}-{\frac{\ln \left ( x \right ) }{4}}-{\frac{1}{8+4\,x}}+{\frac{\ln \left ( 2+x \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/x-1/4*ln(x)-1/4/(2+x)+1/4*ln(2+x)

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Maxima [A]  time = 1.08473, size = 34, normalized size = 1.26 \begin{align*} -\frac{x + 1}{2 \,{\left (x^{2} + 2 \, x\right )}} + \frac{1}{4} \, \log \left (x + 2\right ) - \frac{1}{4} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.23551, size = 99, normalized size = 3.67 \begin{align*} \frac{{\left (x^{2} + 2 \, x\right )} \log \left (x + 2\right ) -{\left (x^{2} + 2 \, x\right )} \log \left (x\right ) - 2 \, x - 2}{4 \,{\left (x^{2} + 2 \, x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.101413, size = 22, normalized size = 0.81 \begin{align*} - \frac{x + 1}{2 x^{2} + 4 x} - \frac{\log{\left (x \right )}}{4} + \frac{\log{\left (x + 2 \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)/(2*x**2 + 4*x) - log(x)/4 + log(x + 2)/4

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Giac [A]  time = 1.08426, size = 36, normalized size = 1.33 \begin{align*} -\frac{x + 1}{2 \,{\left (x^{2} + 2 \, x\right )}} + \frac{1}{4} \, \log \left ({\left | x + 2 \right |}\right ) - \frac{1}{4} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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