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

Optimal. Leaf size=17 \[ \frac {1}{4} \log \left (x^4+2 x^2+4 x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1587} \[ \frac {1}{4} \log \left (x^4+2 x^2+4 x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[4*x + 2*x^2 + x^4]/4

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.18 \[ \frac {1}{4} \log \left (x^3+2 x+4\right )+\frac {\log (x)}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]/4 + Log[4 + 2*x + x^3]/4

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fricas [A]  time = 0.80, size = 15, normalized size = 0.88 \[ \frac {1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(x^4 + 2*x^2 + 4*x)

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giac [A]  time = 0.27, size = 18, normalized size = 1.06 \[ \frac {1}{4} \, \log \left (4 \, {\left | \frac {1}{4} \, x^{4} + \frac {1}{2} \, x^{2} + x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(4*abs(1/4*x^4 + 1/2*x^2 + x))

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ \frac {\ln \left (\left (x^{3}+2 x +4\right ) x \right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(x*(x^3+2*x+4))

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maxima [A]  time = 0.45, size = 15, normalized size = 0.88 \[ \frac {1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(x^4 + 2*x^2 + 4*x)

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mupad [B]  time = 0.07, size = 13, normalized size = 0.76 \[ \frac {\ln \left (x\,\left (x^3+2\,x+4\right )\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x*(2*x + x^3 + 4))/4

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sympy [A]  time = 0.10, size = 14, normalized size = 0.82 \[ \frac {\log {\left (x^{4} + 2 x^{2} + 4 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**4 + 2*x**2 + 4*x)/4

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