3.3.77 \(\int \frac {1+x+x^3}{4 x+2 x^2+x^4} \, dx\) [277]

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

[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 = {1601} \begin {gather*} \frac {1}{4} \log \left (x^4+2 x^2+4 x\right ) \end {gather*}

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 1601

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]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; 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 = 0.00, size = 20, normalized size = 1.18 \begin {gather*} \frac {\log (x)}{4}+\frac {1}{4} \log \left (4+2 x+x^3\right ) \end {gather*}

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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Mathics [A]
time = 1.72, size = 13, normalized size = 0.76 \begin {gather*} \frac {\text {Log}\left [x \left (4+2 x+x^3\right )\right ]}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x (4 + 2 x + x ^ 3)] / 4

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Maple [A]
time = 0.01, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {\ln \left (x \left (x^{3}+2 x +4\right )\right )}{4}\)	\(14\)
risch	\(\frac {\ln \left (x^{4}+2 x^{2}+4 x \right )}{4}\)	\(16\)
norman	\(\frac {\ln \left (x \right )}{4}+\frac {\ln \left (x^{3}+2 x +4\right )}{4}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 4 \, x\right ) \end {gather*}

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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Fricas [A]
time = 0.33, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 4 \, x\right ) \end {gather*}

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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Sympy [A]
time = 0.05, size = 14, normalized size = 0.82 \begin {gather*} \frac {\log {\left (x^{4} + 2 x^{2} + 4 x \right )}}{4} \end {gather*}

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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Giac [A]
time = 0.00, size = 20, normalized size = 1.18 \begin {gather*} \frac {\ln \left |x\right |}{4}+\frac {\ln \left |x^{3}+2 x+4\right |}{4} \end {gather*}

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]

1/4*log(abs(x^3 + 2*x + 4)) + 1/4*log(abs(x))

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

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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