3.3.88 \(\int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx\)

Optimal. Leaf size=23 \[ 81+2 \left (1+e^4\right )+\log \left (\left (3+\frac {1}{2 x}\right )^2+x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2074, 1587} \begin {gather*} \log \left (4 x^3+36 x^2+12 x+1\right )-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 - 12*x + 4*x^3)/(x + 12*x^2 + 36*x^3 + 4*x^4),x]

[Out]

-2*Log[x] + Log[1 + 12*x + 36*x^2 + 4*x^3]

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]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{x}+\frac {12 \left (1+6 x+x^2\right )}{1+12 x+36 x^2+4 x^3}\right ) \, dx\\ &=-2 \log (x)+12 \int \frac {1+6 x+x^2}{1+12 x+36 x^2+4 x^3} \, dx\\ &=-2 \log (x)+\log \left (1+12 x+36 x^2+4 x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} -2 \log (x)+\log \left (1+12 x+36 x^2+4 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - 12*x + 4*x^3)/(x + 12*x^2 + 36*x^3 + 4*x^4),x]

[Out]

-2*Log[x] + Log[1 + 12*x + 36*x^2 + 4*x^3]

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fricas [A]  time = 0.71, size = 21, normalized size = 0.91 \begin {gather*} \log \left (4 \, x^{3} + 36 \, x^{2} + 12 \, x + 1\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-12*x-2)/(4*x^4+36*x^3+12*x^2+x),x, algorithm="fricas")

[Out]

log(4*x^3 + 36*x^2 + 12*x + 1) - 2*log(x)

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giac [A]  time = 0.35, size = 23, normalized size = 1.00 \begin {gather*} \log \left ({\left | 4 \, x^{3} + 36 \, x^{2} + 12 \, x + 1 \right |}\right ) - 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-12*x-2)/(4*x^4+36*x^3+12*x^2+x),x, algorithm="giac")

[Out]

log(abs(4*x^3 + 36*x^2 + 12*x + 1)) - 2*log(abs(x))

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maple [A]  time = 0.03, size = 22, normalized size = 0.96

method	result	size
default	\(-2 \ln \relax (x )+\ln \left (4 x^{3}+36 x^{2}+12 x +1\right )\)	\(22\)
norman	\(-2 \ln \relax (x )+\ln \left (4 x^{3}+36 x^{2}+12 x +1\right )\)	\(22\)
risch	\(-2 \ln \relax (x )+\ln \left (4 x^{3}+36 x^{2}+12 x +1\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3-12*x-2)/(4*x^4+36*x^3+12*x^2+x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x)+ln(4*x^3+36*x^2+12*x+1)

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maxima [A]  time = 0.45, size = 21, normalized size = 0.91 \begin {gather*} \log \left (4 \, x^{3} + 36 \, x^{2} + 12 \, x + 1\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-12*x-2)/(4*x^4+36*x^3+12*x^2+x),x, algorithm="maxima")

[Out]

log(4*x^3 + 36*x^2 + 12*x + 1) - 2*log(x)

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mupad [B]  time = 0.34, size = 19, normalized size = 0.83 \begin {gather*} \ln \left (x^3+9\,x^2+3\,x+\frac {1}{4}\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x - 4*x^3 + 2)/(x + 12*x^2 + 36*x^3 + 4*x^4),x)

[Out]

log(3*x + 9*x^2 + x^3 + 1/4) - 2*log(x)

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sympy [A]  time = 0.11, size = 20, normalized size = 0.87 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (4 x^{3} + 36 x^{2} + 12 x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3-12*x-2)/(4*x**4+36*x**3+12*x**2+x),x)

[Out]

-2*log(x) + log(4*x**3 + 36*x**2 + 12*x + 1)

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