3.44.60 \(\int \frac {-12-4 x}{144+84 x+16 x^2+x^3+(-48-20 x-2 x^2) \log (4+x)+(4+x) \log ^2(4+x)} \, dx\)

Optimal. Leaf size=20 \[ \log \left (2 e^{-2+\frac {4}{6+x-\log (4+x)}}\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 13, normalized size of antiderivative = 0.65, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6688, 12, 6686} \begin {gather*} \frac {4}{x-\log (x+4)+6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 - 4*x)/(144 + 84*x + 16*x^2 + x^3 + (-48 - 20*x - 2*x^2)*Log[4 + x] + (4 + x)*Log[4 + x]^2),x]

[Out]

4/(6 + x - Log[4 + x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.65 \begin {gather*} \frac {4}{6+x-\log (4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 - 4*x)/(144 + 84*x + 16*x^2 + x^3 + (-48 - 20*x - 2*x^2)*Log[4 + x] + (4 + x)*Log[4 + x]^2),x]

[Out]

4/(6 + x - Log[4 + x])

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fricas [A]  time = 0.83, size = 13, normalized size = 0.65 \begin {gather*} \frac {4}{x - \log \left (x + 4\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-12)/((4+x)*log(4+x)^2+(-2*x^2-20*x-48)*log(4+x)+x^3+16*x^2+84*x+144),x, algorithm="fricas")

[Out]

4/(x - log(x + 4) + 6)

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giac [A]  time = 0.18, size = 13, normalized size = 0.65 \begin {gather*} \frac {4}{x - \log \left (x + 4\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-12)/((4+x)*log(4+x)^2+(-2*x^2-20*x-48)*log(4+x)+x^3+16*x^2+84*x+144),x, algorithm="giac")

[Out]

4/(x - log(x + 4) + 6)

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maple [A]  time = 0.04, size = 14, normalized size = 0.70

method	result	size
norman	\(\frac {4}{x +6-\ln \left (4+x \right )}\)	\(14\)
risch	\(\frac {4}{x +6-\ln \left (4+x \right )}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x-12)/((4+x)*ln(4+x)^2+(-2*x^2-20*x-48)*ln(4+x)+x^3+16*x^2+84*x+144),x,method=_RETURNVERBOSE)

[Out]

4/(x+6-ln(4+x))

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maxima [A]  time = 0.41, size = 13, normalized size = 0.65 \begin {gather*} \frac {4}{x - \log \left (x + 4\right ) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-12)/((4+x)*log(4+x)^2+(-2*x^2-20*x-48)*log(4+x)+x^3+16*x^2+84*x+144),x, algorithm="maxima")

[Out]

4/(x - log(x + 4) + 6)

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mupad [B]  time = 3.47, size = 13, normalized size = 0.65 \begin {gather*} \frac {4}{x-\ln \left (x+4\right )+6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 12)/(84*x - log(x + 4)*(20*x + 2*x^2 + 48) + 16*x^2 + x^3 + log(x + 4)^2*(x + 4) + 144),x)

[Out]

4/(x - log(x + 4) + 6)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.50 \begin {gather*} - \frac {4}{- x + \log {\left (x + 4 \right )} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-12)/((4+x)*ln(4+x)**2+(-2*x**2-20*x-48)*ln(4+x)+x**3+16*x**2+84*x+144),x)

[Out]

-4/(-x + log(x + 4) - 6)

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