3.47.8 \(\int \frac {-14+2 x+6 x^2}{48-56 x+4 x^2+8 x^3+(12-14 x+x^2+2 x^3) \log (-12+14 x-x^2-2 x^3)} \, dx\)

Optimal. Leaf size=19 \[ \log (4+\log ((-3+2 x) (4+(-2-x) x))) \]

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Rubi [A]  time = 0.14, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6741, 6684} \begin {gather*} \log \left (\log \left (-2 x^3-x^2+14 x-12\right )+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-14 + 2*x + 6*x^2)/(48 - 56*x + 4*x^2 + 8*x^3 + (12 - 14*x + x^2 + 2*x^3)*Log[-12 + 14*x - x^2 - 2*x^3]),
x]

[Out]

Log[4 + Log[-12 + 14*x - x^2 - 2*x^3]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-14+2 x+6 x^2}{\left (12-14 x+x^2+2 x^3\right ) \left (4+\log \left (-12+14 x-x^2-2 x^3\right )\right )} \, dx\\ &=\log \left (4+\log \left (-12+14 x-x^2-2 x^3\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 19, normalized size = 1.00 \begin {gather*} \log \left (4+\log \left (-12+14 x-x^2-2 x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-14 + 2*x + 6*x^2)/(48 - 56*x + 4*x^2 + 8*x^3 + (12 - 14*x + x^2 + 2*x^3)*Log[-12 + 14*x - x^2 - 2*
x^3]),x]

[Out]

Log[4 + Log[-12 + 14*x - x^2 - 2*x^3]]

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fricas [A]  time = 0.43, size = 19, normalized size = 1.00 \begin {gather*} \log \left (\log \left (-2 \, x^{3} - x^{2} + 14 \, x - 12\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2+2*x-14)/((2*x^3+x^2-14*x+12)*log(-2*x^3-x^2+14*x-12)+8*x^3+4*x^2-56*x+48),x, algorithm="frica
s")

[Out]

log(log(-2*x^3 - x^2 + 14*x - 12) + 4)

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giac [A]  time = 0.18, size = 19, normalized size = 1.00 \begin {gather*} \log \left (\log \left (-2 \, x^{3} - x^{2} + 14 \, x - 12\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2+2*x-14)/((2*x^3+x^2-14*x+12)*log(-2*x^3-x^2+14*x-12)+8*x^3+4*x^2-56*x+48),x, algorithm="giac"
)

[Out]

log(log(-2*x^3 - x^2 + 14*x - 12) + 4)

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maple [A]  time = 0.05, size = 20, normalized size = 1.05

method	result	size
norman	\(\ln \left (\ln \left (-2 x^{3}-x^{2}+14 x -12\right )+4\right )\)	\(20\)
risch	\(\ln \left (\ln \left (-2 x^{3}-x^{2}+14 x -12\right )+4\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^2+2*x-14)/((2*x^3+x^2-14*x+12)*ln(-2*x^3-x^2+14*x-12)+8*x^3+4*x^2-56*x+48),x,method=_RETURNVERBOSE)

[Out]

ln(ln(-2*x^3-x^2+14*x-12)+4)

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maxima [A]  time = 0.40, size = 20, normalized size = 1.05 \begin {gather*} \log \left (\log \left (-x^{2} - 2 \, x + 4\right ) + \log \left (2 \, x - 3\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^2+2*x-14)/((2*x^3+x^2-14*x+12)*log(-2*x^3-x^2+14*x-12)+8*x^3+4*x^2-56*x+48),x, algorithm="maxim
a")

[Out]

log(log(-x^2 - 2*x + 4) + log(2*x - 3) + 4)

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mupad [B]  time = 0.74, size = 19, normalized size = 1.00 \begin {gather*} \ln \left (\ln \left (-\left (2\,x-3\right )\,\left (x^2+2\,x-4\right )\right )+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 6*x^2 - 14)/(log(14*x - x^2 - 2*x^3 - 12)*(x^2 - 14*x + 2*x^3 + 12) - 56*x + 4*x^2 + 8*x^3 + 48),x)

[Out]

log(log(-(2*x - 3)*(2*x + x^2 - 4)) + 4)

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sympy [A]  time = 0.25, size = 17, normalized size = 0.89 \begin {gather*} \log {\left (\log {\left (- 2 x^{3} - x^{2} + 14 x - 12 \right )} + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x**2+2*x-14)/((2*x**3+x**2-14*x+12)*ln(-2*x**3-x**2+14*x-12)+8*x**3+4*x**2-56*x+48),x)

[Out]

log(log(-2*x**3 - x**2 + 14*x - 12) + 4)

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