3.52.42 \(\int \frac {-14-5 x+24 x^2+11 x^3+(-4-2 x+6 x^2+3 x^3) \log (\frac {e^2}{2+x})}{-14 x-7 x^2+8 x^3+4 x^4+(-4 x-2 x^2+2 x^3+x^4) \log (\frac {e^2}{2+x})} \, dx\)

Optimal. Leaf size=22 \[ \log \left (x+x \left (-2+x^2\right ) \left (4+\log \left (\frac {e^2}{2+x}\right )\right )\right ) \]

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Rubi [A]  time = 0.43, antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 2, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6741, 6685} \begin {gather*} \log \left (-x \left (-6 x^2+x^2 \left (-\log \left (\frac {1}{x+2}\right )\right )+2 \log \left (\frac {1}{x+2}\right )+11\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-14 - 5*x + 24*x^2 + 11*x^3 + (-4 - 2*x + 6*x^2 + 3*x^3)*Log[E^2/(2 + x)])/(-14*x - 7*x^2 + 8*x^3 + 4*x^4
+ (-4*x - 2*x^2 + 2*x^3 + x^4)*Log[E^2/(2 + x)]),x]

[Out]

Log[-(x*(11 - 6*x^2 + 2*Log[(2 + x)^(-1)] - x^2*Log[(2 + x)^(-1)]))]

Rule 6685

Int[(u_)/((w_)*(y_)), x_Symbol] :> With[{q = DerivativeDivides[y*w, u, x]}, Simp[q*Log[RemoveContent[y*w, x]],
x] /;  !FalseQ[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+5 x-24 x^2-11 x^3-\left (-4-2 x+6 x^2+3 x^3\right ) \log \left (\frac {e^2}{2+x}\right )}{x (2+x) \left (11-6 x^2+2 \log \left (\frac {1}{2+x}\right )-x^2 \log \left (\frac {1}{2+x}\right )\right )} \, dx\\ &=\log \left (-x \left (11-6 x^2+2 \log \left (\frac {1}{2+x}\right )-x^2 \log \left (\frac {1}{2+x}\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 30, normalized size = 1.36 \begin {gather*} \log (x)+\log \left (11-6 x^2+2 \log \left (\frac {1}{2+x}\right )-x^2 \log \left (\frac {1}{2+x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-14 - 5*x + 24*x^2 + 11*x^3 + (-4 - 2*x + 6*x^2 + 3*x^3)*Log[E^2/(2 + x)])/(-14*x - 7*x^2 + 8*x^3 +
4*x^4 + (-4*x - 2*x^2 + 2*x^3 + x^4)*Log[E^2/(2 + x)]),x]

[Out]

Log[x] + Log[11 - 6*x^2 + 2*Log[(2 + x)^(-1)] - x^2*Log[(2 + x)^(-1)]]

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fricas [A]  time = 0.55, size = 40, normalized size = 1.82 \begin {gather*} \log \left (x^{3} - 2 \, x\right ) + \log \left (\frac {4 \, x^{2} + {\left (x^{2} - 2\right )} \log \left (\frac {e^{2}}{x + 2}\right ) - 7}{x^{2} - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+6*x^2-2*x-4)*log(exp(2)/(2+x))+11*x^3+24*x^2-5*x-14)/((x^4+2*x^3-2*x^2-4*x)*log(exp(2)/(2+x)
)+4*x^4+8*x^3-7*x^2-14*x),x, algorithm="fricas")

[Out]

log(x^3 - 2*x) + log((4*x^2 + (x^2 - 2)*log(e^2/(x + 2)) - 7)/(x^2 - 2))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+6*x^2-2*x-4)*log(exp(2)/(2+x))+11*x^3+24*x^2-5*x-14)/((x^4+2*x^3-2*x^2-4*x)*log(exp(2)/(2+x)
)+4*x^4+8*x^3-7*x^2-14*x),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.16, size = 36, normalized size = 1.64

method	result	size
norman	\(\ln \relax (x )+\ln \left (\ln \left (\frac {{\mathrm e}^{2}}{2+x}\right ) x^{2}+4 x^{2}-2 \ln \left (\frac {{\mathrm e}^{2}}{2+x}\right )-7\right )\)	\(36\)
risch	\(\ln \left (x^{3}-2 x \right )+\ln \left (\ln \left (\frac {{\mathrm e}^{2}}{2+x}\right )+\frac {4 x^{2}-7}{x^{2}-2}\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3+6*x^2-2*x-4)*ln(exp(2)/(2+x))+11*x^3+24*x^2-5*x-14)/((x^4+2*x^3-2*x^2-4*x)*ln(exp(2)/(2+x))+4*x^4+
8*x^3-7*x^2-14*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(ln(exp(2)/(2+x))*x^2+4*x^2-2*ln(exp(2)/(2+x))-7)

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maxima [A]  time = 0.43, size = 37, normalized size = 1.68 \begin {gather*} \log \left (x^{2} - 2\right ) + \log \relax (x) + \log \left (-\frac {6 \, x^{2} - {\left (x^{2} - 2\right )} \log \left (x + 2\right ) - 11}{x^{2} - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+6*x^2-2*x-4)*log(exp(2)/(2+x))+11*x^3+24*x^2-5*x-14)/((x^4+2*x^3-2*x^2-4*x)*log(exp(2)/(2+x)
)+4*x^4+8*x^3-7*x^2-14*x),x, algorithm="maxima")

[Out]

log(x^2 - 2) + log(x) + log(-(6*x^2 - (x^2 - 2)*log(x + 2) - 11)/(x^2 - 2))

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mupad [B]  time = 3.66, size = 44, normalized size = 2.00 \begin {gather*} \ln \left (x^3-2\,x\right )+\ln \left (\frac {2\,\ln \left (\frac {1}{x+2}\right )-6\,x^2-x^2\,\ln \left (\frac {1}{x+2}\right )+11}{x^2-2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + log(exp(2)/(x + 2))*(2*x - 6*x^2 - 3*x^3 + 4) - 24*x^2 - 11*x^3 + 14)/(14*x + log(exp(2)/(x + 2))*(
4*x + 2*x^2 - 2*x^3 - x^4) + 7*x^2 - 8*x^3 - 4*x^4),x)

[Out]

log(x^3 - 2*x) + log((2*log(1/(x + 2)) - 6*x^2 - x^2*log(1/(x + 2)) + 11)/(x^2 - 2))

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sympy [A]  time = 0.42, size = 29, normalized size = 1.32 \begin {gather*} \log {\left (x^{3} - 2 x \right )} + \log {\left (\log {\left (\frac {e^{2}}{x + 2} \right )} + \frac {4 x^{2} - 7}{x^{2} - 2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**3+6*x**2-2*x-4)*ln(exp(2)/(2+x))+11*x**3+24*x**2-5*x-14)/((x**4+2*x**3-2*x**2-4*x)*ln(exp(2)/
(2+x))+4*x**4+8*x**3-7*x**2-14*x),x)

[Out]

log(x**3 - 2*x) + log(log(exp(2)/(x + 2)) + (4*x**2 - 7)/(x**2 - 2))

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