3.41.2 \(\int \frac {-12-24 x-20 e^6 x-4 e^6 x \log (x)}{36 x+24 x^2+4 x^3+(12 x+4 x^2) \log (x)+x \log ^2(x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {3-e^6 x}{1+\frac {1}{4} (2+2 x+\log (x))} \]

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Rubi [F]  time = 0.51, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-12-24 x-20 e^6 x-4 e^6 x \log (x)}{36 x+24 x^2+4 x^3+\left (12 x+4 x^2\right ) \log (x)+x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12 - 24*x - 20*E^6*x - 4*E^6*x*Log[x])/(36*x + 24*x^2 + 4*x^3 + (12*x + 4*x^2)*Log[x] + x*Log[x]^2),x]

[Out]

-4*(6 - E^6)*Defer[Int][(6 + 2*x + Log[x])^(-2), x] - 12*Defer[Int][1/(x*(6 + 2*x + Log[x])^2), x] + 8*E^6*Def
er[Int][x/(6 + 2*x + Log[x])^2, x] - 4*E^6*Defer[Int][(6 + 2*x + Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-12+\left (-24-20 e^6\right ) x-4 e^6 x \log (x)}{36 x+24 x^2+4 x^3+\left (12 x+4 x^2\right ) \log (x)+x \log ^2(x)} \, dx\\ &=\int \frac {4 \left (-3-\left (6+5 e^6\right ) x-e^6 x \log (x)\right )}{x (6+2 x+\log (x))^2} \, dx\\ &=4 \int \frac {-3-\left (6+5 e^6\right ) x-e^6 x \log (x)}{x (6+2 x+\log (x))^2} \, dx\\ &=4 \int \left (\frac {-3-\left (6-e^6\right ) x+2 e^6 x^2}{x (6+2 x+\log (x))^2}-\frac {e^6}{6+2 x+\log (x)}\right ) \, dx\\ &=4 \int \frac {-3-\left (6-e^6\right ) x+2 e^6 x^2}{x (6+2 x+\log (x))^2} \, dx-\left (4 e^6\right ) \int \frac {1}{6+2 x+\log (x)} \, dx\\ &=4 \int \left (-\frac {6 \left (1-\frac {e^6}{6}\right )}{(6+2 x+\log (x))^2}-\frac {3}{x (6+2 x+\log (x))^2}+\frac {2 e^6 x}{(6+2 x+\log (x))^2}\right ) \, dx-\left (4 e^6\right ) \int \frac {1}{6+2 x+\log (x)} \, dx\\ &=-\left (12 \int \frac {1}{x (6+2 x+\log (x))^2} \, dx\right )-\left (4 e^6\right ) \int \frac {1}{6+2 x+\log (x)} \, dx+\left (8 e^6\right ) \int \frac {x}{(6+2 x+\log (x))^2} \, dx-\left (4 \left (6-e^6\right )\right ) \int \frac {1}{(6+2 x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 18, normalized size = 0.75 \begin {gather*} -\frac {4 \left (-3+e^6 x\right )}{6+2 x+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 - 24*x - 20*E^6*x - 4*E^6*x*Log[x])/(36*x + 24*x^2 + 4*x^3 + (12*x + 4*x^2)*Log[x] + x*Log[x]^2
),x]

[Out]

(-4*(-3 + E^6*x))/(6 + 2*x + Log[x])

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fricas [A]  time = 0.52, size = 17, normalized size = 0.71 \begin {gather*} -\frac {4 \, {\left (x e^{6} - 3\right )}}{2 \, x + \log \relax (x) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(6)*log(x)-20*x*exp(6)-24*x-12)/(x*log(x)^2+(4*x^2+12*x)*log(x)+4*x^3+24*x^2+36*x),x, algor
ithm="fricas")

[Out]

-4*(x*e^6 - 3)/(2*x + log(x) + 6)

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giac [A]  time = 0.15, size = 17, normalized size = 0.71 \begin {gather*} -\frac {4 \, {\left (x e^{6} - 3\right )}}{2 \, x + \log \relax (x) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(6)*log(x)-20*x*exp(6)-24*x-12)/(x*log(x)^2+(4*x^2+12*x)*log(x)+4*x^3+24*x^2+36*x),x, algor
ithm="giac")

[Out]

-4*(x*e^6 - 3)/(2*x + log(x) + 6)

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maple [A]  time = 0.05, size = 18, normalized size = 0.75

method	result	size
risch	\(-\frac {4 \left (x \,{\mathrm e}^{6}-3\right )}{6+\ln \relax (x )+2 x}\)	\(18\)
norman	\(\frac {2 \,{\mathrm e}^{6} \ln \relax (x )+12+12 \,{\mathrm e}^{6}}{6+\ln \relax (x )+2 x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x*exp(6)*ln(x)-20*x*exp(6)-24*x-12)/(x*ln(x)^2+(4*x^2+12*x)*ln(x)+4*x^3+24*x^2+36*x),x,method=_RETURNV
ERBOSE)

[Out]

-4*(x*exp(6)-3)/(6+ln(x)+2*x)

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maxima [A]  time = 0.40, size = 17, normalized size = 0.71 \begin {gather*} -\frac {4 \, {\left (x e^{6} - 3\right )}}{2 \, x + \log \relax (x) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(6)*log(x)-20*x*exp(6)-24*x-12)/(x*log(x)^2+(4*x^2+12*x)*log(x)+4*x^3+24*x^2+36*x),x, algor
ithm="maxima")

[Out]

-4*(x*e^6 - 3)/(2*x + log(x) + 6)

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mupad [B]  time = 2.62, size = 18, normalized size = 0.75 \begin {gather*} -\frac {4\,x\,{\mathrm {e}}^6-12}{2\,x+\ln \relax (x)+6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(24*x + 20*x*exp(6) + 4*x*exp(6)*log(x) + 12)/(36*x + x*log(x)^2 + log(x)*(12*x + 4*x^2) + 24*x^2 + 4*x^3
),x)

[Out]

-(4*x*exp(6) - 12)/(2*x + log(x) + 6)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.62 \begin {gather*} \frac {- 4 x e^{6} + 12}{2 x + \log {\relax (x )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(6)*ln(x)-20*x*exp(6)-24*x-12)/(x*ln(x)**2+(4*x**2+12*x)*ln(x)+4*x**3+24*x**2+36*x),x)

[Out]

(-4*x*exp(6) + 12)/(2*x + log(x) + 6)

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