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

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

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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 = {} $$\begin{array}{c}\int \frac{-12-24x-20e^{6}x-4e^{6}x\log (x)}{36x+24x^2+4x^3+(12x+4x^2)\log (x)+x{\log }^2(x)}dx\end{array}$$

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

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

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

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 $$\begin{array}{c}-\frac{4(x{e}^6-3)}{2x+\log (x)+6}\end{array}$$

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 $$\begin{array}{c}-\frac{4(x{e}^6-3)}{2x+\log (x)+6}\end{array}$$

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	$-\frac{4(x{\mathrm{e}}^6-3)}{6+\ln (x)+2x}$	$18$
norman	$\frac{2{\mathrm{e}}^6\ln (x)+12+12{\mathrm{e}}^6}{6+\ln (x)+2x}$	$23$

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 $$\begin{array}{c}-\frac{4(x{e}^6-3)}{2x+\log (x)+6}\end{array}$$

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 $$\begin{array}{c}-\frac{4x{\mathrm{e}}^6-12}{2x+\ln (x)+6}\end{array}$$

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 $$\begin{array}{c}\frac{-4x{e}^6+12}{2x+\log (x)+6}\end{array}$$

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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