3.38.76 \(\int \frac {-3-2 x+2 x \log (x)+\log ^2(x)}{9 x+12 x^2+4 x^3+(6 x+4 x^2) \log ^2(x)+x \log ^4(x)} \, dx\)

Optimal. Leaf size=28 \[ -e^3+\left (-1+e^2\right )^2-\frac {\log (x)}{3+2 x+\log ^2(x)} \]

________________________________________________________________________________________

Rubi [F]  time = 0.48, 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 {-3-2 x+2 x \log (x)+\log ^2(x)}{9 x+12 x^2+4 x^3+\left (6 x+4 x^2\right ) \log ^2(x)+x \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3 - 2*x + 2*x*Log[x] + Log[x]^2)/(9*x + 12*x^2 + 4*x^3 + (6*x + 4*x^2)*Log[x]^2 + x*Log[x]^4),x]

[Out]

-4*Defer[Int][(3 + 2*x + Log[x]^2)^(-2), x] - 6*Defer[Int][1/(x*(3 + 2*x + Log[x]^2)^2), x] + 2*Defer[Int][Log
[x]/(3 + 2*x + Log[x]^2)^2, x] + Defer[Int][1/(x*(3 + 2*x + Log[x]^2)), x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\log (x)}{3+2 x+\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - 2*x + 2*x*Log[x] + Log[x]^2)/(9*x + 12*x^2 + 4*x^3 + (6*x + 4*x^2)*Log[x]^2 + x*Log[x]^4),x]

[Out]

-(Log[x]/(3 + 2*x + Log[x]^2))

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\log \relax (x)}{\log \relax (x)^{2} + 2 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^2+2*x*log(x)-2*x-3)/(x*log(x)^4+(4*x^2+6*x)*log(x)^2+4*x^3+12*x^2+9*x),x, algorithm="fricas"
)

[Out]

-log(x)/(log(x)^2 + 2*x + 3)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\log \relax (x)}{\log \relax (x)^{2} + 2 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^2+2*x*log(x)-2*x-3)/(x*log(x)^4+(4*x^2+6*x)*log(x)^2+4*x^3+12*x^2+9*x),x, algorithm="giac")

[Out]

-log(x)/(log(x)^2 + 2*x + 3)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 16, normalized size = 0.57

method	result	size
norman	\(-\frac {\ln \relax (x )}{3+\ln \relax (x )^{2}+2 x}\)	\(16\)
risch	\(-\frac {\ln \relax (x )}{3+\ln \relax (x )^{2}+2 x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)^2+2*x*ln(x)-2*x-3)/(x*ln(x)^4+(4*x^2+6*x)*ln(x)^2+4*x^3+12*x^2+9*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)/(3+ln(x)^2+2*x)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\log \relax (x)}{\log \relax (x)^{2} + 2 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)^2+2*x*log(x)-2*x-3)/(x*log(x)^4+(4*x^2+6*x)*log(x)^2+4*x^3+12*x^2+9*x),x, algorithm="maxima"
)

[Out]

-log(x)/(log(x)^2 + 2*x + 3)

________________________________________________________________________________________

mupad [B]  time = 2.26, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\ln \relax (x)}{{\ln \relax (x)}^2+2\,x+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - log(x)^2 - 2*x*log(x) + 3)/(9*x + log(x)^2*(6*x + 4*x^2) + x*log(x)^4 + 12*x^2 + 4*x^3),x)

[Out]

-log(x)/(2*x + log(x)^2 + 3)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 14, normalized size = 0.50 \begin {gather*} - \frac {\log {\relax (x )}}{2 x + \log {\relax (x )}^{2} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)**2+2*x*ln(x)-2*x-3)/(x*ln(x)**4+(4*x**2+6*x)*ln(x)**2+4*x**3+12*x**2+9*x),x)

[Out]

-log(x)/(2*x + log(x)**2 + 3)

________________________________________________________________________________________