3.63.90 \(\int \frac {-3 x^2+4 x^3+(-3+4 x-3 x^2) \log (x)+(-3+4 x-4 x \log (x)+3 \log ^2(x)) \log (\frac {1}{3} (3-4 x+3 \log (x)))}{3 x^2-4 x^3+3 x^2 \log (x)} \, dx\)

Optimal. Leaf size=22 \[ 5-x-\frac {\log (x) \log \left (1-\frac {4 x}{3}+\log (x)\right )}{x} \]

________________________________________________________________________________________

Rubi [F]  time = 0.80, 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 x^2+4 x^3+\left (-3+4 x-3 x^2\right ) \log (x)+\left (-3+4 x-4 x \log (x)+3 \log ^2(x)\right ) \log \left (\frac {1}{3} (3-4 x+3 \log (x))\right )}{3 x^2-4 x^3+3 x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x^(-1) - x + (4*Log[x])/3 - (16*Defer[Int][(-3 + 4*x - 3*Log[x])^(-1), x])/3 - 3*Defer[Int][1/(x^2*(-3 + 4*x -
3*Log[x])), x] + 8*Defer[Int][1/(x*(-3 + 4*x - 3*Log[x])), x] - Defer[Int][Log[1 - (4*x)/3 + Log[x]]/x^2, x]
+ Defer[Int][(Log[x]*Log[1 - (4*x)/3 + Log[x]])/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 x^2-4 x^3+3 \log (x)-4 x \log (x)+3 x^2 \log (x)}{x^2 (-3+4 x-3 \log (x))}+\frac {(-1+\log (x)) \log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2}\right ) \, dx\\ &=\int \frac {3 x^2-4 x^3+3 \log (x)-4 x \log (x)+3 x^2 \log (x)}{x^2 (-3+4 x-3 \log (x))} \, dx+\int \frac {(-1+\log (x)) \log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2} \, dx\\ &=\int \left (\frac {-3+4 x-3 x^2}{3 x^2}-\frac {(-3+4 x)^2}{3 x^2 (-3+4 x-3 \log (x))}\right ) \, dx+\int \left (-\frac {\log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2}+\frac {\log (x) \log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {-3+4 x-3 x^2}{x^2} \, dx-\frac {1}{3} \int \frac {(-3+4 x)^2}{x^2 (-3+4 x-3 \log (x))} \, dx-\int \frac {\log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2} \, dx+\int \frac {\log (x) \log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (-3-\frac {3}{x^2}+\frac {4}{x}\right ) \, dx-\frac {1}{3} \int \left (\frac {16}{-3+4 x-3 \log (x)}+\frac {9}{x^2 (-3+4 x-3 \log (x))}-\frac {24}{x (-3+4 x-3 \log (x))}\right ) \, dx-\int \frac {\log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2} \, dx+\int \frac {\log (x) \log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2} \, dx\\ &=\frac {1}{x}-x+\frac {4 \log (x)}{3}-3 \int \frac {1}{x^2 (-3+4 x-3 \log (x))} \, dx-\frac {16}{3} \int \frac {1}{-3+4 x-3 \log (x)} \, dx+8 \int \frac {1}{x (-3+4 x-3 \log (x))} \, dx-\int \frac {\log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2} \, dx+\int \frac {\log (x) \log \left (1-\frac {4 x}{3}+\log (x)\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 21, normalized size = 0.95 \begin {gather*} -x-\frac {\log (x) \log \left (1-\frac {4 x}{3}+\log (x)\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - (Log[x]*Log[1 - (4*x)/3 + Log[x]])/x

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 20, normalized size = 0.91 \begin {gather*} -\frac {x^{2} + \log \relax (x) \log \left (-\frac {4}{3} \, x + \log \relax (x) + 1\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(x)^2-4*x*log(x)+4*x-3)*log(log(x)+1-4/3*x)+(-3*x^2+4*x-3)*log(x)+4*x^3-3*x^2)/(3*x^2*log(x)-
4*x^3+3*x^2),x, algorithm="fricas")

[Out]

-(x^2 + log(x)*log(-4/3*x + log(x) + 1))/x

________________________________________________________________________________________

giac [A]  time = 0.16, size = 29, normalized size = 1.32 \begin {gather*} -x + \frac {\log \relax (3) \log \relax (x)}{x} - \frac {\log \relax (x) \log \left (-4 \, x + 3 \, \log \relax (x) + 3\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(x)^2-4*x*log(x)+4*x-3)*log(log(x)+1-4/3*x)+(-3*x^2+4*x-3)*log(x)+4*x^3-3*x^2)/(3*x^2*log(x)-
4*x^3+3*x^2),x, algorithm="giac")

[Out]

-x + log(3)*log(x)/x - log(x)*log(-4*x + 3*log(x) + 3)/x

________________________________________________________________________________________

maple [A]  time = 0.10, size = 20, normalized size = 0.91

method	result	size
risch	\(-\frac {\ln \relax (x ) \ln \left (\ln \relax (x )+1-\frac {4 x}{3}\right )}{x}-x\)	\(20\)
norman	\(\frac {-x^{2}-\ln \relax (x ) \ln \left (\ln \relax (x )+1-\frac {4 x}{3}\right )}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*ln(x)^2-4*x*ln(x)+4*x-3)*ln(ln(x)+1-4/3*x)+(-3*x^2+4*x-3)*ln(x)+4*x^3-3*x^2)/(3*x^2*ln(x)-4*x^3+3*x^2)
,x,method=_RETURNVERBOSE)

[Out]

-ln(x)*ln(ln(x)+1-4/3*x)/x-x

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 28, normalized size = 1.27 \begin {gather*} -\frac {x^{2} - \log \relax (3) \log \relax (x) + \log \relax (x) \log \left (-4 \, x + 3 \, \log \relax (x) + 3\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(x)^2-4*x*log(x)+4*x-3)*log(log(x)+1-4/3*x)+(-3*x^2+4*x-3)*log(x)+4*x^3-3*x^2)/(3*x^2*log(x)-
4*x^3+3*x^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.48, size = 19, normalized size = 0.86 \begin {gather*} -x-\frac {\ln \left (\ln \relax (x)-\frac {4\,x}{3}+1\right )\,\ln \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x) - (4*x)/3 + 1)*(4*x + 3*log(x)^2 - 4*x*log(x) - 3) - log(x)*(3*x^2 - 4*x + 3) - 3*x^2 + 4*x^3)
/(3*x^2*log(x) + 3*x^2 - 4*x^3),x)

[Out]

- x - (log(log(x) - (4*x)/3 + 1)*log(x))/x

________________________________________________________________________________________

sympy [A]  time = 0.48, size = 19, normalized size = 0.86 \begin {gather*} - x - \frac {\log {\relax (x )} \log {\left (- \frac {4 x}{3} + \log {\relax (x )} + 1 \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*ln(x)**2-4*x*ln(x)+4*x-3)*ln(ln(x)+1-4/3*x)+(-3*x**2+4*x-3)*ln(x)+4*x**3-3*x**2)/(3*x**2*ln(x)-4
*x**3+3*x**2),x)

[Out]

-x - log(x)*log(-4*x/3 + log(x) + 1)/x

________________________________________________________________________________________