∫ 1+log( 3_ 2x)+log2( 3_ 2x) ______________________________________________xlog ( 3 ___2x)+(28+x+log(2)) log2( 3

Optimal. Leaf size=23 \[ \log \left (\frac {28+x+\log (2)+\frac {x}{\log \left (\frac {3}{2 x}\right )}}{\log (4)}\right ) \]

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Rubi [F] time = 1.19, 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 {1+\log \left (\frac {3}{2 x}\right )+\log ^2\left (\frac {3}{2 x}\right )}{x \log \left (\frac {3}{2 x}\right )+(28+x+\log (2)) \log ^2\left (\frac {3}{2 x}\right )} \, dx \end {gather*}

Int[(1 + Log[3/(2*x)] + Log[3/(2*x)]^2)/(x*Log[3/(2*x)] + (28 + x + Log[2])*Log[3/(2*x)]^2),x]

Log[28 + x + Log[2]] - Log[Log[3/(2*x)]] + Defer[Int][(-x - (28 + x + Log[2])*Log[3/(2*x)])^(-1), x] - (28 + L

og[2])*Defer[Int][1/(x*(x + (28 + x + Log[2])*Log[3/(2*x)])), x] + (28 + Log[2])*Defer[Int][1/((28 + x + Log[2

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{28+x+\log (2)}+\frac {1}{x \log \left (\frac {3}{2 x}\right )}+\frac {-x^2-x (28+\log (2))-(28+\log (2))^2}{x (28+x+\log (2)) \left (x+x \log \left (\frac {3}{2 x}\right )+28 \left (1+\frac {\log (2)}{28}\right ) \log \left (\frac {3}{2 x}\right )\right )}\right ) \, dx\\ &=\log (28+x+\log (2))+\int \frac {1}{x \log \left (\frac {3}{2 x}\right )} \, dx+\int \frac {-x^2-x (28+\log (2))-(28+\log (2))^2}{x (28+x+\log (2)) \left (x+x \log \left (\frac {3}{2 x}\right )+28 \left (1+\frac {\log (2)}{28}\right ) \log \left (\frac {3}{2 x}\right )\right )} \, dx\\ &=\log (28+x+\log (2))+\int \frac {-x^2-x (28+\log (2))-(28+\log (2))^2}{x (28+x+\log (2)) \left (x+(28+x+\log (2)) \log \left (\frac {3}{2 x}\right )\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {3}{2 x}\right )\right )\\ &=\log (28+x+\log (2))-\log \left (\log \left (\frac {3}{2 x}\right )\right )+\int \left (\frac {1}{-x-x \log \left (\frac {3}{2 x}\right )-28 \left (1+\frac {\log (2)}{28}\right ) \log \left (\frac {3}{2 x}\right )}+\frac {-28-\log (2)}{x \left (x+x \log \left (\frac {3}{2 x}\right )+28 \left (1+\frac {\log (2)}{28}\right ) \log \left (\frac {3}{2 x}\right )\right )}+\frac {28+\log (2)}{(28+x+\log (2)) \left (x+x \log \left (\frac {3}{2 x}\right )+28 \left (1+\frac {\log (2)}{28}\right ) \log \left (\frac {3}{2 x}\right )\right )}\right ) \, dx\\ &=\log (28+x+\log (2))-\log \left (\log \left (\frac {3}{2 x}\right )\right )+(-28-\log (2)) \int \frac {1}{x \left (x+x \log \left (\frac {3}{2 x}\right )+28 \left (1+\frac {\log (2)}{28}\right ) \log \left (\frac {3}{2 x}\right )\right )} \, dx+(28+\log (2)) \int \frac {1}{(28+x+\log (2)) \left (x+x \log \left (\frac {3}{2 x}\right )+28 \left (1+\frac {\log (2)}{28}\right ) \log \left (\frac {3}{2 x}\right )\right )} \, dx+\int \frac {1}{-x-x \log \left (\frac {3}{2 x}\right )-28 \left (1+\frac {\log (2)}{28}\right ) \log \left (\frac {3}{2 x}\right )} \, dx\\ &=\log (28+x+\log (2))-\log \left (\log \left (\frac {3}{2 x}\right )\right )+(-28-\log (2)) \int \frac {1}{x \left (x+(28+x+\log (2)) \log \left (\frac {3}{2 x}\right )\right )} \, dx+(28+\log (2)) \int \frac {1}{(28+x+\log (2)) \left (x+(28+x+\log (2)) \log \left (\frac {3}{2 x}\right )\right )} \, dx+\int \frac {1}{-x-(28+x+\log (2)) \log \left (\frac {3}{2 x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.48, size = 46, normalized size = 2.00 \begin {gather*} -\log \left (\log \left (\frac {3}{2 x}\right )\right )+\log \left (x+28 \log \left (\frac {3}{2 x}\right )+x \log \left (\frac {3}{2 x}\right )+\log (2) \log \left (\frac {3}{2 x}\right )\right ) \end {gather*}

Integrate[(1 + Log[3/(2*x)] + Log[3/(2*x)]^2)/(x*Log[3/(2*x)] + (28 + x + Log[2])*Log[3/(2*x)]^2),x]

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fricas [A] time = 0.77, size = 39, normalized size = 1.70 \begin {gather*} \log \left (x + \log \relax (2) + 28\right ) + \log \left (\frac {{\left (x + \log \relax (2) + 28\right )} \log \left (\frac {3}{2 \, x}\right ) + x}{x + \log \relax (2) + 28}\right ) - \log \left (\log \left (\frac {3}{2 \, x}\right )\right ) \end {gather*}

integrate((log(3/2/x)^2+log(3/2/x)+1)/((log(2)+x+28)*log(3/2/x)^2+x*log(3/2/x)),x, algorithm="fricas")

log(x + log(2) + 28) + log(((x + log(2) + 28)*log(3/2/x) + x)/(x + log(2) + 28)) - log(log(3/2/x))

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giac [B] time = 0.26, size = 53, normalized size = 2.30 \begin {gather*} \log \left (\frac {3 \, \log \relax (2) \log \left (\frac {3}{2 \, x}\right )}{x} + \frac {84 \, \log \left (\frac {3}{2 \, x}\right )}{x} + 3 \, \log \left (\frac {3}{2 \, x}\right ) + 3\right ) - \log \left (\frac {3}{2 \, x}\right ) - \log \left (\log \left (\frac {3}{2 \, x}\right )\right ) \end {gather*}

integrate((log(3/2/x)^2+log(3/2/x)+1)/((log(2)+x+28)*log(3/2/x)^2+x*log(3/2/x)),x, algorithm="giac")

log(3*log(2)*log(3/2/x)/x + 84*log(3/2/x)/x + 3*log(3/2/x) + 3) - log(3/2/x) - log(log(3/2/x))

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method	result	size

risch	\(\ln \left (\ln \relax (2)+x +28\right )-\ln \left (\ln \left (\frac {3}{2 x}\right )\right )+\ln \left (\ln \left (\frac {3}{2 x}\right )+\frac {x}{\ln \relax (2)+x +28}\right )\)	\(34\)
norman	\(-\ln \left (\ln \left (\frac {3}{2 x}\right )\right )+\ln \left (\ln \left (\frac {3}{2 x}\right ) \ln \relax (2)+x \ln \left (\frac {3}{2 x}\right )+28 \ln \left (\frac {3}{2 x}\right )+x \right )\)	\(39\)

int((ln(3/2/x)^2+ln(3/2/x)+1)/((ln(2)+x+28)*ln(3/2/x)^2+x*ln(3/2/x)),x,method=_RETURNVERBOSE)

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maxima [B] time = 0.56, size = 66, normalized size = 2.87 \begin {gather*} \log \left (x + \log \relax (2) + 28\right ) + \log \left (-\frac {x {\left (\log \relax (3) - \log \relax (2) + 1\right )} + {\left (\log \relax (3) - 28\right )} \log \relax (2) - \log \relax (2)^{2} - {\left (x + \log \relax (2) + 28\right )} \log \relax (x) + 28 \, \log \relax (3)}{x + \log \relax (2) + 28}\right ) - \log \left (-\log \relax (3) + \log \relax (2) + \log \relax (x)\right ) \end {gather*}

integrate((log(3/2/x)^2+log(3/2/x)+1)/((log(2)+x+28)*log(3/2/x)^2+x*log(3/2/x)),x, algorithm="maxima")

log(x + log(2) + 28) + log(-(x*(log(3) - log(2) + 1) + (log(3) - 28)*log(2) - log(2)^2 - (x + log(2) + 28)*log

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mupad [B] time = 2.15, size = 38, normalized size = 1.65 \begin {gather*} \ln \left (x+28\,\ln \left (\frac {3}{2\,x}\right )+\ln \relax (2)\,\ln \left (\frac {3}{2\,x}\right )+x\,\ln \left (\frac {3}{2\,x}\right )\right )-\ln \left (\ln \left (\frac {3}{2\,x}\right )\right ) \end {gather*}

int((log(3/(2*x)) + log(3/(2*x))^2 + 1)/(x*log(3/(2*x)) + log(3/(2*x))^2*(x + log(2) + 28)),x)

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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

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3.29.65 \(\int \frac {1+\log (\frac {3}{2 x})+\log ^2(\frac {3}{2 x})}{x \log (\frac {3}{2 x})+(28+x+\log (2)) \log ^2(\frac {3}{2 x})} \, dx\)