∫ 3−3x+ex(−9x−28x2+23x3+14x4)+(ex(9x+14x2)−3 log(x)) log(1 3(ex(−9x−14x2)+3 log(x))) _________________________________________________________________________________________________________________________

Optimal. Leaf size=22 \[ \left (-1+\frac {1}{x}\right ) \log \left (e^x \left (-3-\frac {14 x}{3}\right ) x+\log (x)\right ) \]

________________________________________________________________________________________

Rubi [F] time = 10.44, 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 {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx \end {gather*}

Int[(3 - 3*x + E^x*(-9*x - 28*x^2 + 23*x^3 + 14*x^4) + (E^x*(9*x + 14*x^2) - 3*Log[x])*Log[(E^x*(-9*x - 14*x^2

) + 3*Log[x])/3])/(E^x*(-9*x^3 - 14*x^4) + 3*x^2*Log[x]),x]

-x^(-1) - x + (14*Log[x])/9 - (23*Log[9 + 14*x])/9 - 3*Defer[Int][1/(x^2*(9*E^x*x + 14*E^x*x^2 - 3*Log[x])), x

] + 3*Defer[Int][1/(x*(9*E^x*x + 14*E^x*x^2 - 3*Log[x])), x] - 3*Defer[Int][Log[x]/(9*E^x*x + 14*E^x*x^2 - 3*L

og[x]), x] + 3*Defer[Int][Log[x]/(x^2*(9*E^x*x + 14*E^x*x^2 - 3*Log[x])), x] + (14*Defer[Int][Log[x]/(x*(9*E^x

*x + 14*E^x*x^2 - 3*Log[x])), x])/3 - (322*Defer[Int][Log[x]/((9 + 14*x)*(9*E^x*x + 14*E^x*x^2 - 3*Log[x])), x

])/3 - Defer[Int][Log[-1/3*(E^x*x*(9 + 14*x)) + Log[x]]/x^2, x]

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3 (-1+x) \left (-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)\right )}{x^2 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {9+28 x-23 x^2-14 x^3-9 \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )-14 x \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2 (9+14 x)}\right ) \, dx\\ &=-\left (3 \int \frac {(-1+x) \left (-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)\right )}{x^2 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx\right )+\int \frac {9+28 x-23 x^2-14 x^3-9 \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )-14 x \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2 (9+14 x)} \, dx\\ &=-\left (3 \int \left (-\frac {-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)}{9 x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {23 \left (-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)\right )}{81 x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {322 \left (-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)\right )}{81 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx\right )+\int \frac {9+28 x-23 x^2-14 x^3-(9+14 x) \log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2 (9+14 x)} \, dx\\ &=\frac {1}{3} \int \frac {-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {23}{27} \int \frac {-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{27} \int \frac {-9-14 x+9 \log (x)+37 x \log (x)+14 x^2 \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\int \left (\frac {9+28 x-23 x^2-14 x^3}{x^2 (9+14 x)}-\frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2}\right ) \, dx\\ &=\frac {1}{3} \int \left (-\frac {9}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {14}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {14 \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)}+\frac {9 \log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {37 \log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx-\frac {23}{27} \int \left (-\frac {14}{9 e^x x+14 e^x x^2-3 \log (x)}-\frac {9}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {37 \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)}+\frac {9 \log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {14 x \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)}\right ) \, dx+\frac {322}{27} \int \left (-\frac {9}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {14 x}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {9 \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {37 x \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {14 x^2 \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx+\int \frac {9+28 x-23 x^2-14 x^3}{x^2 (9+14 x)} \, dx-\int \frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2} \, dx\\ &=-\left (3 \int \frac {1}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx\right )+3 \int \frac {\log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {14}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {14}{3} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {23}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {23}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{27} \int \frac {1}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {322}{27} \int \frac {x \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {37}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {851}{27} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {322}{3} \int \frac {1}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{3} \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {4508}{27} \int \frac {x}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {4508}{27} \int \frac {x^2 \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {11914}{27} \int \frac {x \log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\int \left (-1+\frac {1}{x^2}+\frac {14}{9 x}-\frac {322}{9 (9+14 x)}\right ) \, dx-\int \frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2} \, dx\\ &=-\frac {1}{x}-x+\frac {14 \log (x)}{9}-\frac {23}{9} \log (9+14 x)-3 \int \frac {1}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+3 \int \frac {\log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {14}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {14}{3} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {23}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {23}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{27} \int \frac {1}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {322}{27} \int \frac {x \log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {37}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {851}{27} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {322}{3} \int \frac {1}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{3} \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {4508}{27} \int \left (\frac {1}{14 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {9}{14 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx+\frac {4508}{27} \int \left (-\frac {9 \log (x)}{196 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {x \log (x)}{14 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}+\frac {81 \log (x)}{196 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx+\frac {11914}{27} \int \left (\frac {\log (x)}{14 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}-\frac {9 \log (x)}{14 (9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )}\right ) \, dx-\int \frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2} \, dx\\ &=-\frac {1}{x}-x+\frac {14 \log (x)}{9}-\frac {23}{9} \log (9+14 x)-3 \int \frac {1}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+3 \int \frac {\log (x)}{x^2 \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {14}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {14}{3} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx+\frac {23}{3} \int \frac {1}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {23}{3} \int \frac {\log (x)}{9 e^x x+14 e^x x^2-3 \log (x)} \, dx-\frac {23}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {37}{3} \int \frac {\log (x)}{x \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+69 \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx+\frac {322}{3} \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\frac {851}{3} \int \frac {\log (x)}{(9+14 x) \left (9 e^x x+14 e^x x^2-3 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 1.23, size = 44, normalized size = 2.00 \begin {gather*} -\log \left (9 e^x x+14 e^x x^2-3 \log (x)\right )+\frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x} \end {gather*}

Integrate[(3 - 3*x + E^x*(-9*x - 28*x^2 + 23*x^3 + 14*x^4) + (E^x*(9*x + 14*x^2) - 3*Log[x])*Log[(E^x*(-9*x -

14*x^2) + 3*Log[x])/3])/(E^x*(-9*x^3 - 14*x^4) + 3*x^2*Log[x]),x]

-Log[9*E^x*x + 14*E^x*x^2 - 3*Log[x]] + Log[-1/3*(E^x*x*(9 + 14*x)) + Log[x]]/x

________________________________________________________________________________________

fricas [A] time = 0.63, size = 25, normalized size = 1.14 \begin {gather*} -\frac {{\left (x - 1\right )} \log \left (-\frac {1}{3} \, {\left (14 \, x^{2} + 9 \, x\right )} e^{x} + \log \relax (x)\right )}{x} \end {gather*}

integrate(((-3*log(x)+(14*x^2+9*x)*exp(x))*log(log(x)+1/3*(-14*x^2-9*x)*exp(x))+(14*x^4+23*x^3-28*x^2-9*x)*exp

(x)-3*x+3)/(3*x^2*log(x)+(-14*x^4-9*x^3)*exp(x)),x, algorithm="fricas")

-(x - 1)*log(-1/3*(14*x^2 + 9*x)*e^x + log(x))/x

________________________________________________________________________________________

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

integrate(((-3*log(x)+(14*x^2+9*x)*exp(x))*log(log(x)+1/3*(-14*x^2-9*x)*exp(x))+(14*x^4+23*x^3-28*x^2-9*x)*exp

(x)-3*x+3)/(3*x^2*log(x)+(-14*x^4-9*x^3)*exp(x)),x, algorithm="giac")

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

UT:Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Sign error %%%{ln(` w`),0%%%}

________________________________________________________________________________________


method	result	size

risch	\(\frac {\ln \left (\ln \relax (x )+\frac {\left (-14 x^{2}-9 x \right ) {\mathrm e}^{x}}{3}\right )}{x}-\ln \left (\ln \relax (x )-\frac {14 \,{\mathrm e}^{x} x^{2}}{3}-3 \,{\mathrm e}^{x} x \right )\)	\(41\)

int(((-3*ln(x)+(14*x^2+9*x)*exp(x))*ln(ln(x)+1/3*(-14*x^2-9*x)*exp(x))+(14*x^4+23*x^3-28*x^2-9*x)*exp(x)-3*x+3

)/(3*x^2*ln(x)+(-14*x^4-9*x^3)*exp(x)),x,method=_RETURNVERBOSE)

1/x*ln(ln(x)+1/3*(-14*x^2-9*x)*exp(x))-ln(ln(x)-14/3*exp(x)*x^2-3*exp(x)*x)

________________________________________________________________________________________

maxima [B] time = 0.53, size = 74, normalized size = 3.36 \begin {gather*} -\frac {\log \relax (3) - \log \left (-{\left (14 \, x^{2} + 9 \, x\right )} e^{x} + 3 \, \log \relax (x)\right )}{x} - \log \left (14 \, x + 9\right ) - \log \relax (x) - \log \left (\frac {{\left (14 \, x^{2} + 9 \, x\right )} e^{x} - 3 \, \log \relax (x)}{14 \, x^{2} + 9 \, x}\right ) \end {gather*}

integrate(((-3*log(x)+(14*x^2+9*x)*exp(x))*log(log(x)+1/3*(-14*x^2-9*x)*exp(x))+(14*x^4+23*x^3-28*x^2-9*x)*exp

(x)-3*x+3)/(3*x^2*log(x)+(-14*x^4-9*x^3)*exp(x)),x, algorithm="maxima")

-(log(3) - log(-(14*x^2 + 9*x)*e^x + 3*log(x)))/x - log(14*x + 9) - log(x) - log(((14*x^2 + 9*x)*e^x - 3*log(x

________________________________________________________________________________________

mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {3\,x+\ln \left (\ln \relax (x)-\frac {{\mathrm {e}}^x\,\left (14\,x^2+9\,x\right )}{3}\right )\,\left (3\,\ln \relax (x)-{\mathrm {e}}^x\,\left (14\,x^2+9\,x\right )\right )+{\mathrm {e}}^x\,\left (-14\,x^4-23\,x^3+28\,x^2+9\,x\right )-3}{{\mathrm {e}}^x\,\left (14\,x^4+9\,x^3\right )-3\,x^2\,\ln \relax (x)} \,d x \end {gather*}

int((3*x + log(log(x) - (exp(x)*(9*x + 14*x^2))/3)*(3*log(x) - exp(x)*(9*x + 14*x^2)) + exp(x)*(9*x + 28*x^2 -

23*x^3 - 14*x^4) - 3)/(exp(x)*(9*x^3 + 14*x^4) - 3*x^2*log(x)),x)

int((3*x + log(log(x) - (exp(x)*(9*x + 14*x^2))/3)*(3*log(x) - exp(x)*(9*x + 14*x^2)) + exp(x)*(9*x + 28*x^2 -

23*x^3 - 14*x^4) - 3)/(exp(x)*(9*x^3 + 14*x^4) - 3*x^2*log(x)), x)

________________________________________________________________________________________

sympy [B] time = 1.40, size = 49, normalized size = 2.23 \begin {gather*} - \log {\left (14 x^{2} + 9 x \right )} - \log {\left (e^{x} - \frac {3 \log {\relax (x )}}{14 x^{2} + 9 x} \right )} + \frac {\log {\left (\left (- \frac {14 x^{2}}{3} - 3 x\right ) e^{x} + \log {\relax (x )} \right )}}{x} \end {gather*}

integrate(((-3*ln(x)+(14*x**2+9*x)*exp(x))*ln(ln(x)+1/3*(-14*x**2-9*x)*exp(x))+(14*x**4+23*x**3-28*x**2-9*x)*e

xp(x)-3*x+3)/(3*x**2*ln(x)+(-14*x**4-9*x**3)*exp(x)),x)

-log(14*x**2 + 9*x) - log(exp(x) - 3*log(x)/(14*x**2 + 9*x)) + log((-14*x**2/3 - 3*x)*exp(x) + log(x))/x

________________________________________________________________________________________

3.37.54 \(\int \frac {3-3 x+e^x (-9 x-28 x^2+23 x^3+14 x^4)+(e^x (9 x+14 x^2)-3 \log (x)) \log (\frac {1}{3} (e^x (-9 x-14 x^2)+3 \log (x)))}{e^x (-9 x^3-14 x^4)+3 x^2 \log (x)} \, dx\)