Integrand size = 134, antiderivative size = 30 \[ \int \frac {-168 x^2+48 x^3+e^{3/x} \left (-45 x-20 x^2+10 x^3\right )+\left (72 x-24 x^2+e^{3/x} \left (45-5 x^2\right )\right ) \log (3-x)}{72 x^3-24 x^4+e^{3/x} \left (15 x^3-5 x^4\right )+\left (-72 x^2+24 x^3+e^{3/x} \left (-15 x^2+5 x^3\right )\right ) \log (3-x)} \, dx=\log \left (\frac {-\frac {24}{5}-e^{3/x}}{x (-x+\log (3-x))}\right ) \] Output:
ln((-24/5-exp(3/x))/(ln(3-x)-x)/x)
Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-168 x^2+48 x^3+e^{3/x} \left (-45 x-20 x^2+10 x^3\right )+\left (72 x-24 x^2+e^{3/x} \left (45-5 x^2\right )\right ) \log (3-x)}{72 x^3-24 x^4+e^{3/x} \left (15 x^3-5 x^4\right )+\left (-72 x^2+24 x^3+e^{3/x} \left (-15 x^2+5 x^3\right )\right ) \log (3-x)} \, dx=\log \left (24+5 e^{3/x}\right )-\log (x)-\log (x-\log (3-x)) \] Input:
Integrate[(-168*x^2 + 48*x^3 + E^(3/x)*(-45*x - 20*x^2 + 10*x^3) + (72*x - 24*x^2 + E^(3/x)*(45 - 5*x^2))*Log[3 - x])/(72*x^3 - 24*x^4 + E^(3/x)*(15 *x^3 - 5*x^4) + (-72*x^2 + 24*x^3 + E^(3/x)*(-15*x^2 + 5*x^3))*Log[3 - x]) ,x]
Output:
Log[24 + 5*E^(3/x)] - Log[x] - Log[x - Log[3 - x]]
Time = 4.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {7292, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {48 x^3-168 x^2+\left (-24 x^2+e^{3/x} \left (45-5 x^2\right )+72 x\right ) \log (3-x)+e^{3/x} \left (10 x^3-20 x^2-45 x\right )}{-24 x^4+72 x^3+e^{3/x} \left (15 x^3-5 x^4\right )+\left (24 x^3-72 x^2+e^{3/x} \left (5 x^3-15 x^2\right )\right ) \log (3-x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {48 x^3-168 x^2+\left (-24 x^2+e^{3/x} \left (45-5 x^2\right )+72 x\right ) \log (3-x)+e^{3/x} \left (10 x^3-20 x^2-45 x\right )}{\left (5 e^{3/x}+24\right ) (3-x) x^2 (x-\log (3-x))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {72}{\left (5 e^{3/x}+24\right ) x^2}+\frac {-2 x^3+4 x^2+x^2 \log (3-x)+9 x-9 \log (3-x)}{(x-3) x^2 (x-\log (3-x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (5 e^{3/x}+24\right )-\log (x)-\log (x-\log (3-x))\) |
Input:
Int[(-168*x^2 + 48*x^3 + E^(3/x)*(-45*x - 20*x^2 + 10*x^3) + (72*x - 24*x^ 2 + E^(3/x)*(45 - 5*x^2))*Log[3 - x])/(72*x^3 - 24*x^4 + E^(3/x)*(15*x^3 - 5*x^4) + (-72*x^2 + 24*x^3 + E^(3/x)*(-15*x^2 + 5*x^3))*Log[3 - x]),x]
Output:
Log[24 + 5*E^(3/x)] - Log[x] - Log[x - Log[3 - x]]
Time = 2.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {3}{x}}+\frac {24}{5}\right )-\ln \left (\ln \left (-x +3\right )-x \right )\) | \(28\) |
parallelrisch | \(-\ln \left (x \right )-\ln \left (x -\ln \left (-x +3\right )\right )+\ln \left ({\mathrm e}^{\frac {3}{x}}+\frac {24}{5}\right )\) | \(28\) |
norman | \(-\ln \left (x \right )-\ln \left (x -\ln \left (-x +3\right )\right )+\ln \left (5 \,{\mathrm e}^{\frac {3}{x}}+24\right )\) | \(30\) |
Input:
int((((-5*x^2+45)*exp(3/x)-24*x^2+72*x)*ln(-x+3)+(10*x^3-20*x^2-45*x)*exp( 3/x)+48*x^3-168*x^2)/(((5*x^3-15*x^2)*exp(3/x)+24*x^3-72*x^2)*ln(-x+3)+(-5 *x^4+15*x^3)*exp(3/x)-24*x^4+72*x^3),x,method=_RETURNVERBOSE)
Output:
-ln(x)+ln(exp(3/x)+24/5)-ln(ln(-x+3)-x)
Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-168 x^2+48 x^3+e^{3/x} \left (-45 x-20 x^2+10 x^3\right )+\left (72 x-24 x^2+e^{3/x} \left (45-5 x^2\right )\right ) \log (3-x)}{72 x^3-24 x^4+e^{3/x} \left (15 x^3-5 x^4\right )+\left (-72 x^2+24 x^3+e^{3/x} \left (-15 x^2+5 x^3\right )\right ) \log (3-x)} \, dx=-\log \left (x\right ) - \log \left (-x + \log \left (-x + 3\right )\right ) + \log \left (5 \, e^{\frac {3}{x}} + 24\right ) \] Input:
integrate((((-5*x^2+45)*exp(3/x)-24*x^2+72*x)*log(3-x)+(10*x^3-20*x^2-45*x )*exp(3/x)+48*x^3-168*x^2)/(((5*x^3-15*x^2)*exp(3/x)+24*x^3-72*x^2)*log(3- x)+(-5*x^4+15*x^3)*exp(3/x)-24*x^4+72*x^3),x, algorithm="fricas")
Output:
-log(x) - log(-x + log(-x + 3)) + log(5*e^(3/x) + 24)
Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {-168 x^2+48 x^3+e^{3/x} \left (-45 x-20 x^2+10 x^3\right )+\left (72 x-24 x^2+e^{3/x} \left (45-5 x^2\right )\right ) \log (3-x)}{72 x^3-24 x^4+e^{3/x} \left (15 x^3-5 x^4\right )+\left (-72 x^2+24 x^3+e^{3/x} \left (-15 x^2+5 x^3\right )\right ) \log (3-x)} \, dx=- \log {\left (x \right )} - \log {\left (- x + \log {\left (3 - x \right )} \right )} + \log {\left (e^{\frac {3}{x}} + \frac {24}{5} \right )} \] Input:
integrate((((-5*x**2+45)*exp(3/x)-24*x**2+72*x)*ln(3-x)+(10*x**3-20*x**2-4 5*x)*exp(3/x)+48*x**3-168*x**2)/(((5*x**3-15*x**2)*exp(3/x)+24*x**3-72*x** 2)*ln(3-x)+(-5*x**4+15*x**3)*exp(3/x)-24*x**4+72*x**3),x)
Output:
-log(x) - log(-x + log(3 - x)) + log(exp(3/x) + 24/5)
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {-168 x^2+48 x^3+e^{3/x} \left (-45 x-20 x^2+10 x^3\right )+\left (72 x-24 x^2+e^{3/x} \left (45-5 x^2\right )\right ) \log (3-x)}{72 x^3-24 x^4+e^{3/x} \left (15 x^3-5 x^4\right )+\left (-72 x^2+24 x^3+e^{3/x} \left (-15 x^2+5 x^3\right )\right ) \log (3-x)} \, dx=-\log \left (x\right ) - \log \left (-x + \log \left (-x + 3\right )\right ) + \log \left (e^{\frac {3}{x}} + \frac {24}{5}\right ) \] Input:
integrate((((-5*x^2+45)*exp(3/x)-24*x^2+72*x)*log(3-x)+(10*x^3-20*x^2-45*x )*exp(3/x)+48*x^3-168*x^2)/(((5*x^3-15*x^2)*exp(3/x)+24*x^3-72*x^2)*log(3- x)+(-5*x^4+15*x^3)*exp(3/x)-24*x^4+72*x^3),x, algorithm="maxima")
Output:
-log(x) - log(-x + log(-x + 3)) + log(e^(3/x) + 24/5)
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-168 x^2+48 x^3+e^{3/x} \left (-45 x-20 x^2+10 x^3\right )+\left (72 x-24 x^2+e^{3/x} \left (45-5 x^2\right )\right ) \log (3-x)}{72 x^3-24 x^4+e^{3/x} \left (15 x^3-5 x^4\right )+\left (-72 x^2+24 x^3+e^{3/x} \left (-15 x^2+5 x^3\right )\right ) \log (3-x)} \, dx=-\log \left (x\right ) - \log \left (-x + \log \left (-x + 3\right )\right ) + \log \left (5 \, e^{\frac {3}{x}} + 24\right ) \] Input:
integrate((((-5*x^2+45)*exp(3/x)-24*x^2+72*x)*log(3-x)+(10*x^3-20*x^2-45*x )*exp(3/x)+48*x^3-168*x^2)/(((5*x^3-15*x^2)*exp(3/x)+24*x^3-72*x^2)*log(3- x)+(-5*x^4+15*x^3)*exp(3/x)-24*x^4+72*x^3),x, algorithm="giac")
Output:
-log(x) - log(-x + log(-x + 3)) + log(5*e^(3/x) + 24)
Time = 4.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {-168 x^2+48 x^3+e^{3/x} \left (-45 x-20 x^2+10 x^3\right )+\left (72 x-24 x^2+e^{3/x} \left (45-5 x^2\right )\right ) \log (3-x)}{72 x^3-24 x^4+e^{3/x} \left (15 x^3-5 x^4\right )+\left (-72 x^2+24 x^3+e^{3/x} \left (-15 x^2+5 x^3\right )\right ) \log (3-x)} \, dx=\ln \left ({\mathrm {e}}^{3/x}+\frac {24}{5}\right )-\ln \left (\ln \left (3-x\right )-x\right )-\ln \left (x\right ) \] Input:
int((log(3 - x)*(exp(3/x)*(5*x^2 - 45) - 72*x + 24*x^2) + exp(3/x)*(45*x + 20*x^2 - 10*x^3) + 168*x^2 - 48*x^3)/(log(3 - x)*(exp(3/x)*(15*x^2 - 5*x^ 3) + 72*x^2 - 24*x^3) - exp(3/x)*(15*x^3 - 5*x^4) - 72*x^3 + 24*x^4),x)
Output:
log(exp(3/x) + 24/5) - log(log(3 - x) - x) - log(x)
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {-168 x^2+48 x^3+e^{3/x} \left (-45 x-20 x^2+10 x^3\right )+\left (72 x-24 x^2+e^{3/x} \left (45-5 x^2\right )\right ) \log (3-x)}{72 x^3-24 x^4+e^{3/x} \left (15 x^3-5 x^4\right )+\left (-72 x^2+24 x^3+e^{3/x} \left (-15 x^2+5 x^3\right )\right ) \log (3-x)} \, dx=\mathrm {log}\left (e^{\frac {2}{x}} 5^{\frac {2}{3}}-2 e^{\frac {1}{x}} 15^{\frac {1}{3}}+4 \,3^{\frac {2}{3}}\right )+\mathrm {log}\left (e^{\frac {1}{x}} 5^{\frac {1}{3}}+2 \,3^{\frac {1}{3}}\right )-\mathrm {log}\left (\frac {\mathrm {log}\left (-x +3\right )-x}{x}\right )-2 \,\mathrm {log}\left (x \right ) \] Input:
int((((-5*x^2+45)*exp(3/x)-24*x^2+72*x)*log(3-x)+(10*x^3-20*x^2-45*x)*exp( 3/x)+48*x^3-168*x^2)/(((5*x^3-15*x^2)*exp(3/x)+24*x^3-72*x^2)*log(3-x)+(-5 *x^4+15*x^3)*exp(3/x)-24*x^4+72*x^3),x)
Output:
log(e**(2/x)*5**(2/3) - 2*e**(1/x)*15**(1/3) + 4*3**(2/3)) + log(e**(1/x)* 5**(1/3) + 2*3**(1/3)) - log((log( - x + 3) - x)/x) - 2*log(x)