Integrand size = 67, antiderivative size = 30 \[ \int \frac {1+e^x (3-x)+e^{288-432 x+162 x^2} \left (6485+e^x (15-5 x)-7020 x+1620 x^2\right )}{-3+x+e^{288-432 x+162 x^2} (-15+5 x)} \, dx=-e^x+\log \left (3+5 e^{18 (-4+3 x)^2} (3-x)-x\right ) \] Output:
ln(3-x+exp(3*(-4+3*x)*(9*x-12))^2*(15-5*x))-exp(x)
Time = 7.45 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {1+e^x (3-x)+e^{288-432 x+162 x^2} \left (6485+e^x (15-5 x)-7020 x+1620 x^2\right )}{-3+x+e^{288-432 x+162 x^2} (-15+5 x)} \, dx=-e^x+\log \left (1+5 e^{288-432 x+162 x^2}\right )+\log (3-x) \] Input:
Integrate[(1 + E^x*(3 - x) + E^(288 - 432*x + 162*x^2)*(6485 + E^x*(15 - 5 *x) - 7020*x + 1620*x^2))/(-3 + x + E^(288 - 432*x + 162*x^2)*(-15 + 5*x)) ,x]
Output:
-E^x + Log[1 + 5*E^(288 - 432*x + 162*x^2)] + Log[3 - x]
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.
\(\displaystyle \int \frac {e^{162 x^2-432 x+288} \left (1620 x^2-7020 x+e^x (15-5 x)+6485\right )+e^x (3-x)+1}{e^{162 x^2-432 x+288} (5 x-15)+x-3} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {324 x^2-e^x x-1404 x+3 e^x+1297}{x-3}-\frac {108 e^{432 x} (3 x-4)}{5 e^{162 x^2+288}+e^{432 x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 432 \int \frac {e^{432 x}}{e^{432 x}+5 e^{162 x^2+288}}dx-324 \int \frac {e^{432 x} x}{e^{432 x}+5 e^{162 x^2+288}}dx+162 x^2-432 x-e^x+\log (3-x)\) |
Input:
Int[(1 + E^x*(3 - x) + E^(288 - 432*x + 162*x^2)*(6485 + E^x*(15 - 5*x) - 7020*x + 1620*x^2))/(-3 + x + E^(288 - 432*x + 162*x^2)*(-15 + 5*x)),x]
Output:
$Aborted
Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\ln \left (-3+x \right )-{\mathrm e}^{x}-288+\ln \left ({\mathrm e}^{18 \left (-4+3 x \right )^{2}}+\frac {1}{5}\right )\) | \(24\) |
parallelrisch | \(-{\mathrm e}^{x}+\ln \left (-3+x \right )+\ln \left ({\mathrm e}^{162 x^{2}-432 x +288}+\frac {1}{5}\right )\) | \(26\) |
norman | \(-{\mathrm e}^{x}+\ln \left (-3+x \right )+\ln \left (5 \,{\mathrm e}^{162 x^{2}-432 x +288}+1\right )\) | \(28\) |
parts | \(-{\mathrm e}^{x}+\ln \left (-3+x \right )+\ln \left (5 \,{\mathrm e}^{162 x^{2}-432 x +288}+1\right )\) | \(28\) |
Input:
int((((15-5*x)*exp(x)+1620*x^2-7020*x+6485)*exp(81*x^2-216*x+144)^2+(-x+3) *exp(x)+1)/((5*x-15)*exp(81*x^2-216*x+144)^2+x-3),x,method=_RETURNVERBOSE)
Output:
ln(-3+x)-exp(x)-288+ln(exp(18*(-4+3*x)^2)+1/5)
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1+e^x (3-x)+e^{288-432 x+162 x^2} \left (6485+e^x (15-5 x)-7020 x+1620 x^2\right )}{-3+x+e^{288-432 x+162 x^2} (-15+5 x)} \, dx=-e^{x} + \log \left (x - 3\right ) + \log \left (5 \, e^{\left (162 \, x^{2} - 432 \, x + 288\right )} + 1\right ) \] Input:
integrate((((15-5*x)*exp(x)+1620*x^2-7020*x+6485)*exp(81*x^2-216*x+144)^2+ (3-x)*exp(x)+1)/((5*x-15)*exp(81*x^2-216*x+144)^2+x-3),x, algorithm="frica s")
Output:
-e^x + log(x - 3) + log(5*e^(162*x^2 - 432*x + 288) + 1)
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {1+e^x (3-x)+e^{288-432 x+162 x^2} \left (6485+e^x (15-5 x)-7020 x+1620 x^2\right )}{-3+x+e^{288-432 x+162 x^2} (-15+5 x)} \, dx=- e^{x} + \log {\left (x - 3 \right )} + \log {\left (e^{162 x^{2} - 432 x + 288} + \frac {1}{5} \right )} \] Input:
integrate((((15-5*x)*exp(x)+1620*x**2-7020*x+6485)*exp(81*x**2-216*x+144)* *2+(3-x)*exp(x)+1)/((5*x-15)*exp(81*x**2-216*x+144)**2+x-3),x)
Output:
-exp(x) + log(x - 3) + log(exp(162*x**2 - 432*x + 288) + 1/5)
Time = 1.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1+e^x (3-x)+e^{288-432 x+162 x^2} \left (6485+e^x (15-5 x)-7020 x+1620 x^2\right )}{-3+x+e^{288-432 x+162 x^2} (-15+5 x)} \, dx=-432 \, x - e^{x} + \log \left (\frac {1}{5} \, {\left (5 \, e^{\left (162 \, x^{2} + 288\right )} + e^{\left (432 \, x\right )}\right )} e^{\left (-288\right )}\right ) + \log \left (x - 3\right ) \] Input:
integrate((((15-5*x)*exp(x)+1620*x^2-7020*x+6485)*exp(81*x^2-216*x+144)^2+ (3-x)*exp(x)+1)/((5*x-15)*exp(81*x^2-216*x+144)^2+x-3),x, algorithm="maxim a")
Output:
-432*x - e^x + log(1/5*(5*e^(162*x^2 + 288) + e^(432*x))*e^(-288)) + log(x - 3)
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {1+e^x (3-x)+e^{288-432 x+162 x^2} \left (6485+e^x (15-5 x)-7020 x+1620 x^2\right )}{-3+x+e^{288-432 x+162 x^2} (-15+5 x)} \, dx=-x - e^{x} + \log \left (x - 3\right ) + \log \left (5 \, e^{\left (162 \, x^{2} - 431 \, x + 288\right )} + e^{x}\right ) \] Input:
integrate((((15-5*x)*exp(x)+1620*x^2-7020*x+6485)*exp(81*x^2-216*x+144)^2+ (3-x)*exp(x)+1)/((5*x-15)*exp(81*x^2-216*x+144)^2+x-3),x, algorithm="giac" )
Output:
-x - e^x + log(x - 3) + log(5*e^(162*x^2 - 431*x + 288) + e^x)
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1+e^x (3-x)+e^{288-432 x+162 x^2} \left (6485+e^x (15-5 x)-7020 x+1620 x^2\right )}{-3+x+e^{288-432 x+162 x^2} (-15+5 x)} \, dx=\ln \left (\left (x-3\right )\,\left (5\,{\mathrm {e}}^{-432\,x}\,{\mathrm {e}}^{288}\,{\mathrm {e}}^{162\,x^2}+1\right )\right )-{\mathrm {e}}^x \] Input:
int(-(exp(162*x^2 - 432*x + 288)*(7020*x + exp(x)*(5*x - 15) - 1620*x^2 - 6485) + exp(x)*(x - 3) - 1)/(x + exp(162*x^2 - 432*x + 288)*(5*x - 15) - 3 ),x)
Output:
log((x - 3)*(5*exp(-432*x)*exp(288)*exp(162*x^2) + 1)) - exp(x)
\[ \int \frac {1+e^x (3-x)+e^{288-432 x+162 x^2} \left (6485+e^x (15-5 x)-7020 x+1620 x^2\right )}{-3+x+e^{288-432 x+162 x^2} (-15+5 x)} \, dx=-e^{x}-2160 \left (\int \frac {e^{162 x^{2}}}{5 e^{162 x^{2}} e^{288}+e^{432 x}}d x \right ) e^{288}-324 \left (\int \frac {e^{432 x} x}{5 e^{162 x^{2}} e^{288}+e^{432 x}}d x \right )+\mathrm {log}\left (x -3\right )+162 x^{2} \] Input:
int((((15-5*x)*exp(x)+1620*x^2-7020*x+6485)*exp(81*x^2-216*x+144)^2+(3-x)* exp(x)+1)/((5*x-15)*exp(81*x^2-216*x+144)^2+x-3),x)
Output:
- e**x - 2160*int(e**(162*x**2)/(5*e**(162*x**2)*e**288 + e**(432*x)),x)* e**288 - 324*int((e**(432*x)*x)/(5*e**(162*x**2)*e**288 + e**(432*x)),x) + log(x - 3) + 162*x**2