Integrand size = 127, antiderivative size = 23 \[ \int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx=\left (3+(4-x)^{\frac {1}{4-\frac {3 e^{-x}}{5}}}\right )^2 \] Output:
(3+exp(ln(4-x)/(4-3/5/exp(x))))^2
Time = 3.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx=\left (6+(4-x)^{\frac {5 e^x}{-3+20 e^x}}\right ) (4-x)^{\frac {5 e^x}{-3+20 e^x}} \] Input:
Integrate[((4 - x)^((5*E^x)/(-3 + 20*E^x))*(-90*E^x + 600*E^(2*x) + E^x*(3 60 - 90*x)*Log[4 - x]) + (4 - x)^((10*E^x)/(-3 + 20*E^x))*(-30*E^x + 200*E ^(2*x) + E^x*(120 - 30*x)*Log[4 - x]))/(-36 + E^x*(480 - 120*x) + 9*x + E^ (2*x)*(-1600 + 400*x)),x]
Output:
(6 + (4 - x)^((5*E^x)/(-3 + 20*E^x)))*(4 - x)^((5*E^x)/(-3 + 20*E^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.
\(\displaystyle \int \frac {(4-x)^{\frac {5 e^x}{20 e^x-3}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{20 e^x-3}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{e^x (480-120 x)+9 x+e^{2 x} (400 x-1600)-36} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {10 e^x \left ((4-x)^{\frac {5 e^x}{20 e^x-3}}+3\right ) (4-x)^{\frac {5 e^x}{20 e^x-3}-1} \left (-20 e^x+3 x \log (4-x)-12 \log (4-x)+3\right )}{\left (3-20 e^x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 10 \int \frac {e^x \left ((4-x)^{-\frac {5 e^x}{3-20 e^x}}+3\right ) (4-x)^{-1-\frac {5 e^x}{3-20 e^x}} \left (3 x \log (4-x)-12 \log (4-x)-20 e^x+3\right )}{\left (3-20 e^x\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle 10 \int \frac {e^x \left ((4-x)^{-\frac {5 e^x}{3-20 e^x}}+3\right ) (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \left (3 x \log (4-x)-12 \log (4-x)-20 e^x+3\right )}{\left (3-20 e^x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 10 \int \left (-\frac {3 e^x \left (-3 x \log (4-x)+12 \log (4-x)+20 e^x-3\right ) (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}}}{\left (-3+20 e^x\right )^2}-\frac {e^x \left (-3 x \log (4-x)+12 \log (4-x)+20 e^x-3\right ) (4-x)^{\frac {5 e^x}{-3+20 e^x}-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}}}{\left (-3+20 e^x\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 10 \left (-3 \int \frac {e^x (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}}}{-3+20 e^x}dx-\int \frac {e^x (4-x)^{-\frac {-3+10 e^x}{-3+20 e^x}}}{-3+20 e^x}dx-9 \int \frac {\int \frac {e^x (4-x)^{\frac {5 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2}dx}{4-x}dx-3 \int \frac {\int \frac {e^x (4-x)^{\frac {10 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2}dx}{4-x}dx-9 \log (4-x) \int \frac {e^x (4-x)^{\frac {5 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2}dx-3 \log (4-x) \int \frac {e^x (4-x)^{\frac {10 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2}dx\right )\) |
Input:
Int[((4 - x)^((5*E^x)/(-3 + 20*E^x))*(-90*E^x + 600*E^(2*x) + E^x*(360 - 9 0*x)*Log[4 - x]) + (4 - x)^((10*E^x)/(-3 + 20*E^x))*(-30*E^x + 200*E^(2*x) + E^x*(120 - 30*x)*Log[4 - x]))/(-36 + E^x*(480 - 120*x) + 9*x + E^(2*x)* (-1600 + 400*x)),x]
Output:
$Aborted
Time = 5.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83
method | result | size |
risch | \(\left (-x +4\right )^{\frac {10 \,{\mathrm e}^{x}}{20 \,{\mathrm e}^{x}-3}}+6 \left (-x +4\right )^{\frac {5 \,{\mathrm e}^{x}}{20 \,{\mathrm e}^{x}-3}}\) | \(42\) |
parallelrisch | \({\mathrm e}^{\frac {10 \,{\mathrm e}^{x} \ln \left (-x +4\right )}{20 \,{\mathrm e}^{x}-3}}+6 \,{\mathrm e}^{\frac {5 \,{\mathrm e}^{x} \ln \left (-x +4\right )}{20 \,{\mathrm e}^{x}-3}}\) | \(44\) |
Input:
int((((-30*x+120)*exp(x)*ln(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp(x)*ln( -x+4)/(20*exp(x)-3))^2+((-90*x+360)*exp(x)*ln(-x+4)+600*exp(x)^2-90*exp(x) )*exp(5*exp(x)*ln(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2+(-120*x+480 )*exp(x)+9*x-36),x,method=_RETURNVERBOSE)
Output:
((-x+4)^(5*exp(x)/(20*exp(x)-3)))^2+6*(-x+4)^(5*exp(x)/(20*exp(x)-3))
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx={\left (-x + 4\right )}^{\frac {10 \, e^{x}}{20 \, e^{x} - 3}} + 6 \, {\left (-x + 4\right )}^{\frac {5 \, e^{x}}{20 \, e^{x} - 3}} \] Input:
integrate((((-30*x+120)*exp(x)*log(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp (x)*log(-x+4)/(20*exp(x)-3))^2+((-90*x+360)*exp(x)*log(-x+4)+600*exp(x)^2- 90*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2+( -120*x+480)*exp(x)+9*x-36),x, algorithm="fricas")
Output:
(-x + 4)^(10*e^x/(20*e^x - 3)) + 6*(-x + 4)^(5*e^x/(20*e^x - 3))
Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx=e^{\frac {10 e^{x} \log {\left (4 - x \right )}}{20 e^{x} - 3}} + 6 e^{\frac {5 e^{x} \log {\left (4 - x \right )}}{20 e^{x} - 3}} \] Input:
integrate((((-30*x+120)*exp(x)*ln(-x+4)+200*exp(x)**2-30*exp(x))*exp(5*exp (x)*ln(-x+4)/(20*exp(x)-3))**2+((-90*x+360)*exp(x)*ln(-x+4)+600*exp(x)**2- 90*exp(x))*exp(5*exp(x)*ln(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)**2+( -120*x+480)*exp(x)+9*x-36),x)
Output:
exp(10*exp(x)*log(4 - x)/(20*exp(x) - 3)) + 6*exp(5*exp(x)*log(4 - x)/(20* exp(x) - 3))
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx={\left (-x + 4\right )}^{\frac {10 \, e^{x}}{20 \, e^{x} - 3}} + 6 \, {\left (-x + 4\right )}^{\frac {5 \, e^{x}}{20 \, e^{x} - 3}} \] Input:
integrate((((-30*x+120)*exp(x)*log(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp (x)*log(-x+4)/(20*exp(x)-3))^2+((-90*x+360)*exp(x)*log(-x+4)+600*exp(x)^2- 90*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2+( -120*x+480)*exp(x)+9*x-36),x, algorithm="maxima")
Output:
(-x + 4)^(10*e^x/(20*e^x - 3)) + 6*(-x + 4)^(5*e^x/(20*e^x - 3))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.40 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.30 \[ \int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx={\left (-x + 4\right )}^{\frac {10 \, e^{4}}{20 \, e^{4} - 3 \, e^{\left (-x + 4\right )}}} + 6 \, {\left (-x + 4\right )}^{\frac {5 \, e^{4}}{20 \, e^{4} - 3 \, e^{\left (-x + 4\right )}}} \] Input:
integrate((((-30*x+120)*exp(x)*log(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp (x)*log(-x+4)/(20*exp(x)-3))^2+((-90*x+360)*exp(x)*log(-x+4)+600*exp(x)^2- 90*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2+( -120*x+480)*exp(x)+9*x-36),x, algorithm="giac")
Output:
(-x + 4)^(10*e^4/(20*e^4 - 3*e^(-x + 4))) + 6*(-x + 4)^(5*e^4/(20*e^4 - 3* e^(-x + 4)))
Time = 2.72 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx=\left ({\left (4-x\right )}^{\frac {5\,{\mathrm {e}}^x}{20\,{\mathrm {e}}^x-3}}+6\right )\,{\left (4-x\right )}^{\frac {5\,{\mathrm {e}}^x}{20\,{\mathrm {e}}^x-3}} \] Input:
int(-(exp((10*exp(x)*log(4 - x))/(20*exp(x) - 3))*(30*exp(x) - 200*exp(2*x ) + exp(x)*log(4 - x)*(30*x - 120)) + exp((5*exp(x)*log(4 - x))/(20*exp(x) - 3))*(90*exp(x) - 600*exp(2*x) + exp(x)*log(4 - x)*(90*x - 360)))/(9*x - exp(x)*(120*x - 480) + exp(2*x)*(400*x - 1600) - 36),x)
Output:
((4 - x)^((5*exp(x))/(20*exp(x) - 3)) + 6)*(4 - x)^((5*exp(x))/(20*exp(x) - 3))
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx=e^{\frac {5 e^{x} \mathrm {log}\left (-x +4\right )}{20 e^{x}-3}} \left (e^{\frac {5 e^{x} \mathrm {log}\left (-x +4\right )}{20 e^{x}-3}}+6\right ) \] Input:
int((((-30*x+120)*exp(x)*log(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp(x)*lo g(-x+4)/(20*exp(x)-3))^2+((-90*x+360)*exp(x)*log(-x+4)+600*exp(x)^2-90*exp (x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2+(-120*x +480)*exp(x)+9*x-36),x)
Output:
e**((5*e**x*log( - x + 4))/(20*e**x - 3))*(e**((5*e**x*log( - x + 4))/(20* e**x - 3)) + 6)