Integrand size = 33, antiderivative size = 28 \[ \int \frac {1}{8} \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx=x-\left (7+\frac {1}{4} \left (-3-e^{3-x}-x+\log (3)\right )\right )^2 \] Output:
x-(25/4-1/4*x-1/4*exp(3-x)+1/4*ln(3))^2
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {1}{8} \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx=\frac {1}{8} \left (-\frac {1}{2} e^{6-2 x}+33 x-\frac {x^2}{2}+x \log (3)+e^{-x} \left (-e^3 x+e^3 (25+\log (3))\right )\right ) \] Input:
Integrate[(33 + E^(6 - 2*x) - x + E^(3 - x)*(-26 + x - Log[3]) + Log[3])/8 ,x]
Output:
(-1/2*E^(6 - 2*x) + 33*x - x^2/2 + x*Log[3] + (-(E^3*x) + E^3*(25 + Log[3] ))/E^x)/8
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 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.
\(\displaystyle \int \frac {1}{8} \left (-x+e^{6-2 x}+e^{3-x} (x-26-\log (3))+33+\log (3)\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \int \left (-x+e^{6-2 x}-e^{3-x} (-x+\log (3)+26)+\log (3)+33\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} \left (-\frac {x^2}{2}-\frac {1}{2} e^{6-2 x}-e^{3-x}+x (33+\log (3))+e^{3-x} (-x+26+\log (3))\right )\) |
Input:
Int[(33 + E^(6 - 2*x) - x + E^(3 - x)*(-26 + x - Log[3]) + Log[3])/8,x]
Output:
(-1/2*E^(6 - 2*x) - E^(3 - x) - x^2/2 + x*(33 + Log[3]) + E^(3 - x)*(26 - x + Log[3]))/8
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36
method | result | size |
risch | \(-\frac {{\mathrm e}^{6-2 x}}{16}+\frac {\left (\ln \left (3\right )+25-x \right ) {\mathrm e}^{-x +3}}{8}+\frac {x \ln \left (3\right )}{8}-\frac {x^{2}}{16}+\frac {33 x}{8}\) | \(38\) |
norman | \(\left (\frac {25}{8}+\frac {\ln \left (3\right )}{8}\right ) {\mathrm e}^{-x +3}+\left (\frac {\ln \left (3\right )}{8}+\frac {33}{8}\right ) x -\frac {x^{2}}{16}-\frac {{\mathrm e}^{6-2 x}}{16}-\frac {{\mathrm e}^{-x +3} x}{8}\) | \(47\) |
parallelrisch | \(-\frac {x^{2}}{16}-\frac {{\mathrm e}^{-x +3} x}{8}-\frac {{\mathrm e}^{6-2 x}}{16}+\frac {{\mathrm e}^{-x +3} \ln \left (3\right )}{8}+\frac {25 \,{\mathrm e}^{-x +3}}{8}+\left (\frac {\ln \left (3\right )}{8}+\frac {33}{8}\right ) x\) | \(52\) |
default | \(\frac {33 x}{8}-\frac {{\mathrm e}^{6-2 x}}{16}+\frac {{\mathrm e}^{-x +3} \left (-x +3\right )}{8}+\frac {11 \,{\mathrm e}^{-x +3}}{4}+\frac {{\mathrm e}^{-x +3} \ln \left (3\right )}{8}-\frac {x^{2}}{16}+\frac {x \ln \left (3\right )}{8}\) | \(56\) |
parts | \(\frac {33 x}{8}-\frac {{\mathrm e}^{6-2 x}}{16}+\frac {{\mathrm e}^{-x +3} \left (-x +3\right )}{8}+\frac {11 \,{\mathrm e}^{-x +3}}{4}+\frac {{\mathrm e}^{-x +3} \ln \left (3\right )}{8}-\frac {x^{2}}{16}+\frac {x \ln \left (3\right )}{8}\) | \(56\) |
derivativedivides | \(\frac {15 x}{4}-\frac {45}{4}-\frac {\left (-x +3\right )^{2}}{16}-\frac {{\mathrm e}^{6-2 x}}{16}+\frac {{\mathrm e}^{-x +3} \left (-x +3\right )}{8}+\frac {11 \,{\mathrm e}^{-x +3}}{4}+\frac {{\mathrm e}^{-x +3} \ln \left (3\right )}{8}-\frac {\ln \left (3\right ) \left (-x +3\right )}{8}\) | \(65\) |
orering | \(\frac {\left (4 \ln \left (3\right )^{3}-6 x^{2} \ln \left (3\right )+2 x^{3}+324 \ln \left (3\right )^{2}+16 x \ln \left (3\right )-184 x^{2}+8685 \ln \left (3\right )+467 x +77080\right ) \left (\frac {{\mathrm e}^{6-2 x}}{8}+\frac {\left (-\ln \left (3\right )+x -26\right ) {\mathrm e}^{-x +3}}{8}+\frac {\ln \left (3\right )}{8}-\frac {x}{8}+\frac {33}{8}\right )}{4 \ln \left (3\right )^{2}-8 x \ln \left (3\right )+4 x^{2}+230 \ln \left (3\right )-230 x +3280}+\frac {\left (6 \ln \left (3\right )^{3}-9 x^{2} \ln \left (3\right )+3 x^{3}+489 \ln \left (3\right )^{2}+6 x \ln \left (3\right )-267 x^{2}+13200 \ln \left (3\right )+176 x +118080\right ) \left (-\frac {{\mathrm e}^{6-2 x}}{4}+\frac {{\mathrm e}^{-x +3}}{8}-\frac {\left (-\ln \left (3\right )+x -26\right ) {\mathrm e}^{-x +3}}{8}-\frac {1}{8}\right )}{4 \ln \left (3\right )^{2}-8 x \ln \left (3\right )+4 x^{2}+230 \ln \left (3\right )-230 x +3280}+\frac {\left (2 \ln \left (3\right )^{3}-3 x^{2} \ln \left (3\right )+x^{3}+163 \ln \left (3\right )^{2}-88 x^{2}+4400 \ln \left (3\right )+39360\right ) \left (\frac {{\mathrm e}^{6-2 x}}{2}-\frac {{\mathrm e}^{-x +3}}{4}+\frac {\left (-\ln \left (3\right )+x -26\right ) {\mathrm e}^{-x +3}}{8}\right )}{4 \ln \left (3\right )^{2}-8 x \ln \left (3\right )+4 x^{2}+230 \ln \left (3\right )-230 x +3280}\) | \(311\) |
Input:
int(1/8*exp(-x+3)^2+1/8*(-ln(3)+x-26)*exp(-x+3)+1/8*ln(3)-1/8*x+33/8,x,met hod=_RETURNVERBOSE)
Output:
-1/16*exp(6-2*x)+1/8*(ln(3)+25-x)*exp(-x+3)+1/8*x*ln(3)-1/16*x^2+33/8*x
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {1}{8} \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx=-\frac {1}{16} \, x^{2} - \frac {1}{8} \, {\left (x - \log \left (3\right ) - 25\right )} e^{\left (-x + 3\right )} + \frac {1}{8} \, x \log \left (3\right ) + \frac {33}{8} \, x - \frac {1}{16} \, e^{\left (-2 \, x + 6\right )} \] Input:
integrate(1/8*exp(3-x)^2+1/8*(-log(3)+x-26)*exp(3-x)+1/8*log(3)-1/8*x+33/8 ,x, algorithm="fricas")
Output:
-1/16*x^2 - 1/8*(x - log(3) - 25)*e^(-x + 3) + 1/8*x*log(3) + 33/8*x - 1/1 6*e^(-2*x + 6)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {1}{8} \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx=- \frac {x^{2}}{16} + x \left (\frac {\log {\left (3 \right )}}{8} + \frac {33}{8}\right ) + \frac {\left (- 16 x + 16 \log {\left (3 \right )} + 400\right ) e^{3 - x}}{128} - \frac {e^{6 - 2 x}}{16} \] Input:
integrate(1/8*exp(3-x)**2+1/8*(-ln(3)+x-26)*exp(3-x)+1/8*ln(3)-1/8*x+33/8, x)
Output:
-x**2/16 + x*(log(3)/8 + 33/8) + (-16*x + 16*log(3) + 400)*exp(3 - x)/128 - exp(6 - 2*x)/16
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {1}{8} \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx=-\frac {1}{16} \, x^{2} - \frac {1}{8} \, {\left (x e^{3} - {\left (\log \left (3\right ) + 25\right )} e^{3}\right )} e^{\left (-x\right )} + \frac {1}{8} \, x \log \left (3\right ) + \frac {33}{8} \, x - \frac {1}{16} \, e^{\left (-2 \, x + 6\right )} \] Input:
integrate(1/8*exp(3-x)^2+1/8*(-log(3)+x-26)*exp(3-x)+1/8*log(3)-1/8*x+33/8 ,x, algorithm="maxima")
Output:
-1/16*x^2 - 1/8*(x*e^3 - (log(3) + 25)*e^3)*e^(-x) + 1/8*x*log(3) + 33/8*x - 1/16*e^(-2*x + 6)
Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {1}{8} \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx=-\frac {1}{16} \, x^{2} - \frac {1}{8} \, {\left (x - \log \left (3\right ) - 25\right )} e^{\left (-x + 3\right )} + \frac {1}{8} \, x \log \left (3\right ) + \frac {33}{8} \, x - \frac {1}{16} \, e^{\left (-2 \, x + 6\right )} \] Input:
integrate(1/8*exp(3-x)^2+1/8*(-log(3)+x-26)*exp(3-x)+1/8*log(3)-1/8*x+33/8 ,x, algorithm="giac")
Output:
-1/16*x^2 - 1/8*(x - log(3) - 25)*e^(-x + 3) + 1/8*x*log(3) + 33/8*x - 1/1 6*e^(-2*x + 6)
Time = 3.87 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {1}{8} \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx=x\,\left (\frac {\ln \left (3\right )}{8}+\frac {33}{8}\right )-\frac {{\mathrm {e}}^{6-2\,x}}{16}+{\mathrm {e}}^{3-x}\,\left (\frac {\ln \left (3\right )}{8}+\frac {25}{8}\right )-\frac {x\,{\mathrm {e}}^{3-x}}{8}-\frac {x^2}{16} \] Input:
int(log(3)/8 - x/8 + exp(6 - 2*x)/8 - (exp(3 - x)*(log(3) - x + 26))/8 + 3 3/8,x)
Output:
x*(log(3)/8 + 33/8) - exp(6 - 2*x)/16 + exp(3 - x)*(log(3)/8 + 25/8) - (x* exp(3 - x))/8 - x^2/16
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {1}{8} \left (33+e^{6-2 x}-x+e^{3-x} (-26+x-\log (3))+\log (3)\right ) \, dx=\frac {2 e^{2 x} \mathrm {log}\left (3\right ) x -e^{2 x} x^{2}+66 e^{2 x} x +2 e^{x} \mathrm {log}\left (3\right ) e^{3}-2 e^{x} e^{3} x +50 e^{x} e^{3}-e^{6}}{16 e^{2 x}} \] Input:
int(1/8*exp(3-x)^2+1/8*(-log(3)+x-26)*exp(3-x)+1/8*log(3)-1/8*x+33/8,x)
Output:
(2*e**(2*x)*log(3)*x - e**(2*x)*x**2 + 66*e**(2*x)*x + 2*e**x*log(3)*e**3 - 2*e**x*e**3*x + 50*e**x*e**3 - e**6)/(16*e**(2*x))