Integrand size = 107, antiderivative size = 24 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {e}{3+e^{2 x}+x}+\log \left (\frac {3 x}{3+82 x}\right ) \] Output:
ln(3*x/(82*x+3))+exp(1)/(exp(x)^2+3+x)
Time = 0.91 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {e}{3+e^{2 x}+x}+\log (x)-\log (3+82 x) \] Input:
Integrate[(27 + 3*E^(4*x) + 18*x + 3*x^2 + E*(-3*x - 82*x^2) + E^(2*x)*(18 + 6*x + E*(-6*x - 164*x^2)))/(27*x + 756*x^2 + 495*x^3 + 82*x^4 + E^(4*x) *(3*x + 82*x^2) + E^(2*x)*(18*x + 498*x^2 + 164*x^3)),x]
Output:
E/(3 + E^(2*x) + x) + Log[x] - Log[3 + 82*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.
\(\displaystyle \int \frac {3 x^2+e \left (-82 x^2-3 x\right )+e^{2 x} \left (e \left (-164 x^2-6 x\right )+6 x+18\right )+18 x+3 e^{4 x}+27}{82 x^4+495 x^3+756 x^2+e^{4 x} \left (82 x^2+3 x\right )+e^{2 x} \left (164 x^3+498 x^2+18 x\right )+27 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^2+e \left (-82 x^2-3 x\right )+e^{2 x} \left (e \left (-164 x^2-6 x\right )+6 x+18\right )+18 x+3 e^{4 x}+27}{x \left (x+e^{2 x}+3\right )^2 (82 x+3)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e (2 x+5)}{\left (x+e^{2 x}+3\right )^2}-\frac {2 e}{x+e^{2 x}+3}+\frac {3}{x (82 x+3)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 5 e \int \frac {1}{\left (x+e^{2 x}+3\right )^2}dx+2 e \int \frac {x}{\left (x+e^{2 x}+3\right )^2}dx-2 e \int \frac {1}{x+e^{2 x}+3}dx+\log (x)-\log (82 x+3)\) |
Input:
Int[(27 + 3*E^(4*x) + 18*x + 3*x^2 + E*(-3*x - 82*x^2) + E^(2*x)*(18 + 6*x + E*(-6*x - 164*x^2)))/(27*x + 756*x^2 + 495*x^3 + 82*x^4 + E^(4*x)*(3*x + 82*x^2) + E^(2*x)*(18*x + 498*x^2 + 164*x^3)),x]
Output:
$Aborted
Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {{\mathrm e}}{{\mathrm e}^{2 x}+3+x}-\ln \left (82 x +3\right )+\ln \left (x \right )\) | \(24\) |
risch | \(\frac {{\mathrm e}}{{\mathrm e}^{2 x}+3+x}-\ln \left (82 x +3\right )+\ln \left (x \right )\) | \(24\) |
parallelrisch | \(\frac {\ln \left (x \right ) {\mathrm e}^{2 x}-\ln \left (x +\frac {3}{82}\right ) {\mathrm e}^{2 x}+x \ln \left (x \right )-\ln \left (x +\frac {3}{82}\right ) x +{\mathrm e}+3 \ln \left (x \right )-3 \ln \left (x +\frac {3}{82}\right )}{{\mathrm e}^{2 x}+3+x}\) | \(52\) |
Input:
int((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp( 1)+3*x^2+18*x+27)/((82*x^2+3*x)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+8 2*x^4+495*x^3+756*x^2+27*x),x,method=_RETURNVERBOSE)
Output:
exp(1)/(exp(x)^2+3+x)-ln(82*x+3)+ln(x)
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=-\frac {{\left (x + e^{\left (2 \, x\right )} + 3\right )} \log \left (82 \, x + 3\right ) - {\left (x + e^{\left (2 \, x\right )} + 3\right )} \log \left (x\right ) - e}{x + e^{\left (2 \, x\right )} + 3} \] Input:
integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x )*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp( x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="fricas")
Output:
-((x + e^(2*x) + 3)*log(82*x + 3) - (x + e^(2*x) + 3)*log(x) - e)/(x + e^( 2*x) + 3)
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\log {\left (x \right )} - \log {\left (x + \frac {3}{82} \right )} + \frac {e}{x + e^{2 x} + 3} \] Input:
integrate((3*exp(x)**4+((-164*x**2-6*x)*exp(1)+18+6*x)*exp(x)**2+(-82*x**2 -3*x)*exp(1)+3*x**2+18*x+27)/((82*x**2+3*x)*exp(x)**4+(164*x**3+498*x**2+1 8*x)*exp(x)**2+82*x**4+495*x**3+756*x**2+27*x),x)
Output:
log(x) - log(x + 3/82) + E/(x + exp(2*x) + 3)
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {e}{x + e^{\left (2 \, x\right )} + 3} - \log \left (82 \, x + 3\right ) + \log \left (x\right ) \] Input:
integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x )*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp( x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="maxima")
Output:
e/(x + e^(2*x) + 3) - log(82*x + 3) + log(x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {x \log \left (2 \, x\right ) + e^{\left (2 \, x\right )} \log \left (2 \, x\right ) - x \log \left (82 \, x + 3\right ) - e^{\left (2 \, x\right )} \log \left (82 \, x + 3\right ) + e + 3 \, \log \left (2 \, x\right ) - 3 \, \log \left (82 \, x + 3\right )}{x + e^{\left (2 \, x\right )} + 3} \] Input:
integrate((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x )*exp(1)+3*x^2+18*x+27)/((82*x^2+3*x)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp( x)^2+82*x^4+495*x^3+756*x^2+27*x),x, algorithm="giac")
Output:
(x*log(2*x) + e^(2*x)*log(2*x) - x*log(82*x + 3) - e^(2*x)*log(82*x + 3) + e + 3*log(2*x) - 3*log(82*x + 3))/(x + e^(2*x) + 3)
Time = 3.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\ln \left (x\right )-\ln \left (x+\frac {3}{82}\right )-\frac {\frac {{\mathrm {e}}^{2\,x+1}}{3}+\frac {x\,\mathrm {e}}{3}}{x+{\mathrm {e}}^{2\,x}+3} \] Input:
int((18*x + 3*exp(4*x) + exp(2*x)*(6*x - exp(1)*(6*x + 164*x^2) + 18) - ex p(1)*(3*x + 82*x^2) + 3*x^2 + 27)/(27*x + exp(4*x)*(3*x + 82*x^2) + exp(2* x)*(18*x + 498*x^2 + 164*x^3) + 756*x^2 + 495*x^3 + 82*x^4),x)
Output:
log(x) - log(x + 3/82) - (exp(2*x + 1)/3 + (x*exp(1))/3)/(x + exp(2*x) + 3 )
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {27+3 e^{4 x}+18 x+3 x^2+e \left (-3 x-82 x^2\right )+e^{2 x} \left (18+6 x+e \left (-6 x-164 x^2\right )\right )}{27 x+756 x^2+495 x^3+82 x^4+e^{4 x} \left (3 x+82 x^2\right )+e^{2 x} \left (18 x+498 x^2+164 x^3\right )} \, dx=\frac {-e^{2 x} \mathrm {log}\left (82 x +3\right )+e^{2 x} \mathrm {log}\left (x \right )-\mathrm {log}\left (82 x +3\right ) x -3 \,\mathrm {log}\left (82 x +3\right )+\mathrm {log}\left (x \right ) x +3 \,\mathrm {log}\left (x \right )+e}{e^{2 x}+x +3} \] Input:
int((3*exp(x)^4+((-164*x^2-6*x)*exp(1)+18+6*x)*exp(x)^2+(-82*x^2-3*x)*exp( 1)+3*x^2+18*x+27)/((82*x^2+3*x)*exp(x)^4+(164*x^3+498*x^2+18*x)*exp(x)^2+8 2*x^4+495*x^3+756*x^2+27*x),x)
Output:
( - e**(2*x)*log(82*x + 3) + e**(2*x)*log(x) - log(82*x + 3)*x - 3*log(82* x + 3) + log(x)*x + 3*log(x) + e)/(e**(2*x) + x + 3)