Integrand size = 100, antiderivative size = 26 \[ \int \frac {36 x+348 x^2+841 x^3+e^x \left (-92 x+116 x^2\right )+\left (36 x+348 x^2+841 x^3+e^x (24+116 x)\right ) \log \left (\frac {4 e^x+6 x+29 x^2}{6+29 x}\right )}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx=x \log \left (x+\frac {e^x}{2 x+5 \left (x+\frac {6+x}{20}\right )}\right ) \] Output:
ln(exp(x)/(29/4*x+3/2)+x)*x
Time = 0.84 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {36 x+348 x^2+841 x^3+e^x \left (-92 x+116 x^2\right )+\left (36 x+348 x^2+841 x^3+e^x (24+116 x)\right ) \log \left (\frac {4 e^x+6 x+29 x^2}{6+29 x}\right )}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx=x \log \left (x+\frac {4 e^x}{6+29 x}\right ) \] Input:
Integrate[(36*x + 348*x^2 + 841*x^3 + E^x*(-92*x + 116*x^2) + (36*x + 348* x^2 + 841*x^3 + E^x*(24 + 116*x))*Log[(4*E^x + 6*x + 29*x^2)/(6 + 29*x)])/ (36*x + 348*x^2 + 841*x^3 + E^x*(24 + 116*x)),x]
Output:
x*Log[x + (4*E^x)/(6 + 29*x)]
Time = 1.46 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {7239, 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.
\(\displaystyle \int \frac {841 x^3+348 x^2+e^x \left (116 x^2-92 x\right )+\left (841 x^3+348 x^2+36 x+e^x (116 x+24)\right ) \log \left (\frac {29 x^2+6 x+4 e^x}{29 x+6}\right )+36 x}{841 x^3+348 x^2+36 x+e^x (116 x+24)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\frac {x \left ((29 x+6)^2+4 e^x (29 x-23)\right )}{(29 x+6) \left (x (29 x+6)+4 e^x\right )}+\log \left (x+\frac {4 e^x}{29 x+6}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x \log \left (x+\frac {4 e^x}{29 x+6}\right )\) |
Input:
Int[(36*x + 348*x^2 + 841*x^3 + E^x*(-92*x + 116*x^2) + (36*x + 348*x^2 + 841*x^3 + E^x*(24 + 116*x))*Log[(4*E^x + 6*x + 29*x^2)/(6 + 29*x)])/(36*x + 348*x^2 + 841*x^3 + E^x*(24 + 116*x)),x]
Output:
x*Log[x + (4*E^x)/(6 + 29*x)]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\ln \left (\frac {4 \,{\mathrm e}^{x}+29 x^{2}+6 x}{29 x +6}\right ) x\) | \(25\) |
parallelrisch | \(\ln \left (\frac {4 \,{\mathrm e}^{x}+29 x^{2}+6 x}{29 x +6}\right ) x\) | \(25\) |
risch | \(x \ln \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )-x \ln \left (x +\frac {6}{29}\right )-\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x +\frac {6}{29}}\right ) \operatorname {csgn}\left (i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )}{x +\frac {6}{29}}\right )}{2}+\frac {i \pi x \,\operatorname {csgn}\left (\frac {i}{x +\frac {6}{29}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )}{x +\frac {6}{29}}\right )}^{2}}{2}+\frac {i \pi x \,\operatorname {csgn}\left (i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )}{x +\frac {6}{29}}\right )}^{2}}{2}-\frac {i \pi x {\operatorname {csgn}\left (\frac {i \left (x^{2}+\frac {6 x}{29}+\frac {4 \,{\mathrm e}^{x}}{29}\right )}{x +\frac {6}{29}}\right )}^{3}}{2}\) | \(177\) |
Input:
int((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*ln((4*exp(x)+29*x^2+6*x)/(29 *x+6))+(116*x^2-92*x)*exp(x)+841*x^3+348*x^2+36*x)/((116*x+24)*exp(x)+841* x^3+348*x^2+36*x),x,method=_RETURNVERBOSE)
Output:
ln((4*exp(x)+29*x^2+6*x)/(29*x+6))*x
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {36 x+348 x^2+841 x^3+e^x \left (-92 x+116 x^2\right )+\left (36 x+348 x^2+841 x^3+e^x (24+116 x)\right ) \log \left (\frac {4 e^x+6 x+29 x^2}{6+29 x}\right )}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx=x \log \left (\frac {29 \, x^{2} + 6 \, x + 4 \, e^{x}}{29 \, x + 6}\right ) \] Input:
integrate((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*log((4*exp(x)+29*x^2+6 *x)/(29*x+6))+(116*x^2-92*x)*exp(x)+841*x^3+348*x^2+36*x)/((116*x+24)*exp( x)+841*x^3+348*x^2+36*x),x, algorithm="fricas")
Output:
x*log((29*x^2 + 6*x + 4*e^x)/(29*x + 6))
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (15) = 30\).
Time = 0.42 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {36 x+348 x^2+841 x^3+e^x \left (-92 x+116 x^2\right )+\left (36 x+348 x^2+841 x^3+e^x (24+116 x)\right ) \log \left (\frac {4 e^x+6 x+29 x^2}{6+29 x}\right )}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx=\left (x + \frac {3}{29}\right ) \log {\left (\frac {29 x^{2} + 6 x + 4 e^{x}}{29 x + 6} \right )} + \frac {3 \log {\left (29 x + 6 \right )}}{29} - \frac {3 \log {\left (\frac {29 x^{2}}{4} + \frac {3 x}{2} + e^{x} \right )}}{29} \] Input:
integrate((((116*x+24)*exp(x)+841*x**3+348*x**2+36*x)*ln((4*exp(x)+29*x**2 +6*x)/(29*x+6))+(116*x**2-92*x)*exp(x)+841*x**3+348*x**2+36*x)/((116*x+24) *exp(x)+841*x**3+348*x**2+36*x),x)
Output:
(x + 3/29)*log((29*x**2 + 6*x + 4*exp(x))/(29*x + 6)) + 3*log(29*x + 6)/29 - 3*log(29*x**2/4 + 3*x/2 + exp(x))/29
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {36 x+348 x^2+841 x^3+e^x \left (-92 x+116 x^2\right )+\left (36 x+348 x^2+841 x^3+e^x (24+116 x)\right ) \log \left (\frac {4 e^x+6 x+29 x^2}{6+29 x}\right )}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx=x \log \left (29 \, x^{2} + 6 \, x + 4 \, e^{x}\right ) - x \log \left (29 \, x + 6\right ) \] Input:
integrate((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*log((4*exp(x)+29*x^2+6 *x)/(29*x+6))+(116*x^2-92*x)*exp(x)+841*x^3+348*x^2+36*x)/((116*x+24)*exp( x)+841*x^3+348*x^2+36*x),x, algorithm="maxima")
Output:
x*log(29*x^2 + 6*x + 4*e^x) - x*log(29*x + 6)
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {36 x+348 x^2+841 x^3+e^x \left (-92 x+116 x^2\right )+\left (36 x+348 x^2+841 x^3+e^x (24+116 x)\right ) \log \left (\frac {4 e^x+6 x+29 x^2}{6+29 x}\right )}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx=x \log \left (\frac {29 \, x^{2} + 6 \, x + 4 \, e^{x}}{29 \, x + 6}\right ) \] Input:
integrate((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*log((4*exp(x)+29*x^2+6 *x)/(29*x+6))+(116*x^2-92*x)*exp(x)+841*x^3+348*x^2+36*x)/((116*x+24)*exp( x)+841*x^3+348*x^2+36*x),x, algorithm="giac")
Output:
x*log((29*x^2 + 6*x + 4*e^x)/(29*x + 6))
Time = 2.91 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {36 x+348 x^2+841 x^3+e^x \left (-92 x+116 x^2\right )+\left (36 x+348 x^2+841 x^3+e^x (24+116 x)\right ) \log \left (\frac {4 e^x+6 x+29 x^2}{6+29 x}\right )}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx=x\,\ln \left (\frac {6\,x+4\,{\mathrm {e}}^x+29\,x^2}{29\,x+6}\right ) \] Input:
int((36*x + log((6*x + 4*exp(x) + 29*x^2)/(29*x + 6))*(36*x + exp(x)*(116* x + 24) + 348*x^2 + 841*x^3) - exp(x)*(92*x - 116*x^2) + 348*x^2 + 841*x^3 )/(36*x + exp(x)*(116*x + 24) + 348*x^2 + 841*x^3),x)
Output:
x*log((6*x + 4*exp(x) + 29*x^2)/(29*x + 6))
\[ \int \frac {36 x+348 x^2+841 x^3+e^x \left (-92 x+116 x^2\right )+\left (36 x+348 x^2+841 x^3+e^x (24+116 x)\right ) \log \left (\frac {4 e^x+6 x+29 x^2}{6+29 x}\right )}{36 x+348 x^2+841 x^3+e^x (24+116 x)} \, dx=1682 \left (\int \frac {x^{2}}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+116 \left (\int \frac {e^{x} x^{2}}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+116 \left (\int \frac {e^{x} \mathrm {log}\left (\frac {4 e^{x}+29 x^{2}+6 x}{29 x +6}\right ) x}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+24 \left (\int \frac {e^{x} \mathrm {log}\left (\frac {4 e^{x}+29 x^{2}+6 x}{29 x +6}\right )}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+841 \left (\int \frac {\mathrm {log}\left (\frac {4 e^{x}+29 x^{2}+6 x}{29 x +6}\right ) x^{3}}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+348 \left (\int \frac {\mathrm {log}\left (\frac {4 e^{x}+29 x^{2}+6 x}{29 x +6}\right ) x^{2}}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+36 \left (\int \frac {\mathrm {log}\left (\frac {4 e^{x}+29 x^{2}+6 x}{29 x +6}\right ) x}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+522 \left (\int \frac {x}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+\frac {1196 \left (\int \frac {x}{4 e^{x}+29 x^{2}+6 x}d x \right )}{29}+36 \left (\int \frac {1}{116 e^{x} x +24 e^{x}+841 x^{3}+348 x^{2}+36 x}d x \right )+\frac {138 \left (\int \frac {1}{4 e^{x}+29 x^{2}+6 x}d x \right )}{29}+\frac {138 \,\mathrm {log}\left (29 x +6\right )}{841}-\frac {52 \,\mathrm {log}\left (4 e^{x}+29 x^{2}+6 x \right )}{29}+x \] Input:
int((((116*x+24)*exp(x)+841*x^3+348*x^2+36*x)*log((4*exp(x)+29*x^2+6*x)/(2 9*x+6))+(116*x^2-92*x)*exp(x)+841*x^3+348*x^2+36*x)/((116*x+24)*exp(x)+841 *x^3+348*x^2+36*x),x)
Output:
(1414562*int(x**2/(116*e**x*x + 24*e**x + 841*x**3 + 348*x**2 + 36*x),x) + 97556*int((e**x*x**2)/(116*e**x*x + 24*e**x + 841*x**3 + 348*x**2 + 36*x) ,x) + 97556*int((e**x*log((4*e**x + 29*x**2 + 6*x)/(29*x + 6))*x)/(116*e** x*x + 24*e**x + 841*x**3 + 348*x**2 + 36*x),x) + 20184*int((e**x*log((4*e* *x + 29*x**2 + 6*x)/(29*x + 6)))/(116*e**x*x + 24*e**x + 841*x**3 + 348*x* *2 + 36*x),x) + 707281*int((log((4*e**x + 29*x**2 + 6*x)/(29*x + 6))*x**3) /(116*e**x*x + 24*e**x + 841*x**3 + 348*x**2 + 36*x),x) + 292668*int((log( (4*e**x + 29*x**2 + 6*x)/(29*x + 6))*x**2)/(116*e**x*x + 24*e**x + 841*x** 3 + 348*x**2 + 36*x),x) + 30276*int((log((4*e**x + 29*x**2 + 6*x)/(29*x + 6))*x)/(116*e**x*x + 24*e**x + 841*x**3 + 348*x**2 + 36*x),x) + 439002*int (x/(116*e**x*x + 24*e**x + 841*x**3 + 348*x**2 + 36*x),x) + 34684*int(x/(4 *e**x + 29*x**2 + 6*x),x) + 30276*int(1/(116*e**x*x + 24*e**x + 841*x**3 + 348*x**2 + 36*x),x) + 4002*int(1/(4*e**x + 29*x**2 + 6*x),x) + 138*log(29 *x + 6) - 1508*log(4*e**x + 29*x**2 + 6*x) + 841*x)/841