Integrand size = 87, antiderivative size = 18 \[ \int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+234 x+156 e^4 x+26 e^8 x} \, dx=x \log \left (-\frac {25}{\left (3+e^4\right )^2}-\frac {26 x}{69}\right ) \] Output:
ln(-26/69*x-25/(exp(4)+3)^2)*x
Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(18)=36\).
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.11 \[ \int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+234 x+156 e^4 x+26 e^8 x} \, dx=\frac {\left (1725+26 \left (3+e^4\right )^2 x\right ) \log \left (-\frac {25}{\left (3+e^4\right )^2}-\frac {26 x}{69}\right )-1725 \log \left (1725+26 \left (3+e^4\right )^2 x\right )}{26 \left (3+e^4\right )^2} \] Input:
Integrate[(234*x + 156*E^4*x + 26*E^8*x + (1725 + 234*x + 156*E^4*x + 26*E ^8*x)*Log[(-1725 - 234*x - 156*E^4*x - 26*E^8*x)/(621 + 414*E^4 + 69*E^8)] )/(1725 + 234*x + 156*E^4*x + 26*E^8*x),x]
Output:
((1725 + 26*(3 + E^4)^2*x)*Log[-25/(3 + E^4)^2 - (26*x)/69] - 1725*Log[172 5 + 26*(3 + E^4)^2*x])/(26*(3 + E^4)^2)
Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(18)=36\).
Time = 0.35 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6, 6, 6, 6, 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.
\(\displaystyle \int \frac {26 e^8 x+156 e^4 x+234 x+\left (26 e^8 x+156 e^4 x+234 x+1725\right ) \log \left (\frac {-26 e^8 x-156 e^4 x-234 x-1725}{621+414 e^4+69 e^8}\right )}{26 e^8 x+156 e^4 x+234 x+1725} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {26 e^8 x+156 e^4 x+234 x+\left (26 e^8 x+156 e^4 x+234 x+1725\right ) \log \left (\frac {-26 e^8 x-156 e^4 x-234 x-1725}{621+414 e^4+69 e^8}\right )}{\left (234+156 e^4\right ) x+26 e^8 x+1725}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {26 e^8 x+156 e^4 x+234 x+\left (26 e^8 x+156 e^4 x+234 x+1725\right ) \log \left (\frac {-26 e^8 x-156 e^4 x-234 x-1725}{621+414 e^4+69 e^8}\right )}{\left (234+156 e^4+26 e^8\right ) x+1725}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (234+156 e^4\right ) x+26 e^8 x+\left (26 e^8 x+156 e^4 x+234 x+1725\right ) \log \left (\frac {-26 e^8 x-156 e^4 x-234 x-1725}{621+414 e^4+69 e^8}\right )}{\left (234+156 e^4+26 e^8\right ) x+1725}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (234+156 e^4+26 e^8\right ) x+\left (26 e^8 x+156 e^4 x+234 x+1725\right ) \log \left (\frac {-26 e^8 x-156 e^4 x-234 x-1725}{621+414 e^4+69 e^8}\right )}{\left (234+156 e^4+26 e^8\right ) x+1725}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\frac {26 \left (3+e^4\right )^2 x}{26 \left (3+e^4\right )^2 x+1725}+\log \left (-\frac {26 x}{69}-\frac {25}{\left (3+e^4\right )^2}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{26} \left (26 x+\frac {1725}{\left (3+e^4\right )^2}\right ) \log \left (-\frac {26 x}{69}-\frac {25}{\left (3+e^4\right )^2}\right )-\frac {1725 \log \left (26 \left (3+e^4\right )^2 x+1725\right )}{26 \left (3+e^4\right )^2}\) |
Input:
Int[(234*x + 156*E^4*x + 26*E^8*x + (1725 + 234*x + 156*E^4*x + 26*E^8*x)* Log[(-1725 - 234*x - 156*E^4*x - 26*E^8*x)/(621 + 414*E^4 + 69*E^8)])/(172 5 + 234*x + 156*E^4*x + 26*E^8*x),x]
Output:
((1725/(3 + E^4)^2 + 26*x)*Log[-25/(3 + E^4)^2 - (26*x)/69])/26 - (1725*Lo g[1725 + 26*(3 + E^4)^2*x])/(26*(3 + E^4)^2)
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs. \(2(15)=30\).
Time = 0.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78
method | result | size |
risch | \(x \ln \left (\frac {-26 \,{\mathrm e}^{8} x -156 x \,{\mathrm e}^{4}-234 x -1725}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}\right )\) | \(32\) |
parallelrisch | \(\ln \left (-\frac {26 \,{\mathrm e}^{8} x +156 x \,{\mathrm e}^{4}+234 x +1725}{69 \left (9+{\mathrm e}^{8}+6 \,{\mathrm e}^{4}\right )}\right ) x\) | \(35\) |
norman | \(x \ln \left (\frac {-26 \,{\mathrm e}^{8} x -156 x \,{\mathrm e}^{4}-234 x -1725}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}\right )\) | \(36\) |
parts | \(\left (26 \,{\mathrm e}^{8}+156 \,{\mathrm e}^{4}+234\right ) \left (\frac {x}{26 \,{\mathrm e}^{8}+156 \,{\mathrm e}^{4}+234}-\frac {1725 \ln \left (26 \,{\mathrm e}^{8} x +156 x \,{\mathrm e}^{4}+234 x +1725\right )}{\left (26 \,{\mathrm e}^{8}+156 \,{\mathrm e}^{4}+234\right )^{2}}\right )+\frac {\left (69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621\right ) \left (\left (\frac {\left (-26 \,{\mathrm e}^{8}-156 \,{\mathrm e}^{4}-234\right ) x}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}-\frac {1725}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}\right ) \ln \left (\frac {\left (-26 \,{\mathrm e}^{8}-156 \,{\mathrm e}^{4}-234\right ) x}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}-\frac {1725}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}\right )-\frac {\left (-26 \,{\mathrm e}^{8}-156 \,{\mathrm e}^{4}-234\right ) x}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}+\frac {1725}{69 \,{\mathrm e}^{8}+414 \,{\mathrm e}^{4}+621}\right )}{-26 \,{\mathrm e}^{8}-156 \,{\mathrm e}^{4}-234}\) | \(224\) |
derivativedivides | \(\text {Expression too large to display}\) | \(731\) |
default | \(\text {Expression too large to display}\) | \(731\) |
Input:
int(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*ln((-26*x*exp(4)^2-156*x*exp( 4)-234*x-1725)/(69*exp(4)^2+414*exp(4)+621))+26*x*exp(4)^2+156*x*exp(4)+23 4*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x,method=_RETURNVERBOSE)
Output:
x*ln((-26*exp(8)*x-156*x*exp(4)-234*x-1725)/(69*exp(8)+414*exp(4)+621))
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+234 x+156 e^4 x+26 e^8 x} \, dx=x \log \left (-\frac {26 \, x e^{8} + 156 \, x e^{4} + 234 \, x + 1725}{69 \, {\left (e^{8} + 6 \, e^{4} + 9\right )}}\right ) \] Input:
integrate(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*log((-26*x*exp(4)^2-156 *x*exp(4)-234*x-1725)/(69*exp(4)^2+414*exp(4)+621))+26*x*exp(4)^2+156*x*ex p(4)+234*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x, algorithm="fricas")
Output:
x*log(-1/69*(26*x*e^8 + 156*x*e^4 + 234*x + 1725)/(e^8 + 6*e^4 + 9))
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+234 x+156 e^4 x+26 e^8 x} \, dx=x \log {\left (\frac {- 26 x e^{8} - 156 x e^{4} - 234 x - 1725}{621 + 414 e^{4} + 69 e^{8}} \right )} \] Input:
integrate(((26*x*exp(4)**2+156*x*exp(4)+234*x+1725)*ln((-26*x*exp(4)**2-15 6*x*exp(4)-234*x-1725)/(69*exp(4)**2+414*exp(4)+621))+26*x*exp(4)**2+156*x *exp(4)+234*x)/(26*x*exp(4)**2+156*x*exp(4)+234*x+1725),x)
Output:
x*log((-26*x*exp(8) - 156*x*exp(4) - 234*x - 1725)/(621 + 414*exp(4) + 69* exp(8)))
Leaf count of result is larger than twice the leaf count of optimal. 858 vs. \(2 (15) = 30\).
Time = 0.15 (sec) , antiderivative size = 858, normalized size of antiderivative = 47.67 \[ \int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+234 x+156 e^4 x+26 e^8 x} \, dx=\text {Too large to display} \] Input:
integrate(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*log((-26*x*exp(4)^2-156 *x*exp(4)-234*x-1725)/(69*exp(4)^2+414*exp(4)+621))+26*x*exp(4)^2+156*x*ex p(4)+234*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x, algorithm="maxima")
Output:
1/26*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^ 16 + 12*e^12 + 54*e^8 + 108*e^4 + 81))*e^8*log(-26/69*x*e^8/(e^8 + 6*e^4 + 9) - 52/23*x*e^4/(e^8 + 6*e^4 + 9) - 78/23*x/(e^8 + 6*e^4 + 9) - 25/(e^8 + 6*e^4 + 9)) + 3/13*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81))*e^4*log(-26/69*x*e^ 8/(e^8 + 6*e^4 + 9) - 52/23*x*e^4/(e^8 + 6*e^4 + 9) - 78/23*x/(e^8 + 6*e^4 + 9) - 25/(e^8 + 6*e^4 + 9)) + 1/26*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26 *x*(e^8 + 6*e^4 + 9) + 1725)/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81))*e^8 - 1/52*(52*x*(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)^ 2 - 3450*log(26*x*(e^8 + 6*e^4 + 9) + 1725))*(e^8/(e^8 + 6*e^4 + 9) + 6*e^ 4/(e^8 + 6*e^4 + 9) + 9/(e^8 + 6*e^4 + 9))*e^8/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81) + 3/13*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81))*e^4 - 3/26*(52*x*(e ^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)^2 - 3450*log(26* x*(e^8 + 6*e^4 + 9) + 1725))*(e^8/(e^8 + 6*e^4 + 9) + 6*e^4/(e^8 + 6*e^4 + 9) + 9/(e^8 + 6*e^4 + 9))*e^4/(e^16 + 12*e^12 + 54*e^8 + 108*e^4 + 81) + 9/26*(26*x/(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 + 9) + 1725)/(e^ 16 + 12*e^12 + 54*e^8 + 108*e^4 + 81))*log(-26/69*x*e^8/(e^8 + 6*e^4 + 9) - 52/23*x*e^4/(e^8 + 6*e^4 + 9) - 78/23*x/(e^8 + 6*e^4 + 9) - 25/(e^8 + 6* e^4 + 9)) - 9/52*(52*x*(e^8 + 6*e^4 + 9) - 1725*log(26*x*(e^8 + 6*e^4 +...
Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+234 x+156 e^4 x+26 e^8 x} \, dx=x \log \left (-\frac {26 \, x e^{8} + 156 \, x e^{4} + 234 \, x + 1725}{69 \, {\left (e^{8} + 6 \, e^{4} + 9\right )}}\right ) \] Input:
integrate(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*log((-26*x*exp(4)^2-156 *x*exp(4)-234*x-1725)/(69*exp(4)^2+414*exp(4)+621))+26*x*exp(4)^2+156*x*ex p(4)+234*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x, algorithm="giac")
Output:
x*log(-1/69*(26*x*e^8 + 156*x*e^4 + 234*x + 1725)/(e^8 + 6*e^4 + 9))
Time = 0.65 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+234 x+156 e^4 x+26 e^8 x} \, dx=x\,\left (\ln \left (-234\,x-156\,x\,{\mathrm {e}}^4-26\,x\,{\mathrm {e}}^8-1725\right )-\ln \left (414\,{\mathrm {e}}^4+69\,{\mathrm {e}}^8+621\right )\right ) \] Input:
int((234*x + 156*x*exp(4) + 26*x*exp(8) + log(-(234*x + 156*x*exp(4) + 26* x*exp(8) + 1725)/(414*exp(4) + 69*exp(8) + 621))*(234*x + 156*x*exp(4) + 2 6*x*exp(8) + 1725))/(234*x + 156*x*exp(4) + 26*x*exp(8) + 1725),x)
Output:
x*(log(- 234*x - 156*x*exp(4) - 26*x*exp(8) - 1725) - log(414*exp(4) + 69* exp(8) + 621))
Time = 0.16 (sec) , antiderivative size = 185, normalized size of antiderivative = 10.28 \[ \int \frac {234 x+156 e^4 x+26 e^8 x+\left (1725+234 x+156 e^4 x+26 e^8 x\right ) \log \left (\frac {-1725-234 x-156 e^4 x-26 e^8 x}{621+414 e^4+69 e^8}\right )}{1725+234 x+156 e^4 x+26 e^8 x} \, dx=\frac {-1725 \,\mathrm {log}\left (26 e^{8} x +156 e^{4} x +234 x +1725\right )+26 \,\mathrm {log}\left (\frac {-26 e^{8} x -156 e^{4} x -234 x -1725}{69 e^{8}+414 e^{4}+621}\right ) e^{8} x +156 \,\mathrm {log}\left (\frac {-26 e^{8} x -156 e^{4} x -234 x -1725}{69 e^{8}+414 e^{4}+621}\right ) e^{4} x +234 \,\mathrm {log}\left (\frac {-26 e^{8} x -156 e^{4} x -234 x -1725}{69 e^{8}+414 e^{4}+621}\right ) x +1725 \,\mathrm {log}\left (\frac {-26 e^{8} x -156 e^{4} x -234 x -1725}{69 e^{8}+414 e^{4}+621}\right )}{26 e^{8}+156 e^{4}+234} \] Input:
int(((26*x*exp(4)^2+156*x*exp(4)+234*x+1725)*log((-26*x*exp(4)^2-156*x*exp (4)-234*x-1725)/(69*exp(4)^2+414*exp(4)+621))+26*x*exp(4)^2+156*x*exp(4)+2 34*x)/(26*x*exp(4)^2+156*x*exp(4)+234*x+1725),x)
Output:
( - 1725*log(26*e**8*x + 156*e**4*x + 234*x + 1725) + 26*log(( - 26*e**8*x - 156*e**4*x - 234*x - 1725)/(69*e**8 + 414*e**4 + 621))*e**8*x + 156*log (( - 26*e**8*x - 156*e**4*x - 234*x - 1725)/(69*e**8 + 414*e**4 + 621))*e* *4*x + 234*log(( - 26*e**8*x - 156*e**4*x - 234*x - 1725)/(69*e**8 + 414*e **4 + 621))*x + 1725*log(( - 26*e**8*x - 156*e**4*x - 234*x - 1725)/(69*e* *8 + 414*e**4 + 621)))/(26*(e**8 + 6*e**4 + 9))