Integrand size = 60, antiderivative size = 24 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=x+e^{x+x \log (3)} \left (\frac {1}{2} \left (4+e^5\right )+x\right )^5 \]
Leaf count is larger than twice the leaf count of optimal. \(909\) vs. \(2(24)=48\).
Time = 7.33 (sec) , antiderivative size = 909, normalized size of antiderivative = 37.88 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx =\text {Too large to display} \]
Integrate[(4 + E^5 + 2*x + (E^(x + x*Log[3])*(4 + E^5 + 2*x)^5*(14 + E^5 + 2*x + (4 + E^5 + 2*x)*Log[3]))/32)/(4 + E^5 + 2*x),x]
(3^x*E^(25 + x)*(1 + Log[3])^6 + 32*3^(1 + x)*E^x*(5*x^4*(1 + Log[3])^5 + 4*x^3*(1 + Log[3])^4*(-5 + (1 + Log[3])*Log[9]) - 4*(40 + 20*Log[3]^4 - 2* Log[3]^2*(-80 + Log[9]) + Log[3]*(130 - 3*Log[81]) - 2*Log[3]^3*(-40 + Log [81]) + Log[9]*(1 + Log[81])) + 4*x*(1 + Log[3])*(40 + 20*Log[3]^4 - 2*Log [3]^2*(-80 + Log[9]) + Log[3]*(130 - 3*Log[81]) - 2*Log[3]^3*(-40 + Log[81 ]) + Log[9]*(1 + Log[81])) + 2*x^2*(1 + Log[3])^2*(-20 - Log[9] + Log[3]^2 *(-40 + Log[81]) + 2*Log[3]^3*Log[81] + Log[27]*Log[81] - 2*Log[3]*(25 + L og[81]))) + 8*3^x*E^(15 + x)*(1 + Log[3])^3*(20 + 50*Log[3] + 12*Log[3]^3 - Log[9]^2 + x^2*(5 + 13*Log[3] + 3*Log[3]^3 + Log[9] + Log[9]^2 + Log[3]^ 2*(11 + Log[9])) + Log[27]*Log[81] + 2*Log[3]^2*(26 + Log[81]) + x*(20 + 5 6*Log[3] + 12*Log[3]^3 + Log[9]^2 + Log[81] + Log[9]*Log[81] + 2*Log[3]^2* (24 + Log[81])) + Log[59049]) + 3^x*E^(20 + x)*(1 + Log[3])^4*(20 + 16*Log [3]^2 + Log[9] + Log[3]*(38 + Log[81]) + x*(1 + Log[3])*(10 + Log[59049])) + 16*(1 + Log[3])^3*((3*E)^x*x^5*(1 + Log[3])^2*(2 + Log[9]) + 16*(3*E)^x *x^2*(1 + Log[3])^2*(25 + Log[81]) + 16*(3*E)^x*(64 - Log[3]*(-28 + Log[9] ) + Log[9] + Log[3]^2*(14 + Log[81])) + 4*(3*E)^x*x^3*(50 - 8*Log[3]*(-10 + Log[9]) - Log[9] + Log[27]*Log[81] + 2*Log[3]^2*(20 + Log[81])) + 2*x*(1 + Log[3])*(1 + Log[3]^2 + Log[9] - 8*3^(1 + x)*E^x*Log[9] + 8*(3*E)^x*(-5 0 + Log[3]*Log[9] + Log[9]*Log[81] + Log[6561])) + (3*E)^x*x^4*(1 + Log[3] )*(-10 + Log[3]*(8*Log[9] + Log[81]) + Log[59049])) + 16*3^x*E^(5 + x)*...
Time = 8.73 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {7292, 7293, 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.
\(\displaystyle \int \frac {2 x+\frac {1}{32} \left (2 x+e^5+4\right )^5 e^{x+x \log (3)} \left (2 x+\left (2 x+e^5+4\right ) \log (3)+e^5+14\right )+e^5+4}{2 x+e^5+4} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x+\frac {1}{32} \left (2 x+e^5+4\right )^5 e^{x+x \log (3)} \left (2 x+\left (2 x+e^5+4\right ) \log (3)+e^5+14\right )+4 \left (1+\frac {e^5}{4}\right )}{2 x+e^5+4}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{32} (3 e)^x \left (2 x+e^5+4\right )^4 \left (x (2+\log (9))+14+\log (81)+e^5 (1+\log (3))\right )+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{32} (3 e)^x \left (2 x+e^5+4\right )^5+x\) |
Int[(4 + E^5 + 2*x + (E^(x + x*Log[3])*(4 + E^5 + 2*x)^5*(14 + E^5 + 2*x + (4 + E^5 + 2*x)*Log[3]))/32)/(4 + E^5 + 2*x),x]
3.8.97.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs. \(2(19)=38\).
Time = 0.73 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.75
method | result | size |
risch | \(x +\left (32+\frac {{\mathrm e}^{25}}{32}+\frac {5 x \,{\mathrm e}^{20}}{16}+\frac {5 x^{2} {\mathrm e}^{15}}{4}+\frac {5 x^{3} {\mathrm e}^{10}}{2}+\frac {5 x^{4} {\mathrm e}^{5}}{2}+\frac {5 \,{\mathrm e}^{20}}{8}+20 \,{\mathrm e}^{10}+30 x \,{\mathrm e}^{10}+5 x \,{\mathrm e}^{15}+15 x^{2} {\mathrm e}^{10}+20 x^{3} {\mathrm e}^{5}+5 \,{\mathrm e}^{15}+80 x +40 \,{\mathrm e}^{5}+60 x^{2} {\mathrm e}^{5}+80 x \,{\mathrm e}^{5}+x^{5}+10 x^{4}+40 x^{3}+80 x^{2}\right ) 3^{x} {\mathrm e}^{x}\) | \(114\) |
norman | \(x +\left (32+40 \,{\mathrm e}^{5}+\frac {{\mathrm e}^{25}}{32}+\frac {5 \,{\mathrm e}^{20}}{8}+5 \,{\mathrm e}^{15}+20 \,{\mathrm e}^{10}\right ) {\mathrm e}^{x \ln \left (3\right )+x}+{\mathrm e}^{x \ln \left (3\right )+x} x^{5}+\left (\frac {5 \,{\mathrm e}^{5}}{2}+10\right ) x^{4} {\mathrm e}^{x \ln \left (3\right )+x}+\left (\frac {5 \,{\mathrm e}^{10}}{2}+20 \,{\mathrm e}^{5}+40\right ) x^{3} {\mathrm e}^{x \ln \left (3\right )+x}+\left (\frac {5 \,{\mathrm e}^{15}}{4}+15 \,{\mathrm e}^{10}+60 \,{\mathrm e}^{5}+80\right ) x^{2} {\mathrm e}^{x \ln \left (3\right )+x}+\left (\frac {5 \,{\mathrm e}^{20}}{16}+5 \,{\mathrm e}^{15}+30 \,{\mathrm e}^{10}+80 \,{\mathrm e}^{5}+80\right ) x \,{\mathrm e}^{x \ln \left (3\right )+x}\) | \(154\) |
parallelrisch | \(x +\frac {{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{25}}{32}+\frac {5 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{20}}{8}+5 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{15}+20 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{10}+40 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} {\mathrm e}^{5}+32 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x}+\frac {5 \,{\mathrm e}^{15} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{2}}{4}+15 \,{\mathrm e}^{10} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{2}+60 \,{\mathrm e}^{5} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{2}+80 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{2}+\frac {5 \,{\mathrm e}^{10} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{3}}{2}+20 \,{\mathrm e}^{5} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{3}+40 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{3}+\frac {5 \,{\mathrm e}^{5} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{4}}{2}+10 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{4}+{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x^{5}+\frac {5 \,{\mathrm e}^{20} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x}{16}+5 \,{\mathrm e}^{15} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x +30 \,{\mathrm e}^{10} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x +80 \,{\mathrm e}^{5} {\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x +80 \,{\mathrm e}^{\left (\ln \left (3\right )+1\right ) x} x\) | \(276\) |
parts | \(\text {Expression too large to display}\) | \(1910\) |
derivativedivides | \(\text {Expression too large to display}\) | \(34958\) |
default | \(\text {Expression too large to display}\) | \(34958\) |
int((((exp(5)+2*x+4)*ln(3)+exp(5)+2*x+14)*exp(x*ln(3)+x)*(1/2*exp(5)+x+2)^ 5+exp(5)+2*x+4)/(exp(5)+2*x+4),x,method=_RETURNVERBOSE)
x+(32+1/32*exp(25)+5/16*x*exp(20)+5/4*x^2*exp(15)+5/2*x^3*exp(10)+5/2*x^4* exp(5)+5/8*exp(20)+20*exp(10)+30*x*exp(10)+5*x*exp(15)+15*x^2*exp(10)+20*x ^3*exp(5)+5*exp(15)+80*x+40*exp(5)+60*x^2*exp(5)+80*x*exp(5)+x^5+10*x^4+40 *x^3+80*x^2)*3^x*exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.00 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=\frac {1}{32} \, {\left (32 \, x^{5} + 320 \, x^{4} + 1280 \, x^{3} + 2560 \, x^{2} + 10 \, {\left (x + 2\right )} e^{20} + 40 \, {\left (x^{2} + 4 \, x + 4\right )} e^{15} + 80 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} e^{10} + 80 \, {\left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 32 \, x + 16\right )} e^{5} + 2560 \, x + e^{25} + 1024\right )} e^{\left (x \log \left (3\right ) + x\right )} + x \]
integrate((((exp(5)+2*x+4)*log(3)+exp(5)+2*x+14)*exp(x*log(3)+x)*(1/2*exp( 5)+x+2)^5+exp(5)+2*x+4)/(exp(5)+2*x+4),x, algorithm=\
1/32*(32*x^5 + 320*x^4 + 1280*x^3 + 2560*x^2 + 10*(x + 2)*e^20 + 40*(x^2 + 4*x + 4)*e^15 + 80*(x^3 + 6*x^2 + 12*x + 8)*e^10 + 80*(x^4 + 8*x^3 + 24*x ^2 + 32*x + 16)*e^5 + 2560*x + e^25 + 1024)*e^(x*log(3) + x) + x
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (19) = 38\).
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 5.75 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=x + \frac {\left (32 x^{5} + 320 x^{4} + 80 x^{4} e^{5} + 1280 x^{3} + 640 x^{3} e^{5} + 80 x^{3} e^{10} + 2560 x^{2} + 1920 x^{2} e^{5} + 480 x^{2} e^{10} + 40 x^{2} e^{15} + 2560 x + 2560 x e^{5} + 960 x e^{10} + 160 x e^{15} + 10 x e^{20} + 1024 + 1280 e^{5} + 640 e^{10} + 160 e^{15} + 20 e^{20} + e^{25}\right ) e^{x + x \log {\left (3 \right )}}}{32} \]
integrate((((exp(5)+2*x+4)*ln(3)+exp(5)+2*x+14)*exp(x*ln(3)+x)*(1/2*exp(5) +x+2)**5+exp(5)+2*x+4)/(exp(5)+2*x+4),x)
x + (32*x**5 + 320*x**4 + 80*x**4*exp(5) + 1280*x**3 + 640*x**3*exp(5) + 8 0*x**3*exp(10) + 2560*x**2 + 1920*x**2*exp(5) + 480*x**2*exp(10) + 40*x**2 *exp(15) + 2560*x + 2560*x*exp(5) + 960*x*exp(10) + 160*x*exp(15) + 10*x*e xp(20) + 1024 + 1280*exp(5) + 640*exp(10) + 160*exp(15) + 20*exp(20) + exp (25))*exp(x + x*log(3))/32
\[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=\int { \frac {{\left ({\left (2 \, x + e^{5} + 4\right )} \log \left (3\right ) + 2 \, x + e^{5} + 14\right )} {\left (2 \, x + e^{5} + 4\right )}^{5} e^{\left (x \log \left (3\right ) + x\right )} + 64 \, x + 32 \, e^{5} + 128}{32 \, {\left (2 \, x + e^{5} + 4\right )}} \,d x } \]
integrate((((exp(5)+2*x+4)*log(3)+exp(5)+2*x+14)*exp(x*log(3)+x)*(1/2*exp( 5)+x+2)^5+exp(5)+2*x+4)/(exp(5)+2*x+4),x, algorithm=\
-64*e^(-1/2*(e^5 + 4)*(log(3) + 1))*exp_integral_e(1, -1/2*(2*x + e^5 + 4) *(log(3) + 1))*log(3) - 224*e^(-1/2*(e^5 + 4)*(log(3) + 1))*exp_integral_e (1, -1/2*(2*x + e^5 + 4)*(log(3) + 1)) - 1/2*(e^5 + 4)*log(2*x + e^5 + 4) + 1/2*e^5*log(2*x + e^5 + 4) + x + 1/32*integrate((64*x^6*(log(3) + 1) + 6 4*(3*(log(3) + 1)*e^5 + 12*log(3) + 17)*x^5 + 80*(3*(log(3) + 1)*e^10 + 2* (12*log(3) + 17)*e^5 + 48*log(3) + 88)*x^4 + 160*((log(3) + 1)*e^15 + (12* log(3) + 17)*e^10 + 8*(6*log(3) + 11)*e^5 + 64*log(3) + 144)*x^3 + 20*(3*( log(3) + 1)*e^20 + 4*(12*log(3) + 17)*e^15 + 48*(6*log(3) + 11)*e^10 + 192 *(4*log(3) + 9)*e^5 + 768*log(3) + 2048)*x^2 + 4*(3*(log(3) + 1)*e^25 + 5* (12*log(3) + 17)*e^20 + 80*(6*log(3) + 11)*e^15 + 480*(4*log(3) + 9)*e^10 + 1280*(3*log(3) + 8)*e^5 + 3072*log(3) + 9472)*x + (log(3) + 1)*e^30 + 2* (12*log(3) + 17)*e^25 + 40*(6*log(3) + 11)*e^20 + 320*(4*log(3) + 9)*e^15 + 1280*(3*log(3) + 8)*e^10 + 512*(12*log(3) + 37)*e^5)*e^(x*log(3) + x)/(2 *x + e^5 + 4), x) + 2*log(2*x + e^5 + 4)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (20) = 40\).
Time = 0.37 (sec) , antiderivative size = 240, normalized size of antiderivative = 10.00 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=x^{5} e^{\left (x \log \left (3\right ) + x\right )} + \frac {5}{2} \, x^{4} e^{\left (x \log \left (3\right ) + x + 5\right )} + 10 \, x^{4} e^{\left (x \log \left (3\right ) + x\right )} + \frac {5}{2} \, x^{3} e^{\left (x \log \left (3\right ) + x + 10\right )} + 20 \, x^{3} e^{\left (x \log \left (3\right ) + x + 5\right )} + 40 \, x^{3} e^{\left (x \log \left (3\right ) + x\right )} + \frac {5}{4} \, x^{2} e^{\left (x \log \left (3\right ) + x + 15\right )} + 15 \, x^{2} e^{\left (x \log \left (3\right ) + x + 10\right )} + 60 \, x^{2} e^{\left (x \log \left (3\right ) + x + 5\right )} + 80 \, x^{2} e^{\left (x \log \left (3\right ) + x\right )} + \frac {5}{16} \, x e^{\left (x \log \left (3\right ) + x + 20\right )} + 5 \, x e^{\left (x \log \left (3\right ) + x + 15\right )} + 30 \, x e^{\left (x \log \left (3\right ) + x + 10\right )} + 80 \, x e^{\left (x \log \left (3\right ) + x + 5\right )} + 80 \, x e^{\left (x \log \left (3\right ) + x\right )} + x + \frac {1}{32} \, e^{\left (x \log \left (3\right ) + x + 25\right )} + \frac {5}{8} \, e^{\left (x \log \left (3\right ) + x + 20\right )} + 5 \, e^{\left (x \log \left (3\right ) + x + 15\right )} + 20 \, e^{\left (x \log \left (3\right ) + x + 10\right )} + 40 \, e^{\left (x \log \left (3\right ) + x + 5\right )} + 32 \, e^{\left (x \log \left (3\right ) + x\right )} \]
integrate((((exp(5)+2*x+4)*log(3)+exp(5)+2*x+14)*exp(x*log(3)+x)*(1/2*exp( 5)+x+2)^5+exp(5)+2*x+4)/(exp(5)+2*x+4),x, algorithm=\
x^5*e^(x*log(3) + x) + 5/2*x^4*e^(x*log(3) + x + 5) + 10*x^4*e^(x*log(3) + x) + 5/2*x^3*e^(x*log(3) + x + 10) + 20*x^3*e^(x*log(3) + x + 5) + 40*x^3 *e^(x*log(3) + x) + 5/4*x^2*e^(x*log(3) + x + 15) + 15*x^2*e^(x*log(3) + x + 10) + 60*x^2*e^(x*log(3) + x + 5) + 80*x^2*e^(x*log(3) + x) + 5/16*x*e^ (x*log(3) + x + 20) + 5*x*e^(x*log(3) + x + 15) + 30*x*e^(x*log(3) + x + 1 0) + 80*x*e^(x*log(3) + x + 5) + 80*x*e^(x*log(3) + x) + x + 1/32*e^(x*log (3) + x + 25) + 5/8*e^(x*log(3) + x + 20) + 5*e^(x*log(3) + x + 15) + 20*e ^(x*log(3) + x + 10) + 40*e^(x*log(3) + x + 5) + 32*e^(x*log(3) + x)
Time = 12.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.17 \[ \int \frac {4+e^5+2 x+\frac {1}{32} e^{x+x \log (3)} \left (4+e^5+2 x\right )^5 \left (14+e^5+2 x+\left (4+e^5+2 x\right ) \log (3)\right )}{4+e^5+2 x} \, dx=x+3^x\,x^5\,{\mathrm {e}}^x+\frac {3^x\,{\mathrm {e}}^x\,\left (1280\,{\mathrm {e}}^5+640\,{\mathrm {e}}^{10}+160\,{\mathrm {e}}^{15}+20\,{\mathrm {e}}^{20}+{\mathrm {e}}^{25}+1024\right )}{32}+\frac {3^x\,x^4\,{\mathrm {e}}^x\,\left (80\,{\mathrm {e}}^5+320\right )}{32}+\frac {5\,3^x\,x^2\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^3}{4}+\frac {5\,3^x\,x^3\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^2}{2}+\frac {5\,3^x\,x\,{\mathrm {e}}^x\,{\left ({\mathrm {e}}^5+4\right )}^4}{16} \]
int((2*x + exp(5) + exp(x + x*log(3))*(x + exp(5)/2 + 2)^5*(2*x + exp(5) + log(3)*(2*x + exp(5) + 4) + 14) + 4)/(2*x + exp(5) + 4),x)