Integrand size = 185, antiderivative size = 36 \[ \int \frac {-192-672 x-912 x^2-600 x^3-192 x^4-24 x^5+e^x \left (-12-30 x-18 x^2\right )+\left (96+144 x+72 x^2+12 x^3+e^x (6+3 x)\right ) \log \left (\frac {e^{-2 x} \left (6400+25 e^{2 x}+12800 x+9600 x^2+3200 x^3+400 x^4+e^x \left (800+800 x+200 x^2\right )\right )}{16 x^2+32 x^3+24 x^4+8 x^5+x^6}\right )}{160+560 x+760 x^2+500 x^3+160 x^4+20 x^5+e^x \left (10+25 x+20 x^2+5 x^3\right )} \, dx=\frac {3 \log \left (\frac {25 \left (4 e^{-x}+\frac {4}{(4+2 x)^2}\right )^2}{x^2}\right )}{5+\frac {5}{x}} \]
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(36)=72\).
Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.58 \[ \int \frac {-192-672 x-912 x^2-600 x^3-192 x^4-24 x^5+e^x \left (-12-30 x-18 x^2\right )+\left (96+144 x+72 x^2+12 x^3+e^x (6+3 x)\right ) \log \left (\frac {e^{-2 x} \left (6400+25 e^{2 x}+12800 x+9600 x^2+3200 x^3+400 x^4+e^x \left (800+800 x+200 x^2\right )\right )}{16 x^2+32 x^3+24 x^4+8 x^5+x^6}\right )}{160+560 x+760 x^2+500 x^3+160 x^4+20 x^5+e^x \left (10+25 x+20 x^2+5 x^3\right )} \, dx=-\frac {3 \left (2+4 x+2 x^2+2 (1+x) \log (x)+4 (1+x) \log (2+x)-2 \log \left (e^x+4 (2+x)^2\right )-2 x \log \left (e^x+4 (2+x)^2\right )+\log \left (\frac {25 e^{-2 x} \left (e^x+4 (2+x)^2\right )^2}{x^2 (2+x)^4}\right )\right )}{5 (1+x)} \]
Integrate[(-192 - 672*x - 912*x^2 - 600*x^3 - 192*x^4 - 24*x^5 + E^x*(-12 - 30*x - 18*x^2) + (96 + 144*x + 72*x^2 + 12*x^3 + E^x*(6 + 3*x))*Log[(640 0 + 25*E^(2*x) + 12800*x + 9600*x^2 + 3200*x^3 + 400*x^4 + E^x*(800 + 800* x + 200*x^2))/(E^(2*x)*(16*x^2 + 32*x^3 + 24*x^4 + 8*x^5 + x^6))])/(160 + 560*x + 760*x^2 + 500*x^3 + 160*x^4 + 20*x^5 + E^x*(10 + 25*x + 20*x^2 + 5 *x^3)),x]
(-3*(2 + 4*x + 2*x^2 + 2*(1 + x)*Log[x] + 4*(1 + x)*Log[2 + x] - 2*Log[E^x + 4*(2 + x)^2] - 2*x*Log[E^x + 4*(2 + x)^2] + Log[(25*(E^x + 4*(2 + x)^2) ^2)/(E^(2*x)*x^2*(2 + x)^4)]))/(5*(1 + 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.
| \(\displaystyle \int \frac {-24 x^5-192 x^4-600 x^3-912 x^2+e^x \left (-18 x^2-30 x-12\right )+\left (12 x^3+72 x^2+144 x+e^x (3 x+6)+96\right ) \log \left (\frac {e^{-2 x} \left (400 x^4+3200 x^3+9600 x^2+e^x \left (200 x^2+800 x+800\right )+12800 x+25 e^{2 x}+6400\right )}{x^6+8 x^5+24 x^4+32 x^3+16 x^2}\right )-672 x-192}{20 x^5+160 x^4+500 x^3+760 x^2+e^x \left (5 x^3+20 x^2+25 x+10\right )+560 x+160} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-24 x^5-192 x^4-600 x^3-912 x^2+e^x \left (-18 x^2-30 x-12\right )+\left (12 x^3+72 x^2+144 x+e^x (3 x+6)+96\right ) \log \left (\frac {e^{-2 x} \left (400 x^4+3200 x^3+9600 x^2+e^x \left (200 x^2+800 x+800\right )+12800 x+25 e^{2 x}+6400\right )}{x^6+8 x^5+24 x^4+32 x^3+16 x^2}\right )-672 x-192}{5 (x+1)^2 (x+2) \left (4 x^2+16 x+e^x+16\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int -\frac {3 \left (8 x^5+64 x^4+200 x^3+304 x^2+224 x+2 e^x \left (3 x^2+5 x+2\right )-\left (4 x^3+24 x^2+48 x+e^x (x+2)+32\right ) \log \left (\frac {25 e^{-2 x} \left (16 x^4+128 x^3+384 x^2+512 x+e^{2 x}+8 e^x \left (x^2+4 x+4\right )+256\right )}{x^6+8 x^5+24 x^4+32 x^3+16 x^2}\right )+64\right )}{(x+1)^2 (x+2) \left (4 x^2+16 x+e^x+16\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{5} \int \frac {8 x^5+64 x^4+200 x^3+304 x^2+224 x+2 e^x \left (3 x^2+5 x+2\right )-\left (4 x^3+24 x^2+48 x+e^x (x+2)+32\right ) \log \left (\frac {25 e^{-2 x} \left (16 x^4+128 x^3+384 x^2+512 x+e^{2 x}+8 e^x \left (x^2+4 x+4\right )+256\right )}{x^6+8 x^5+24 x^4+32 x^3+16 x^2}\right )+64}{(x+1)^2 (x+2) \left (4 x^2+16 x+e^x+16\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{5} \int \left (\frac {8 (x+2) x^2}{(x+1) \left (4 x^2+16 x+e^x+16\right )}+\frac {6 x^2-\log \left (\frac {25 e^{-2 x} \left (4 (x+2)^2+e^x\right )^2}{x^2 (x+2)^4}\right ) x+10 x-2 \log \left (\frac {25 e^{-2 x} \left (4 (x+2)^2+e^x\right )^2}{x^2 (x+2)^4}\right )+4}{(x+1)^2 (x+2)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{5} \left (16 \int \frac {x}{4 x^2+16 x+e^x+16}dx+8 \int \frac {x^2}{4 x^2+16 x+e^x+16}dx+\frac {\log \left (\frac {25 e^{-2 x} \left (4 (x+2)^2+e^x\right )^2}{x^2 (x+2)^4}\right )}{x+1}+2 \log (x)+4 \log (x+2)\right )\) |
Int[(-192 - 672*x - 912*x^2 - 600*x^3 - 192*x^4 - 24*x^5 + E^x*(-12 - 30*x - 18*x^2) + (96 + 144*x + 72*x^2 + 12*x^3 + E^x*(6 + 3*x))*Log[(6400 + 25 *E^(2*x) + 12800*x + 9600*x^2 + 3200*x^3 + 400*x^4 + E^x*(800 + 800*x + 20 0*x^2))/(E^(2*x)*(16*x^2 + 32*x^3 + 24*x^4 + 8*x^5 + x^6))])/(160 + 560*x + 760*x^2 + 500*x^3 + 160*x^4 + 20*x^5 + E^x*(10 + 25*x + 20*x^2 + 5*x^3)) ,x]
3.23.82.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(35)=70\).
Time = 1.64 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.14
| method | result | size |
| parallelrisch | \(\frac {3 \ln \left (\frac {\left (25 \,{\mathrm e}^{2 x}+\left (200 x^{2}+800 x +800\right ) {\mathrm e}^{x}+400 x^{4}+3200 x^{3}+9600 x^{2}+12800 x +6400\right ) {\mathrm e}^{-2 x}}{x^{2} \left (x^{4}+8 x^{3}+24 x^{2}+32 x +16\right )}\right ) x}{5 \left (1+x \right )}\) | \(77\) |
| risch | \(\text {Expression too large to display}\) | \(1110\) |
int(((exp(x)*(6+3*x)+12*x^3+72*x^2+144*x+96)*ln((25*exp(x)^2+(200*x^2+800* x+800)*exp(x)+400*x^4+3200*x^3+9600*x^2+12800*x+6400)/(x^6+8*x^5+24*x^4+32 *x^3+16*x^2)/exp(x)^2)+(-18*x^2-30*x-12)*exp(x)-24*x^5-192*x^4-600*x^3-912 *x^2-672*x-192)/((5*x^3+20*x^2+25*x+10)*exp(x)+20*x^5+160*x^4+500*x^3+760* x^2+560*x+160),x,method=_RETURNVERBOSE)
3/5*ln((25*exp(x)^2+(200*x^2+800*x+800)*exp(x)+400*x^4+3200*x^3+9600*x^2+1 2800*x+6400)/x^2/(x^4+8*x^3+24*x^2+32*x+16)/exp(x)^2)*x/(1+x)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.14 \[ \int \frac {-192-672 x-912 x^2-600 x^3-192 x^4-24 x^5+e^x \left (-12-30 x-18 x^2\right )+\left (96+144 x+72 x^2+12 x^3+e^x (6+3 x)\right ) \log \left (\frac {e^{-2 x} \left (6400+25 e^{2 x}+12800 x+9600 x^2+3200 x^3+400 x^4+e^x \left (800+800 x+200 x^2\right )\right )}{16 x^2+32 x^3+24 x^4+8 x^5+x^6}\right )}{160+560 x+760 x^2+500 x^3+160 x^4+20 x^5+e^x \left (10+25 x+20 x^2+5 x^3\right )} \, dx=\frac {3 \, x \log \left (\frac {25 \, {\left (16 \, x^{4} + 128 \, x^{3} + 384 \, x^{2} + 8 \, {\left (x^{2} + 4 \, x + 4\right )} e^{x} + 512 \, x + e^{\left (2 \, x\right )} + 256\right )} e^{\left (-2 \, x\right )}}{x^{6} + 8 \, x^{5} + 24 \, x^{4} + 32 \, x^{3} + 16 \, x^{2}}\right )}{5 \, {\left (x + 1\right )}} \]
integrate(((exp(x)*(6+3*x)+12*x^3+72*x^2+144*x+96)*log((25*exp(x)^2+(200*x ^2+800*x+800)*exp(x)+400*x^4+3200*x^3+9600*x^2+12800*x+6400)/(x^6+8*x^5+24 *x^4+32*x^3+16*x^2)/exp(x)^2)+(-18*x^2-30*x-12)*exp(x)-24*x^5-192*x^4-600* x^3-912*x^2-672*x-192)/((5*x^3+20*x^2+25*x+10)*exp(x)+20*x^5+160*x^4+500*x ^3+760*x^2+560*x+160),x, algorithm=\
3/5*x*log(25*(16*x^4 + 128*x^3 + 384*x^2 + 8*(x^2 + 4*x + 4)*e^x + 512*x + e^(2*x) + 256)*e^(-2*x)/(x^6 + 8*x^5 + 24*x^4 + 32*x^3 + 16*x^2))/(x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (27) = 54\).
Time = 0.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.22 \[ \int \frac {-192-672 x-912 x^2-600 x^3-192 x^4-24 x^5+e^x \left (-12-30 x-18 x^2\right )+\left (96+144 x+72 x^2+12 x^3+e^x (6+3 x)\right ) \log \left (\frac {e^{-2 x} \left (6400+25 e^{2 x}+12800 x+9600 x^2+3200 x^3+400 x^4+e^x \left (800+800 x+200 x^2\right )\right )}{16 x^2+32 x^3+24 x^4+8 x^5+x^6}\right )}{160+560 x+760 x^2+500 x^3+160 x^4+20 x^5+e^x \left (10+25 x+20 x^2+5 x^3\right )} \, dx=- \frac {6 x}{5} - \frac {6 \log {\left (x \right )}}{5} - \frac {12 \log {\left (x + 2 \right )}}{5} + \frac {6 \log {\left (4 x^{2} + 16 x + e^{x} + 16 \right )}}{5} - \frac {3 \log {\left (\frac {\left (400 x^{4} + 3200 x^{3} + 9600 x^{2} + 12800 x + \left (200 x^{2} + 800 x + 800\right ) e^{x} + 25 e^{2 x} + 6400\right ) e^{- 2 x}}{x^{6} + 8 x^{5} + 24 x^{4} + 32 x^{3} + 16 x^{2}} \right )}}{5 x + 5} \]
integrate(((exp(x)*(6+3*x)+12*x**3+72*x**2+144*x+96)*ln((25*exp(x)**2+(200 *x**2+800*x+800)*exp(x)+400*x**4+3200*x**3+9600*x**2+12800*x+6400)/(x**6+8 *x**5+24*x**4+32*x**3+16*x**2)/exp(x)**2)+(-18*x**2-30*x-12)*exp(x)-24*x** 5-192*x**4-600*x**3-912*x**2-672*x-192)/((5*x**3+20*x**2+25*x+10)*exp(x)+2 0*x**5+160*x**4+500*x**3+760*x**2+560*x+160),x)
-6*x/5 - 6*log(x)/5 - 12*log(x + 2)/5 + 6*log(4*x**2 + 16*x + exp(x) + 16) /5 - 3*log((400*x**4 + 3200*x**3 + 9600*x**2 + 12800*x + (200*x**2 + 800*x + 800)*exp(x) + 25*exp(2*x) + 6400)*exp(-2*x)/(x**6 + 8*x**5 + 24*x**4 + 32*x**3 + 16*x**2))/(5*x + 5)
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {-192-672 x-912 x^2-600 x^3-192 x^4-24 x^5+e^x \left (-12-30 x-18 x^2\right )+\left (96+144 x+72 x^2+12 x^3+e^x (6+3 x)\right ) \log \left (\frac {e^{-2 x} \left (6400+25 e^{2 x}+12800 x+9600 x^2+3200 x^3+400 x^4+e^x \left (800+800 x+200 x^2\right )\right )}{16 x^2+32 x^3+24 x^4+8 x^5+x^6}\right )}{160+560 x+760 x^2+500 x^3+160 x^4+20 x^5+e^x \left (10+25 x+20 x^2+5 x^3\right )} \, dx=-\frac {6 \, {\left (x^{2} - x \log \left (4 \, x^{2} + 16 \, x + e^{x} + 16\right ) + 2 \, x \log \left (x + 2\right ) + x \log \left (x\right ) + x + \log \left (5\right ) + 1\right )}}{5 \, {\left (x + 1\right )}} \]
integrate(((exp(x)*(6+3*x)+12*x^3+72*x^2+144*x+96)*log((25*exp(x)^2+(200*x ^2+800*x+800)*exp(x)+400*x^4+3200*x^3+9600*x^2+12800*x+6400)/(x^6+8*x^5+24 *x^4+32*x^3+16*x^2)/exp(x)^2)+(-18*x^2-30*x-12)*exp(x)-24*x^5-192*x^4-600* x^3-912*x^2-672*x-192)/((5*x^3+20*x^2+25*x+10)*exp(x)+20*x^5+160*x^4+500*x ^3+760*x^2+560*x+160),x, algorithm=\
-6/5*(x^2 - x*log(4*x^2 + 16*x + e^x + 16) + 2*x*log(x + 2) + x*log(x) + x + log(5) + 1)/(x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (29) = 58\).
Time = 0.47 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.53 \[ \int \frac {-192-672 x-912 x^2-600 x^3-192 x^4-24 x^5+e^x \left (-12-30 x-18 x^2\right )+\left (96+144 x+72 x^2+12 x^3+e^x (6+3 x)\right ) \log \left (\frac {e^{-2 x} \left (6400+25 e^{2 x}+12800 x+9600 x^2+3200 x^3+400 x^4+e^x \left (800+800 x+200 x^2\right )\right )}{16 x^2+32 x^3+24 x^4+8 x^5+x^6}\right )}{160+560 x+760 x^2+500 x^3+160 x^4+20 x^5+e^x \left (10+25 x+20 x^2+5 x^3\right )} \, dx=-\frac {3 \, {\left (2 \, x^{2} - 2 \, x \log \left (-4 \, x^{2} - 16 \, x - e^{x} - 16\right ) + 4 \, x \log \left (x + 2\right ) + 2 \, x \log \left (x\right ) + 2 \, x - 2 \, \log \left (-4 \, x^{2} - 16 \, x - e^{x} - 16\right ) + 4 \, \log \left (x + 2\right ) + 2 \, \log \left (x\right ) + \log \left (\frac {25 \, {\left (16 \, x^{4} + 128 \, x^{3} + 8 \, x^{2} e^{x} + 384 \, x^{2} + 32 \, x e^{x} + 512 \, x + e^{\left (2 \, x\right )} + 32 \, e^{x} + 256\right )}}{x^{6} e^{\left (2 \, x\right )} + 8 \, x^{5} e^{\left (2 \, x\right )} + 24 \, x^{4} e^{\left (2 \, x\right )} + 32 \, x^{3} e^{\left (2 \, x\right )} + 16 \, x^{2} e^{\left (2 \, x\right )}}\right )\right )}}{5 \, {\left (x + 1\right )}} \]
integrate(((exp(x)*(6+3*x)+12*x^3+72*x^2+144*x+96)*log((25*exp(x)^2+(200*x ^2+800*x+800)*exp(x)+400*x^4+3200*x^3+9600*x^2+12800*x+6400)/(x^6+8*x^5+24 *x^4+32*x^3+16*x^2)/exp(x)^2)+(-18*x^2-30*x-12)*exp(x)-24*x^5-192*x^4-600* x^3-912*x^2-672*x-192)/((5*x^3+20*x^2+25*x+10)*exp(x)+20*x^5+160*x^4+500*x ^3+760*x^2+560*x+160),x, algorithm=\
-3/5*(2*x^2 - 2*x*log(-4*x^2 - 16*x - e^x - 16) + 4*x*log(x + 2) + 2*x*log (x) + 2*x - 2*log(-4*x^2 - 16*x - e^x - 16) + 4*log(x + 2) + 2*log(x) + lo g(25*(16*x^4 + 128*x^3 + 8*x^2*e^x + 384*x^2 + 32*x*e^x + 512*x + e^(2*x) + 32*e^x + 256)/(x^6*e^(2*x) + 8*x^5*e^(2*x) + 24*x^4*e^(2*x) + 32*x^3*e^( 2*x) + 16*x^2*e^(2*x))))/(x + 1)
Timed out. \[ \int \frac {-192-672 x-912 x^2-600 x^3-192 x^4-24 x^5+e^x \left (-12-30 x-18 x^2\right )+\left (96+144 x+72 x^2+12 x^3+e^x (6+3 x)\right ) \log \left (\frac {e^{-2 x} \left (6400+25 e^{2 x}+12800 x+9600 x^2+3200 x^3+400 x^4+e^x \left (800+800 x+200 x^2\right )\right )}{16 x^2+32 x^3+24 x^4+8 x^5+x^6}\right )}{160+560 x+760 x^2+500 x^3+160 x^4+20 x^5+e^x \left (10+25 x+20 x^2+5 x^3\right )} \, dx=\int -\frac {672\,x-\ln \left (\frac {{\mathrm {e}}^{-2\,x}\,\left (12800\,x+25\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (200\,x^2+800\,x+800\right )+9600\,x^2+3200\,x^3+400\,x^4+6400\right )}{x^6+8\,x^5+24\,x^4+32\,x^3+16\,x^2}\right )\,\left (144\,x+{\mathrm {e}}^x\,\left (3\,x+6\right )+72\,x^2+12\,x^3+96\right )+{\mathrm {e}}^x\,\left (18\,x^2+30\,x+12\right )+912\,x^2+600\,x^3+192\,x^4+24\,x^5+192}{560\,x+760\,x^2+500\,x^3+160\,x^4+20\,x^5+{\mathrm {e}}^x\,\left (5\,x^3+20\,x^2+25\,x+10\right )+160} \,d x \]
int(-(672*x - log((exp(-2*x)*(12800*x + 25*exp(2*x) + exp(x)*(800*x + 200* x^2 + 800) + 9600*x^2 + 3200*x^3 + 400*x^4 + 6400))/(16*x^2 + 32*x^3 + 24* x^4 + 8*x^5 + x^6))*(144*x + exp(x)*(3*x + 6) + 72*x^2 + 12*x^3 + 96) + ex p(x)*(30*x + 18*x^2 + 12) + 912*x^2 + 600*x^3 + 192*x^4 + 24*x^5 + 192)/(5 60*x + 760*x^2 + 500*x^3 + 160*x^4 + 20*x^5 + exp(x)*(25*x + 20*x^2 + 5*x^ 3 + 10) + 160),x)
int(-(672*x - log((exp(-2*x)*(12800*x + 25*exp(2*x) + exp(x)*(800*x + 200* x^2 + 800) + 9600*x^2 + 3200*x^3 + 400*x^4 + 6400))/(16*x^2 + 32*x^3 + 24* x^4 + 8*x^5 + x^6))*(144*x + exp(x)*(3*x + 6) + 72*x^2 + 12*x^3 + 96) + ex p(x)*(30*x + 18*x^2 + 12) + 912*x^2 + 600*x^3 + 192*x^4 + 24*x^5 + 192)/(5 60*x + 760*x^2 + 500*x^3 + 160*x^4 + 20*x^5 + exp(x)*(25*x + 20*x^2 + 5*x^ 3 + 10) + 160), x)