Integrand size = 280, antiderivative size = 28 \[ \int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx=2 x+\left (\log (5)-4 \left (x^2+\log \left (e^{e^3}+\log (2+x)\right )\right )^2\right )^2 \] Output:
2*x+(ln(5)-2*(ln(ln(2+x)+exp(exp(3)))+x^2)*(2*ln(ln(2+x)+exp(exp(3)))+2*x^ 2))^2
Leaf count is larger than twice the leaf count of optimal. \(100\) vs. \(2(28)=56\).
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.57 \[ \int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx=2 \left (x+8 x^8-4 x^4 \log (5)+8 x^2 \left (4 x^4-\log (5)\right ) \log \left (e^{e^3}+\log (2+x)\right )+4 \left (12 x^4-\log (5)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+32 x^2 \log ^3\left (e^{e^3}+\log (2+x)\right )+8 \log ^4\left (e^{e^3}+\log (2+x)\right )\right ) \] Input:
Integrate[(64*x^6 - 16*x^2*Log[5] + E^E^3*(4 + 2*x + 256*x^7 + 128*x^8 + ( -64*x^3 - 32*x^4)*Log[5]) + (4 + 2*x + 256*x^7 + 128*x^8 + (-64*x^3 - 32*x ^4)*Log[5])*Log[2 + x] + (192*x^4 - 16*Log[5] + E^E^3*(768*x^5 + 384*x^6 + (-64*x - 32*x^2)*Log[5]) + (768*x^5 + 384*x^6 + (-64*x - 32*x^2)*Log[5])* Log[2 + x])*Log[E^E^3 + Log[2 + x]] + (192*x^2 + E^E^3*(768*x^3 + 384*x^4) + (768*x^3 + 384*x^4)*Log[2 + x])*Log[E^E^3 + Log[2 + x]]^2 + (64 + E^E^3 *(256*x + 128*x^2) + (256*x + 128*x^2)*Log[2 + x])*Log[E^E^3 + Log[2 + x]] ^3)/(E^E^3*(2 + x) + (2 + x)*Log[2 + x]),x]
Output:
2*(x + 8*x^8 - 4*x^4*Log[5] + 8*x^2*(4*x^4 - Log[5])*Log[E^E^3 + Log[2 + x ]] + 4*(12*x^4 - Log[5])*Log[E^E^3 + Log[2 + x]]^2 + 32*x^2*Log[E^E^3 + Lo g[2 + x]]^3 + 8*Log[E^E^3 + Log[2 + x]]^4)
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.
| \(\displaystyle \int \frac {64 x^6+\left (e^{e^3} \left (128 x^2+256 x\right )+\left (128 x^2+256 x\right ) \log (x+2)+64\right ) \log ^3\left (\log (x+2)+e^{e^3}\right )-16 x^2 \log (5)+\left (192 x^2+e^{e^3} \left (384 x^4+768 x^3\right )+\left (384 x^4+768 x^3\right ) \log (x+2)\right ) \log ^2\left (\log (x+2)+e^{e^3}\right )+e^{e^3} \left (128 x^8+256 x^7+\left (-32 x^4-64 x^3\right ) \log (5)+2 x+4\right )+\left (128 x^8+256 x^7+\left (-32 x^4-64 x^3\right ) \log (5)+2 x+4\right ) \log (x+2)+\left (192 x^4+e^{e^3} \left (384 x^6+768 x^5+\left (-32 x^2-64 x\right ) \log (5)\right )+\left (384 x^6+768 x^5+\left (-32 x^2-64 x\right ) \log (5)\right ) \log (x+2)-16 \log (5)\right ) \log \left (\log (x+2)+e^{e^3}\right )}{e^{e^3} (x+2)+(x+2) \log (x+2)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {64 x^6+\left (e^{e^3} \left (128 x^2+256 x\right )+\left (128 x^2+256 x\right ) \log (x+2)+64\right ) \log ^3\left (\log (x+2)+e^{e^3}\right )-16 x^2 \log (5)+\left (192 x^2+e^{e^3} \left (384 x^4+768 x^3\right )+\left (384 x^4+768 x^3\right ) \log (x+2)\right ) \log ^2\left (\log (x+2)+e^{e^3}\right )+e^{e^3} \left (128 x^8+256 x^7+\left (-32 x^4-64 x^3\right ) \log (5)+2 x+4\right )+\left (128 x^8+256 x^7+\left (-32 x^4-64 x^3\right ) \log (5)+2 x+4\right ) \log (x+2)+\left (192 x^4+e^{e^3} \left (384 x^6+768 x^5+\left (-32 x^2-64 x\right ) \log (5)\right )+\left (384 x^6+768 x^5+\left (-32 x^2-64 x\right ) \log (5)\right ) \log (x+2)-16 \log (5)\right ) \log \left (\log (x+2)+e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (2 e^{e^3} x^2+2 x^2 \log (x+2)+4 e^{e^3} x+4 x \log (x+2)+1\right ) \log ^3\left (\log (x+2)+e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}+\frac {192 x^2 \left (2 e^{e^3} x^2+2 x^2 \log (x+2)+4 e^{e^3} x+4 x \log (x+2)+1\right ) \log ^2\left (\log (x+2)+e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}+\frac {16 \left (12 x^4-\log (5)\right ) \left (2 e^{e^3} x^2+2 x^2 \log (x+2)+4 e^{e^3} x+4 x \log (x+2)+1\right ) \log \left (\log (x+2)+e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}+\frac {2 \left (64 e^{e^3} x^8+64 x^8 \log (x+2)+128 e^{e^3} x^7+128 x^7 \log (x+2)+32 x^6-16 x^4 \log (5) \log (x+2)-16 e^{e^3} x^4 \log (5)-32 x^3 \log (5) \log (x+2)-32 e^{e^3} x^3 \log (5)-8 x^2 \log (5)+e^{e^3} x+x \log (x+2)+2 \log (x+2)+2 e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {64 \left (2 e^{e^3} x^2+2 x^2 \log (x+2)+4 e^{e^3} x+4 x \log (x+2)+1\right ) \log ^3\left (\log (x+2)+e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}+\frac {192 x^2 \left (2 e^{e^3} x^2+2 x^2 \log (x+2)+4 e^{e^3} x+4 x \log (x+2)+1\right ) \log ^2\left (\log (x+2)+e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}+\frac {16 \left (12 x^4-\log (5)\right ) \left (2 e^{e^3} x^2+2 x^2 \log (x+2)+4 e^{e^3} x+4 x \log (x+2)+1\right ) \log \left (\log (x+2)+e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}+\frac {2 \left (64 e^{e^3} x^8+64 x^8 \log (x+2)+128 e^{e^3} x^7+128 x^7 \log (x+2)+32 x^6-16 x^4 \log (5) \log (x+2)-16 e^{e^3} x^4 \log (5)-32 x^3 \log (5) \log (x+2)-32 e^{e^3} x^3 \log (5)-8 x^2 \log (5)+e^{e^3} x+x \log (x+2)+2 \log (x+2)+2 e^{e^3}\right )}{(x+2) \left (\log (x+2)+e^{e^3}\right )}\right )dx\) |
Input:
Int[(64*x^6 - 16*x^2*Log[5] + E^E^3*(4 + 2*x + 256*x^7 + 128*x^8 + (-64*x^ 3 - 32*x^4)*Log[5]) + (4 + 2*x + 256*x^7 + 128*x^8 + (-64*x^3 - 32*x^4)*Lo g[5])*Log[2 + x] + (192*x^4 - 16*Log[5] + E^E^3*(768*x^5 + 384*x^6 + (-64* x - 32*x^2)*Log[5]) + (768*x^5 + 384*x^6 + (-64*x - 32*x^2)*Log[5])*Log[2 + x])*Log[E^E^3 + Log[2 + x]] + (192*x^2 + E^E^3*(768*x^3 + 384*x^4) + (76 8*x^3 + 384*x^4)*Log[2 + x])*Log[E^E^3 + Log[2 + x]]^2 + (64 + E^E^3*(256* x + 128*x^2) + (256*x + 128*x^2)*Log[2 + x])*Log[E^E^3 + Log[2 + x]]^3)/(E ^E^3*(2 + x) + (2 + x)*Log[2 + x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(41)=82\).
Time = 4.67 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.25
| method | result | size |
| risch | \(16 \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{4}+64 x^{2} \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{3}+\left (96 x^{4}-8 \ln \left (5\right )\right ) \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{2}+\left (64 x^{6}-16 x^{2} \ln \left (5\right )\right ) \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )+16 x^{8}-8 x^{4} \ln \left (5\right )+2 x\) | \(91\) |
| parallelrisch | \(16 x^{8}+64 \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right ) x^{6}+96 \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{2} x^{4}-8 x^{4} \ln \left (5\right )+64 x^{2} \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{3}-16 \ln \left (5\right ) \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right ) x^{2}+16 \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{4}-8 \ln \left (5\right ) \ln \left (\ln \left (2+x \right )+{\mathrm e}^{{\mathrm e}^{3}}\right )^{2}-8+2 x\) | \(108\) |
Input:
int((((128*x^2+256*x)*ln(2+x)+(128*x^2+256*x)*exp(exp(3))+64)*ln(ln(2+x)+e xp(exp(3)))^3+((384*x^4+768*x^3)*ln(2+x)+(384*x^4+768*x^3)*exp(exp(3))+192 *x^2)*ln(ln(2+x)+exp(exp(3)))^2+(((-32*x^2-64*x)*ln(5)+384*x^6+768*x^5)*ln (2+x)+((-32*x^2-64*x)*ln(5)+384*x^6+768*x^5)*exp(exp(3))-16*ln(5)+192*x^4) *ln(ln(2+x)+exp(exp(3)))+((-32*x^4-64*x^3)*ln(5)+128*x^8+256*x^7+2*x+4)*ln (2+x)+((-32*x^4-64*x^3)*ln(5)+128*x^8+256*x^7+2*x+4)*exp(exp(3))-16*x^2*ln (5)+64*x^6)/((2+x)*ln(2+x)+(2+x)*exp(exp(3))),x,method=_RETURNVERBOSE)
Output:
16*ln(ln(2+x)+exp(exp(3)))^4+64*x^2*ln(ln(2+x)+exp(exp(3)))^3+(96*x^4-8*ln (5))*ln(ln(2+x)+exp(exp(3)))^2+(64*x^6-16*x^2*ln(5))*ln(ln(2+x)+exp(exp(3) ))+16*x^8-8*x^4*ln(5)+2*x
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (28) = 56\).
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.29 \[ \int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx=16 \, x^{8} - 8 \, x^{4} \log \left (5\right ) + 64 \, x^{2} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{3} + 16 \, \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{4} + 8 \, {\left (12 \, x^{4} - \log \left (5\right )\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{2} + 16 \, {\left (4 \, x^{6} - x^{2} \log \left (5\right )\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 2 \, x \] Input:
integrate((((128*x^2+256*x)*log(2+x)+(128*x^2+256*x)*exp(exp(3))+64)*log(l og(2+x)+exp(exp(3)))^3+((384*x^4+768*x^3)*log(2+x)+(384*x^4+768*x^3)*exp(e xp(3))+192*x^2)*log(log(2+x)+exp(exp(3)))^2+(((-32*x^2-64*x)*log(5)+384*x^ 6+768*x^5)*log(2+x)+((-32*x^2-64*x)*log(5)+384*x^6+768*x^5)*exp(exp(3))-16 *log(5)+192*x^4)*log(log(2+x)+exp(exp(3)))+((-32*x^4-64*x^3)*log(5)+128*x^ 8+256*x^7+2*x+4)*log(2+x)+((-32*x^4-64*x^3)*log(5)+128*x^8+256*x^7+2*x+4)* exp(exp(3))-16*x^2*log(5)+64*x^6)/((2+x)*log(2+x)+(2+x)*exp(exp(3))),x, al gorithm="fricas")
Output:
16*x^8 - 8*x^4*log(5) + 64*x^2*log(e^(e^3) + log(x + 2))^3 + 16*log(e^(e^3 ) + log(x + 2))^4 + 8*(12*x^4 - log(5))*log(e^(e^3) + log(x + 2))^2 + 16*( 4*x^6 - x^2*log(5))*log(e^(e^3) + log(x + 2)) + 2*x
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (27) = 54\).
Time = 0.51 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.54 \[ \int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx=16 x^{8} - 8 x^{4} \log {\left (5 \right )} + 64 x^{2} \log {\left (\log {\left (x + 2 \right )} + e^{e^{3}} \right )}^{3} + 2 x + \left (96 x^{4} - 8 \log {\left (5 \right )}\right ) \log {\left (\log {\left (x + 2 \right )} + e^{e^{3}} \right )}^{2} + \left (64 x^{6} - 16 x^{2} \log {\left (5 \right )}\right ) \log {\left (\log {\left (x + 2 \right )} + e^{e^{3}} \right )} + 16 \log {\left (\log {\left (x + 2 \right )} + e^{e^{3}} \right )}^{4} \] Input:
integrate((((128*x**2+256*x)*ln(2+x)+(128*x**2+256*x)*exp(exp(3))+64)*ln(l n(2+x)+exp(exp(3)))**3+((384*x**4+768*x**3)*ln(2+x)+(384*x**4+768*x**3)*ex p(exp(3))+192*x**2)*ln(ln(2+x)+exp(exp(3)))**2+(((-32*x**2-64*x)*ln(5)+384 *x**6+768*x**5)*ln(2+x)+((-32*x**2-64*x)*ln(5)+384*x**6+768*x**5)*exp(exp( 3))-16*ln(5)+192*x**4)*ln(ln(2+x)+exp(exp(3)))+((-32*x**4-64*x**3)*ln(5)+1 28*x**8+256*x**7+2*x+4)*ln(2+x)+((-32*x**4-64*x**3)*ln(5)+128*x**8+256*x** 7+2*x+4)*exp(exp(3))-16*x**2*ln(5)+64*x**6)/((2+x)*ln(2+x)+(2+x)*exp(exp(3 ))),x)
Output:
16*x**8 - 8*x**4*log(5) + 64*x**2*log(log(x + 2) + exp(exp(3)))**3 + 2*x + (96*x**4 - 8*log(5))*log(log(x + 2) + exp(exp(3)))**2 + (64*x**6 - 16*x** 2*log(5))*log(log(x + 2) + exp(exp(3))) + 16*log(log(x + 2) + exp(exp(3))) **4
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (28) = 56\).
Time = 0.16 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.96 \[ \int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx=16 \, x^{8} - 8 \, x^{4} \log \left (5\right ) + 64 \, x^{2} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{3} + 16 \, \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{4} + 8 \, {\left (12 \, x^{4} - \log \left (5\right )\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{2} + 4 \, {\left (16 \, x^{6} - 4 \, x^{2} \log \left (5\right ) - e^{\left (e^{3}\right )}\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 4 \, e^{\left (e^{3}\right )} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 2 \, x \] Input:
integrate((((128*x^2+256*x)*log(2+x)+(128*x^2+256*x)*exp(exp(3))+64)*log(l og(2+x)+exp(exp(3)))^3+((384*x^4+768*x^3)*log(2+x)+(384*x^4+768*x^3)*exp(e xp(3))+192*x^2)*log(log(2+x)+exp(exp(3)))^2+(((-32*x^2-64*x)*log(5)+384*x^ 6+768*x^5)*log(2+x)+((-32*x^2-64*x)*log(5)+384*x^6+768*x^5)*exp(exp(3))-16 *log(5)+192*x^4)*log(log(2+x)+exp(exp(3)))+((-32*x^4-64*x^3)*log(5)+128*x^ 8+256*x^7+2*x+4)*log(2+x)+((-32*x^4-64*x^3)*log(5)+128*x^8+256*x^7+2*x+4)* exp(exp(3))-16*x^2*log(5)+64*x^6)/((2+x)*log(2+x)+(2+x)*exp(exp(3))),x, al gorithm="maxima")
Output:
16*x^8 - 8*x^4*log(5) + 64*x^2*log(e^(e^3) + log(x + 2))^3 + 16*log(e^(e^3 ) + log(x + 2))^4 + 8*(12*x^4 - log(5))*log(e^(e^3) + log(x + 2))^2 + 4*(1 6*x^6 - 4*x^2*log(5) - e^(e^3))*log(e^(e^3) + log(x + 2)) + 4*e^(e^3)*log( e^(e^3) + log(x + 2)) + 2*x
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (28) = 56\).
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79 \[ \int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx=16 \, x^{8} + 64 \, x^{6} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 96 \, x^{4} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{2} - 8 \, x^{4} \log \left (5\right ) + 64 \, x^{2} \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{3} - 16 \, x^{2} \log \left (5\right ) \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right ) + 16 \, \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{4} - 8 \, \log \left (5\right ) \log \left (e^{\left (e^{3}\right )} + \log \left (x + 2\right )\right )^{2} + 2 \, x \] Input:
integrate((((128*x^2+256*x)*log(2+x)+(128*x^2+256*x)*exp(exp(3))+64)*log(l og(2+x)+exp(exp(3)))^3+((384*x^4+768*x^3)*log(2+x)+(384*x^4+768*x^3)*exp(e xp(3))+192*x^2)*log(log(2+x)+exp(exp(3)))^2+(((-32*x^2-64*x)*log(5)+384*x^ 6+768*x^5)*log(2+x)+((-32*x^2-64*x)*log(5)+384*x^6+768*x^5)*exp(exp(3))-16 *log(5)+192*x^4)*log(log(2+x)+exp(exp(3)))+((-32*x^4-64*x^3)*log(5)+128*x^ 8+256*x^7+2*x+4)*log(2+x)+((-32*x^4-64*x^3)*log(5)+128*x^8+256*x^7+2*x+4)* exp(exp(3))-16*x^2*log(5)+64*x^6)/((2+x)*log(2+x)+(2+x)*exp(exp(3))),x, al gorithm="giac")
Output:
16*x^8 + 64*x^6*log(e^(e^3) + log(x + 2)) + 96*x^4*log(e^(e^3) + log(x + 2 ))^2 - 8*x^4*log(5) + 64*x^2*log(e^(e^3) + log(x + 2))^3 - 16*x^2*log(5)*l og(e^(e^3) + log(x + 2)) + 16*log(e^(e^3) + log(x + 2))^4 - 8*log(5)*log(e ^(e^3) + log(x + 2))^2 + 2*x
Time = 3.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.29 \[ \int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx=2\,x+16\,{\ln \left (\ln \left (x+2\right )+{\mathrm {e}}^{{\mathrm {e}}^3}\right )}^4+64\,x^2\,{\ln \left (\ln \left (x+2\right )+{\mathrm {e}}^{{\mathrm {e}}^3}\right )}^3-\ln \left (\ln \left (x+2\right )+{\mathrm {e}}^{{\mathrm {e}}^3}\right )\,\left (16\,x^2\,\ln \left (5\right )-64\,x^6\right )-8\,x^4\,\ln \left (5\right )+16\,x^8-{\ln \left (\ln \left (x+2\right )+{\mathrm {e}}^{{\mathrm {e}}^3}\right )}^2\,\left (8\,\ln \left (5\right )-96\,x^4\right ) \] Input:
int((exp(exp(3))*(2*x - log(5)*(64*x^3 + 32*x^4) + 256*x^7 + 128*x^8 + 4) + log(log(x + 2) + exp(exp(3)))*(exp(exp(3))*(768*x^5 - log(5)*(64*x + 32* x^2) + 384*x^6) - 16*log(5) + log(x + 2)*(768*x^5 - log(5)*(64*x + 32*x^2) + 384*x^6) + 192*x^4) + log(log(x + 2) + exp(exp(3)))^2*(log(x + 2)*(768* x^3 + 384*x^4) + 192*x^2 + exp(exp(3))*(768*x^3 + 384*x^4)) + log(log(x + 2) + exp(exp(3)))^3*(log(x + 2)*(256*x + 128*x^2) + exp(exp(3))*(256*x + 1 28*x^2) + 64) - 16*x^2*log(5) + 64*x^6 + log(x + 2)*(2*x - log(5)*(64*x^3 + 32*x^4) + 256*x^7 + 128*x^8 + 4))/(exp(exp(3))*(x + 2) + log(x + 2)*(x + 2)),x)
Output:
2*x + 16*log(log(x + 2) + exp(exp(3)))^4 + 64*x^2*log(log(x + 2) + exp(exp (3)))^3 - log(log(x + 2) + exp(exp(3)))*(16*x^2*log(5) - 64*x^6) - 8*x^4*l og(5) + 16*x^8 - log(log(x + 2) + exp(exp(3)))^2*(8*log(5) - 96*x^4)
Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.21 \[ \int \frac {64 x^6-16 x^2 \log (5)+e^{e^3} \left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right )+\left (4+2 x+256 x^7+128 x^8+\left (-64 x^3-32 x^4\right ) \log (5)\right ) \log (2+x)+\left (192 x^4-16 \log (5)+e^{e^3} \left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right )+\left (768 x^5+384 x^6+\left (-64 x-32 x^2\right ) \log (5)\right ) \log (2+x)\right ) \log \left (e^{e^3}+\log (2+x)\right )+\left (192 x^2+e^{e^3} \left (768 x^3+384 x^4\right )+\left (768 x^3+384 x^4\right ) \log (2+x)\right ) \log ^2\left (e^{e^3}+\log (2+x)\right )+\left (64+e^{e^3} \left (256 x+128 x^2\right )+\left (256 x+128 x^2\right ) \log (2+x)\right ) \log ^3\left (e^{e^3}+\log (2+x)\right )}{e^{e^3} (2+x)+(2+x) \log (2+x)} \, dx=16 \mathrm {log}\left (e^{e^{3}}+\mathrm {log}\left (x +2\right )\right )^{4}+64 \mathrm {log}\left (e^{e^{3}}+\mathrm {log}\left (x +2\right )\right )^{3} x^{2}-8 \mathrm {log}\left (e^{e^{3}}+\mathrm {log}\left (x +2\right )\right )^{2} \mathrm {log}\left (5\right )+96 \mathrm {log}\left (e^{e^{3}}+\mathrm {log}\left (x +2\right )\right )^{2} x^{4}-16 \,\mathrm {log}\left (e^{e^{3}}+\mathrm {log}\left (x +2\right )\right ) \mathrm {log}\left (5\right ) x^{2}+64 \,\mathrm {log}\left (e^{e^{3}}+\mathrm {log}\left (x +2\right )\right ) x^{6}-8 \,\mathrm {log}\left (5\right ) x^{4}+16 x^{8}+2 x \] Input:
int((((128*x^2+256*x)*log(2+x)+(128*x^2+256*x)*exp(exp(3))+64)*log(log(2+x )+exp(exp(3)))^3+((384*x^4+768*x^3)*log(2+x)+(384*x^4+768*x^3)*exp(exp(3)) +192*x^2)*log(log(2+x)+exp(exp(3)))^2+(((-32*x^2-64*x)*log(5)+384*x^6+768* x^5)*log(2+x)+((-32*x^2-64*x)*log(5)+384*x^6+768*x^5)*exp(exp(3))-16*log(5 )+192*x^4)*log(log(2+x)+exp(exp(3)))+((-32*x^4-64*x^3)*log(5)+128*x^8+256* x^7+2*x+4)*log(2+x)+((-32*x^4-64*x^3)*log(5)+128*x^8+256*x^7+2*x+4)*exp(ex p(3))-16*x^2*log(5)+64*x^6)/((2+x)*log(2+x)+(2+x)*exp(exp(3))),x)
Output:
2*(8*log(e**(e**3) + log(x + 2))**4 + 32*log(e**(e**3) + log(x + 2))**3*x* *2 - 4*log(e**(e**3) + log(x + 2))**2*log(5) + 48*log(e**(e**3) + log(x + 2))**2*x**4 - 8*log(e**(e**3) + log(x + 2))*log(5)*x**2 + 32*log(e**(e**3) + log(x + 2))*x**6 - 4*log(5)*x**4 + 8*x**8 + x)