Integrand size = 141, antiderivative size = 23 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=x \left (x+\left (e^x+\left (4-\frac {2 x^2}{\log (x)}\right )^2\right )^2\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(66\) vs. \(2(23)=46\).
Time = 1.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.87 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=256 x+32 e^x x+e^{2 x} x+x^2+\frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {8 \left (48+e^x\right ) x^5}{\log ^2(x)}-\frac {32 \left (16+e^x\right ) x^3}{\log (x)} \]
Integrate[(-64*x^8 + (384*x^6 + 144*x^8)*Log[x] + (-768*x^4 - 16*E^x*x^4 - 896*x^6)*Log[x]^2 + (512*x^2 + 1920*x^4 + E^x*(32*x^2 + 40*x^4 + 8*x^5))* Log[x]^3 + (-1536*x^2 + E^x*(-96*x^2 - 32*x^3))*Log[x]^4 + (256 + 2*x + E^ (2*x)*(1 + 2*x) + E^x*(32 + 32*x))*Log[x]^5)/Log[x]^5,x]
256*x + 32*E^x*x + E^(2*x)*x + x^2 + (16*x^9)/Log[x]^4 - (128*x^7)/Log[x]^ 3 + (8*(48 + E^x)*x^5)/Log[x]^2 - (32*(16 + E^x)*x^3)/Log[x]
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(23)=46\).
Time = 1.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {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.
\(\displaystyle \int \frac {-64 x^8+\left (144 x^8+384 x^6\right ) \log (x)+\left (-896 x^6-16 e^x x^4-768 x^4\right ) \log ^2(x)+\left (e^x \left (-32 x^3-96 x^2\right )-1536 x^2\right ) \log ^4(x)+\left (1920 x^4+512 x^2+e^x \left (8 x^5+40 x^4+32 x^2\right )\right ) \log ^3(x)+\left (2 x+e^{2 x} (2 x+1)+e^x (32 x+32)+256\right ) \log ^5(x)}{\log ^5(x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (-32 x^8+72 x^8 \log (x)-448 x^6 \log ^2(x)+192 x^6 \log (x)+960 x^4 \log ^3(x)-384 x^4 \log ^2(x)-768 x^2 \log ^4(x)+256 x^2 \log ^3(x)+x \log ^5(x)+128 \log ^5(x)\right )}{\log ^5(x)}+\frac {8 e^x \left (x^5 \log (x)-2 x^4+5 x^4 \log (x)-4 x^3 \log ^2(x)-12 x^2 \log ^2(x)+4 x^2 \log (x)+4 x \log ^3(x)+4 \log ^3(x)\right )}{\log ^3(x)}+e^{2 x} (2 x+1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {384 x^5}{\log ^2(x)}-\frac {512 x^3}{\log (x)}+x^2+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+256 x-\frac {e^{2 x}}{2}+\frac {1}{2} e^{2 x} (2 x+1)\) |
Int[(-64*x^8 + (384*x^6 + 144*x^8)*Log[x] + (-768*x^4 - 16*E^x*x^4 - 896*x ^6)*Log[x]^2 + (512*x^2 + 1920*x^4 + E^x*(32*x^2 + 40*x^4 + 8*x^5))*Log[x] ^3 + (-1536*x^2 + E^x*(-96*x^2 - 32*x^3))*Log[x]^4 + (256 + 2*x + E^(2*x)* (1 + 2*x) + E^x*(32 + 32*x))*Log[x]^5)/Log[x]^5,x]
-1/2*E^(2*x) + 256*x + x^2 + (E^(2*x)*(1 + 2*x))/2 + (16*x^9)/Log[x]^4 - ( 128*x^7)/Log[x]^3 + (384*x^5)/Log[x]^2 - (512*x^3)/Log[x] + (8*E^x*(x^5*Lo g[x] - 4*x^3*Log[x]^2 + 4*x*Log[x]^3))/Log[x]^3
3.24.52.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(22)=44\).
Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22
method | result | size |
risch | \(x \,{\mathrm e}^{2 x}+x^{2}+32 \,{\mathrm e}^{x} x +256 x +\frac {8 x^{3} \left (2 x^{6}-16 x^{4} \ln \left (x \right )+x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}+48 x^{2} \ln \left (x \right )^{2}-4 \,{\mathrm e}^{x} \ln \left (x \right )^{3}-64 \ln \left (x \right )^{3}\right )}{\ln \left (x \right )^{4}}\) | \(74\) |
parallelrisch | \(\frac {16 x^{9}-128 x^{7} \ln \left (x \right )+8 \,{\mathrm e}^{x} x^{5} \ln \left (x \right )^{2}+384 x^{5} \ln \left (x \right )^{2}-32 \ln \left (x \right )^{3} {\mathrm e}^{x} x^{3}+x \,{\mathrm e}^{2 x} \ln \left (x \right )^{4}-512 x^{3} \ln \left (x \right )^{3}+x^{2} \ln \left (x \right )^{4}+32 \ln \left (x \right )^{4} {\mathrm e}^{x} x +256 x \ln \left (x \right )^{4}}{\ln \left (x \right )^{4}}\) | \(93\) |
int((((1+2*x)*exp(x)^2+(32*x+32)*exp(x)+2*x+256)*ln(x)^5+((-32*x^3-96*x^2) *exp(x)-1536*x^2)*ln(x)^4+((8*x^5+40*x^4+32*x^2)*exp(x)+1920*x^4+512*x^2)* ln(x)^3+(-16*exp(x)*x^4-896*x^6-768*x^4)*ln(x)^2+(144*x^8+384*x^6)*ln(x)-6 4*x^8)/ln(x)^5,x,method=_RETURNVERBOSE)
x*exp(x)^2+x^2+32*exp(x)*x+256*x+8*x^3*(2*x^6-16*x^4*ln(x)+x^2*exp(x)*ln(x )^2+48*x^2*ln(x)^2-4*exp(x)*ln(x)^3-64*ln(x)^3)/ln(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.35 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=\frac {16 \, x^{9} - 128 \, x^{7} \log \left (x\right ) + {\left (x^{2} + x e^{\left (2 \, x\right )} + 32 \, x e^{x} + 256 \, x\right )} \log \left (x\right )^{4} - 32 \, {\left (x^{3} e^{x} + 16 \, x^{3}\right )} \log \left (x\right )^{3} + 8 \, {\left (x^{5} e^{x} + 48 \, x^{5}\right )} \log \left (x\right )^{2}}{\log \left (x\right )^{4}} \]
integrate((((1+2*x)*exp(x)^2+(32*x+32)*exp(x)+2*x+256)*log(x)^5+((-32*x^3- 96*x^2)*exp(x)-1536*x^2)*log(x)^4+((8*x^5+40*x^4+32*x^2)*exp(x)+1920*x^4+5 12*x^2)*log(x)^3+(-16*exp(x)*x^4-896*x^6-768*x^4)*log(x)^2+(144*x^8+384*x^ 6)*log(x)-64*x^8)/log(x)^5,x, algorithm=\
(16*x^9 - 128*x^7*log(x) + (x^2 + x*e^(2*x) + 32*x*e^x + 256*x)*log(x)^4 - 32*(x^3*e^x + 16*x^3)*log(x)^3 + 8*(x^5*e^x + 48*x^5)*log(x)^2)/log(x)^4
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.78 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=x^{2} + 256 x + \frac {x e^{2 x} \log {\left (x \right )}^{2} + \left (8 x^{5} - 32 x^{3} \log {\left (x \right )} + 32 x \log {\left (x \right )}^{2}\right ) e^{x}}{\log {\left (x \right )}^{2}} + \frac {16 x^{9} - 128 x^{7} \log {\left (x \right )} + 384 x^{5} \log {\left (x \right )}^{2} - 512 x^{3} \log {\left (x \right )}^{3}}{\log {\left (x \right )}^{4}} \]
integrate((((1+2*x)*exp(x)**2+(32*x+32)*exp(x)+2*x+256)*ln(x)**5+((-32*x** 3-96*x**2)*exp(x)-1536*x**2)*ln(x)**4+((8*x**5+40*x**4+32*x**2)*exp(x)+192 0*x**4+512*x**2)*ln(x)**3+(-16*exp(x)*x**4-896*x**6-768*x**4)*ln(x)**2+(14 4*x**8+384*x**6)*ln(x)-64*x**8)/ln(x)**5,x)
x**2 + 256*x + (x*exp(2*x)*log(x)**2 + (8*x**5 - 32*x**3*log(x) + 32*x*log (x)**2)*exp(x))/log(x)**2 + (16*x**9 - 128*x**7*log(x) + 384*x**5*log(x)** 2 - 512*x**3*log(x)**3)/log(x)**4
\[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=\int { -\frac {64 \, x^{8} - {\left ({\left (2 \, x + 1\right )} e^{\left (2 \, x\right )} + 32 \, {\left (x + 1\right )} e^{x} + 2 \, x + 256\right )} \log \left (x\right )^{5} + 32 \, {\left (48 \, x^{2} + {\left (x^{3} + 3 \, x^{2}\right )} e^{x}\right )} \log \left (x\right )^{4} - 8 \, {\left (240 \, x^{4} + 64 \, x^{2} + {\left (x^{5} + 5 \, x^{4} + 4 \, x^{2}\right )} e^{x}\right )} \log \left (x\right )^{3} + 16 \, {\left (56 \, x^{6} + x^{4} e^{x} + 48 \, x^{4}\right )} \log \left (x\right )^{2} - 48 \, {\left (3 \, x^{8} + 8 \, x^{6}\right )} \log \left (x\right )}{\log \left (x\right )^{5}} \,d x } \]
integrate((((1+2*x)*exp(x)^2+(32*x+32)*exp(x)+2*x+256)*log(x)^5+((-32*x^3- 96*x^2)*exp(x)-1536*x^2)*log(x)^4+((8*x^5+40*x^4+32*x^2)*exp(x)+1920*x^4+5 12*x^2)*log(x)^3+(-16*exp(x)*x^4-896*x^6-768*x^4)*log(x)^2+(144*x^8+384*x^ 6)*log(x)-64*x^8)/log(x)^5,x, algorithm=\
x^2 + 1/2*(2*x - 1)*e^(2*x) + 32*(x - 1)*e^x + 256*x + 8*((x^5 - 4*x^3*log (x))*e^x - 16*(15*x^5 + 4*x^3)*log(x))/log(x)^2 + 1/2*e^(2*x) + 32*e^x + 1 9200*gamma(-2, -5*log(x)) + 43904*gamma(-2, -7*log(x)) + 131712*gamma(-3, -7*log(x)) + 104976*gamma(-3, -9*log(x)) + 419904*gamma(-4, -9*log(x)) + 9 600*integrate(x^4/log(x), x)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (23) = 46\).
Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=\frac {16 \, x^{9} - 128 \, x^{7} \log \left (x\right ) + 8 \, x^{5} e^{x} \log \left (x\right )^{2} + 384 \, x^{5} \log \left (x\right )^{2} - 32 \, x^{3} e^{x} \log \left (x\right )^{3} - 512 \, x^{3} \log \left (x\right )^{3} + x^{2} \log \left (x\right )^{4} + x e^{\left (2 \, x\right )} \log \left (x\right )^{4} + 32 \, x e^{x} \log \left (x\right )^{4} + 256 \, x \log \left (x\right )^{4}}{\log \left (x\right )^{4}} \]
integrate((((1+2*x)*exp(x)^2+(32*x+32)*exp(x)+2*x+256)*log(x)^5+((-32*x^3- 96*x^2)*exp(x)-1536*x^2)*log(x)^4+((8*x^5+40*x^4+32*x^2)*exp(x)+1920*x^4+5 12*x^2)*log(x)^3+(-16*exp(x)*x^4-896*x^6-768*x^4)*log(x)^2+(144*x^8+384*x^ 6)*log(x)-64*x^8)/log(x)^5,x, algorithm=\
(16*x^9 - 128*x^7*log(x) + 8*x^5*e^x*log(x)^2 + 384*x^5*log(x)^2 - 32*x^3* e^x*log(x)^3 - 512*x^3*log(x)^3 + x^2*log(x)^4 + x*e^(2*x)*log(x)^4 + 32*x *e^x*log(x)^4 + 256*x*log(x)^4)/log(x)^4
Time = 9.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.30 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=256\,x+x\,{\mathrm {e}}^{2\,x}-\frac {512\,x^3}{\ln \left (x\right )}+\frac {384\,x^5}{{\ln \left (x\right )}^2}-\frac {128\,x^7}{{\ln \left (x\right )}^3}+\frac {16\,x^9}{{\ln \left (x\right )}^4}+32\,x\,{\mathrm {e}}^x+x^2-\frac {32\,x^3\,{\mathrm {e}}^x}{\ln \left (x\right )}+\frac {8\,x^5\,{\mathrm {e}}^x}{{\ln \left (x\right )}^2} \]
int((log(x)*(384*x^6 + 144*x^8) + log(x)^5*(2*x + exp(x)*(32*x + 32) + exp (2*x)*(2*x + 1) + 256) - log(x)^4*(exp(x)*(96*x^2 + 32*x^3) + 1536*x^2) - log(x)^2*(16*x^4*exp(x) + 768*x^4 + 896*x^6) + log(x)^3*(exp(x)*(32*x^2 + 40*x^4 + 8*x^5) + 512*x^2 + 1920*x^4) - 64*x^8)/log(x)^5,x)