Integrand size = 192, antiderivative size = 28 \[ \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx=\log \left (-\frac {e^5}{x}+x+4 x \left (-e^{\frac {x}{\log (4)}}+x\right )^4\right ) \] Output:
ln(x+4*x*(x-exp(1/2*x/ln(2)))^4-exp(5)/x)
\[ \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx=\int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx \] Input:
Integrate[((E^5 + x^2 + 20*x^6)*Log[4] + E^((4*x)/Log[4])*(16*x^3 + 4*x^2* Log[4]) + E^((3*x)/Log[4])*(-48*x^4 - 32*x^3*Log[4]) + E^((2*x)/Log[4])*(4 8*x^5 + 72*x^4*Log[4]) + E^(x/Log[4])*(-16*x^6 - 64*x^5*Log[4]))/(4*E^((4* x)/Log[4])*x^3*Log[4] - 16*E^((3*x)/Log[4])*x^4*Log[4] + 24*E^((2*x)/Log[4 ])*x^5*Log[4] - 16*E^(x/Log[4])*x^6*Log[4] + (-(E^5*x) + x^3 + 4*x^7)*Log[ 4]),x]
Output:
Integrate[((E^5 + x^2 + 20*x^6)*Log[4] + E^((4*x)/Log[4])*(16*x^3 + 4*x^2* Log[4]) + E^((3*x)/Log[4])*(-48*x^4 - 32*x^3*Log[4]) + E^((2*x)/Log[4])*(4 8*x^5 + 72*x^4*Log[4]) + E^(x/Log[4])*(-16*x^6 - 64*x^5*Log[4]))/(4*E^((4* x)/Log[4])*x^3*Log[4] - 16*E^((3*x)/Log[4])*x^4*Log[4] + 24*E^((2*x)/Log[4 ])*x^5*Log[4] - 16*E^(x/Log[4])*x^6*Log[4] + (-(E^5*x) + x^3 + 4*x^7)*Log[ 4]), 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.
| \(\displaystyle \int \frac {e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )+\left (20 x^6+x^2+e^5\right ) \log (4)+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )}{-16 x^6 \log (4) e^{\frac {x}{\log (4)}}+24 x^5 \log (4) e^{\frac {2 x}{\log (4)}}-16 x^4 \log (4) e^{\frac {3 x}{\log (4)}}+4 x^3 \log (4) e^{\frac {4 x}{\log (4)}}+\left (4 x^7+x^3-e^5 x\right ) \log (4)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )+\left (20 x^6+x^2+e^5\right ) (-\log (4))-e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )-e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )-e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )}{x \log (4) \left (-4 x^6+16 x^5 e^{\frac {x}{\log (4)}}-24 x^4 e^{\frac {2 x}{\log (4)}}+16 x^3 e^{\frac {3 x}{\log (4)}}-x^2-4 x^2 e^{\frac {4 x}{\log (4)}}+e^5\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {\log (4) \left (20 x^6+x^2+e^5\right )+4 e^{\frac {4 x}{\log (4)}} \left (4 x^3+\log (4) x^2\right )-16 e^{\frac {3 x}{\log (4)}} \left (3 x^4+2 \log (4) x^3\right )+24 e^{\frac {2 x}{\log (4)}} \left (2 x^5+3 \log (4) x^4\right )-16 e^{\frac {x}{\log (4)}} \left (x^6+4 \log (4) x^5\right )}{x \left (-4 x^6+16 e^{\frac {x}{\log (4)}} x^5-24 e^{\frac {2 x}{\log (4)}} x^4+16 e^{\frac {3 x}{\log (4)}} x^3-4 e^{\frac {4 x}{\log (4)}} x^2-x^2+e^5\right )}dx}{\log (4)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\log (4) \left (20 x^6+x^2+e^5\right )+4 e^{\frac {4 x}{\log (4)}} \left (4 x^3+\log (4) x^2\right )-16 e^{\frac {3 x}{\log (4)}} \left (3 x^4+2 \log (4) x^3\right )+24 e^{\frac {2 x}{\log (4)}} \left (2 x^5+3 \log (4) x^4\right )-16 e^{\frac {x}{\log (4)}} \left (x^6+4 \log (4) x^5\right )}{x \left (-4 x^6+16 e^{\frac {x}{\log (4)}} x^5-24 e^{\frac {2 x}{\log (4)}} x^4+16 e^{\frac {3 x}{\log (4)}} x^3-4 e^{\frac {4 x}{\log (4)}} x^2-x^2+e^5\right )}dx}{\log (4)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (\frac {-4 x-\log (4)}{x}+\frac {2 \left (8 x^7-24 e^{\frac {x}{\log (4)}} x^6-8 \log (4) x^6+24 e^{\frac {2 x}{\log (4)}} x^5+24 e^{\frac {x}{\log (4)}} \log (4) x^5-8 e^{\frac {3 x}{\log (4)}} x^4-24 e^{\frac {2 x}{\log (4)}} \log (4) x^4+8 e^{\frac {3 x}{\log (4)}} \log (4) x^3+2 x^3-2 e^5 x-e^5 \log (4)\right )}{x \left (4 x^6-16 e^{\frac {x}{\log (4)}} x^5+24 e^{\frac {2 x}{\log (4)}} x^4-16 e^{\frac {3 x}{\log (4)}} x^3+4 e^{\frac {4 x}{\log (4)}} x^2+x^2-e^5\right )}\right )dx}{\log (4)}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {\int \left (\frac {-4 x-\log (4)}{x}+\frac {2 \left (8 x^7-24 e^{\frac {x}{\log (4)}} x^6-8 \log (4) x^6+24 e^{\frac {2 x}{\log (4)}} x^5+24 e^{\frac {x}{\log (4)}} \log (4) x^5-8 e^{\frac {3 x}{\log (4)}} x^4-24 e^{\frac {2 x}{\log (4)}} \log (4) x^4+8 e^{\frac {3 x}{\log (4)}} \log (4) x^3+2 x^3-2 e^5 x-e^5 \log (4)\right )}{x \left (4 x^6-16 e^{\frac {x}{\log (4)}} x^5+24 e^{\frac {2 x}{\log (4)}} x^4-16 e^{\frac {3 x}{\log (4)}} x^3+4 e^{\frac {4 x}{\log (4)}} x^2+x^2-e^5\right )}\right )dx}{\log (4)}\) |
Input:
Int[((E^5 + x^2 + 20*x^6)*Log[4] + E^((4*x)/Log[4])*(16*x^3 + 4*x^2*Log[4] ) + E^((3*x)/Log[4])*(-48*x^4 - 32*x^3*Log[4]) + E^((2*x)/Log[4])*(48*x^5 + 72*x^4*Log[4]) + E^(x/Log[4])*(-16*x^6 - 64*x^5*Log[4]))/(4*E^((4*x)/Log [4])*x^3*Log[4] - 16*E^((3*x)/Log[4])*x^4*Log[4] + 24*E^((2*x)/Log[4])*x^5 *Log[4] - 16*E^(x/Log[4])*x^6*Log[4] + (-(E^5*x) + x^3 + 4*x^7)*Log[4]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(27)=54\).
Time = 1.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43
| method | result | size |
| risch | \(\ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2 x}{\ln \left (2\right )}}-4 x \,{\mathrm e}^{\frac {3 x}{2 \ln \left (2\right )}}+6 x^{2} {\mathrm e}^{\frac {x}{\ln \left (2\right )}}-4 x^{3} {\mathrm e}^{\frac {x}{2 \ln \left (2\right )}}-\frac {-4 x^{6}-x^{2}+{\mathrm e}^{5}}{4 x^{2}}\right )\) | \(68\) |
| parallelrisch | \(-\frac {\ln \left (2\right ) \ln \left (x \right )-\ln \left ({\mathrm e}^{\frac {2 x}{\ln \left (2\right )}} x^{2}-4 \,{\mathrm e}^{\frac {3 x}{2 \ln \left (2\right )}} x^{3}+6 \,{\mathrm e}^{\frac {x}{\ln \left (2\right )}} x^{4}-4 x^{5} {\mathrm e}^{\frac {x}{2 \ln \left (2\right )}}+x^{6}+\frac {x^{2}}{4}-\frac {{\mathrm e}^{5}}{4}\right ) \ln \left (2\right )}{\ln \left (2\right )}\) | \(88\) |
Input:
int(((8*x^2*ln(2)+16*x^3)*exp(1/2*x/ln(2))^4+(-64*x^3*ln(2)-48*x^4)*exp(1/ 2*x/ln(2))^3+(144*x^4*ln(2)+48*x^5)*exp(1/2*x/ln(2))^2+(-128*x^5*ln(2)-16* x^6)*exp(1/2*x/ln(2))+2*(exp(5)+20*x^6+x^2)*ln(2))/(8*x^3*ln(2)*exp(1/2*x/ ln(2))^4-32*x^4*ln(2)*exp(1/2*x/ln(2))^3+48*x^5*ln(2)*exp(1/2*x/ln(2))^2-3 2*x^6*ln(2)*exp(1/2*x/ln(2))+2*(-x*exp(5)+4*x^7+x^3)*ln(2)),x,method=_RETU RNVERBOSE)
Output:
ln(x)+ln(exp(2*x/ln(2))-4*x*exp(3/2*x/ln(2))+6*x^2*exp(x/ln(2))-4*x^3*exp( 1/2*x/ln(2))-1/4*(-4*x^6-x^2+exp(5))/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (27) = 54\).
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.57 \[ \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx=\log \left (x\right ) + \log \left (\frac {4 \, x^{6} - 16 \, x^{5} e^{\left (\frac {x}{2 \, \log \left (2\right )}\right )} + 24 \, x^{4} e^{\frac {x}{\log \left (2\right )}} - 16 \, x^{3} e^{\left (\frac {3 \, x}{2 \, \log \left (2\right )}\right )} + 4 \, x^{2} e^{\left (\frac {2 \, x}{\log \left (2\right )}\right )} + x^{2} - e^{5}}{x^{2}}\right ) \] Input:
integrate(((8*x^2*log(2)+16*x^3)*exp(1/2*x/log(2))^4+(-64*x^3*log(2)-48*x^ 4)*exp(1/2*x/log(2))^3+(144*x^4*log(2)+48*x^5)*exp(1/2*x/log(2))^2+(-128*x ^5*log(2)-16*x^6)*exp(1/2*x/log(2))+2*(exp(5)+20*x^6+x^2)*log(2))/(8*x^3*l og(2)*exp(1/2*x/log(2))^4-32*x^4*log(2)*exp(1/2*x/log(2))^3+48*x^5*log(2)* exp(1/2*x/log(2))^2-32*x^6*log(2)*exp(1/2*x/log(2))+2*(-x*exp(5)+4*x^7+x^3 )*log(2)),x, algorithm="fricas")
Output:
log(x) + log((4*x^6 - 16*x^5*e^(1/2*x/log(2)) + 24*x^4*e^(x/log(2)) - 16*x ^3*e^(3/2*x/log(2)) + 4*x^2*e^(2*x/log(2)) + x^2 - e^5)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).
Time = 0.56 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43 \[ \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx=\log {\left (x \right )} + \log {\left (- 4 x^{3} e^{\frac {x}{2 \log {\left (2 \right )}}} + 6 x^{2} e^{\frac {x}{\log {\left (2 \right )}}} - 4 x e^{\frac {3 x}{2 \log {\left (2 \right )}}} + e^{\frac {2 x}{\log {\left (2 \right )}}} + \frac {4 x^{6} + x^{2} - e^{5}}{4 x^{2}} \right )} \] Input:
integrate(((8*x**2*ln(2)+16*x**3)*exp(1/2*x/ln(2))**4+(-64*x**3*ln(2)-48*x **4)*exp(1/2*x/ln(2))**3+(144*x**4*ln(2)+48*x**5)*exp(1/2*x/ln(2))**2+(-12 8*x**5*ln(2)-16*x**6)*exp(1/2*x/ln(2))+2*(exp(5)+20*x**6+x**2)*ln(2))/(8*x **3*ln(2)*exp(1/2*x/ln(2))**4-32*x**4*ln(2)*exp(1/2*x/ln(2))**3+48*x**5*ln (2)*exp(1/2*x/ln(2))**2-32*x**6*ln(2)*exp(1/2*x/ln(2))+2*(-x*exp(5)+4*x**7 +x**3)*ln(2)),x)
Output:
log(x) + log(-4*x**3*exp(x/(2*log(2))) + 6*x**2*exp(x/log(2)) - 4*x*exp(3* x/(2*log(2))) + exp(2*x/log(2)) + (4*x**6 + x**2 - exp(5))/(4*x**2))
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (27) = 54\).
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx=2 \, \log \left (x\right ) + \log \left (-\frac {4 \, x^{6} - 16 \, x^{5} e^{\left (\frac {x}{2 \, \log \left (2\right )}\right )} + 24 \, x^{4} e^{\frac {x}{\log \left (2\right )}} - 16 \, x^{3} e^{\left (\frac {3 \, x}{2 \, \log \left (2\right )}\right )} + 4 \, x^{2} e^{\left (\frac {2 \, x}{\log \left (2\right )}\right )} + x^{2} - e^{5}}{16 \, x^{3}}\right ) \] Input:
integrate(((8*x^2*log(2)+16*x^3)*exp(1/2*x/log(2))^4+(-64*x^3*log(2)-48*x^ 4)*exp(1/2*x/log(2))^3+(144*x^4*log(2)+48*x^5)*exp(1/2*x/log(2))^2+(-128*x ^5*log(2)-16*x^6)*exp(1/2*x/log(2))+2*(exp(5)+20*x^6+x^2)*log(2))/(8*x^3*l og(2)*exp(1/2*x/log(2))^4-32*x^4*log(2)*exp(1/2*x/log(2))^3+48*x^5*log(2)* exp(1/2*x/log(2))^2-32*x^6*log(2)*exp(1/2*x/log(2))+2*(-x*exp(5)+4*x^7+x^3 )*log(2)),x, algorithm="maxima")
Output:
2*log(x) + log(-1/16*(4*x^6 - 16*x^5*e^(1/2*x/log(2)) + 24*x^4*e^(x/log(2) ) - 16*x^3*e^(3/2*x/log(2)) + 4*x^2*e^(2*x/log(2)) + x^2 - e^5)/x^3)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
Time = 0.48 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx=\log \left (4 \, x^{6} - 16 \, x^{5} e^{\left (\frac {x}{2 \, \log \left (2\right )}\right )} + 24 \, x^{4} e^{\frac {x}{\log \left (2\right )}} - 16 \, x^{3} e^{\left (\frac {3 \, x}{2 \, \log \left (2\right )}\right )} + 4 \, x^{2} e^{\left (\frac {2 \, x}{\log \left (2\right )}\right )} + x^{2} - e^{5}\right ) - \log \left (x\right ) \] Input:
integrate(((8*x^2*log(2)+16*x^3)*exp(1/2*x/log(2))^4+(-64*x^3*log(2)-48*x^ 4)*exp(1/2*x/log(2))^3+(144*x^4*log(2)+48*x^5)*exp(1/2*x/log(2))^2+(-128*x ^5*log(2)-16*x^6)*exp(1/2*x/log(2))+2*(exp(5)+20*x^6+x^2)*log(2))/(8*x^3*l og(2)*exp(1/2*x/log(2))^4-32*x^4*log(2)*exp(1/2*x/log(2))^3+48*x^5*log(2)* exp(1/2*x/log(2))^2-32*x^6*log(2)*exp(1/2*x/log(2))+2*(-x*exp(5)+4*x^7+x^3 )*log(2)),x, algorithm="giac")
Output:
log(4*x^6 - 16*x^5*e^(1/2*x/log(2)) + 24*x^4*e^(x/log(2)) - 16*x^3*e^(3/2* x/log(2)) + 4*x^2*e^(2*x/log(2)) + x^2 - e^5) - log(x)
Timed out. \[ \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx=\int \frac {2\,\ln \left (2\right )\,\left (20\,x^6+x^2+{\mathrm {e}}^5\right )+{\mathrm {e}}^{\frac {2\,x}{\ln \left (2\right )}}\,\left (16\,x^3+8\,\ln \left (2\right )\,x^2\right )-{\mathrm {e}}^{\frac {3\,x}{2\,\ln \left (2\right )}}\,\left (48\,x^4+64\,\ln \left (2\right )\,x^3\right )-{\mathrm {e}}^{\frac {x}{2\,\ln \left (2\right )}}\,\left (16\,x^6+128\,\ln \left (2\right )\,x^5\right )+{\mathrm {e}}^{\frac {x}{\ln \left (2\right )}}\,\left (48\,x^5+144\,\ln \left (2\right )\,x^4\right )}{2\,\ln \left (2\right )\,\left (4\,x^7+x^3-{\mathrm {e}}^5\,x\right )+8\,x^3\,{\mathrm {e}}^{\frac {2\,x}{\ln \left (2\right )}}\,\ln \left (2\right )+48\,x^5\,{\mathrm {e}}^{\frac {x}{\ln \left (2\right )}}\,\ln \left (2\right )-32\,x^4\,{\mathrm {e}}^{\frac {3\,x}{2\,\ln \left (2\right )}}\,\ln \left (2\right )-32\,x^6\,{\mathrm {e}}^{\frac {x}{2\,\ln \left (2\right )}}\,\ln \left (2\right )} \,d x \] Input:
int((2*log(2)*(exp(5) + x^2 + 20*x^6) + exp((2*x)/log(2))*(8*x^2*log(2) + 16*x^3) - exp((3*x)/(2*log(2)))*(64*x^3*log(2) + 48*x^4) - exp(x/(2*log(2) ))*(128*x^5*log(2) + 16*x^6) + exp(x/log(2))*(144*x^4*log(2) + 48*x^5))/(2 *log(2)*(x^3 - x*exp(5) + 4*x^7) + 8*x^3*exp((2*x)/log(2))*log(2) + 48*x^5 *exp(x/log(2))*log(2) - 32*x^4*exp((3*x)/(2*log(2)))*log(2) - 32*x^6*exp(x /(2*log(2)))*log(2)),x)
Output:
int((2*log(2)*(exp(5) + x^2 + 20*x^6) + exp((2*x)/log(2))*(8*x^2*log(2) + 16*x^3) - exp((3*x)/(2*log(2)))*(64*x^3*log(2) + 48*x^4) - exp(x/(2*log(2) ))*(128*x^5*log(2) + 16*x^6) + exp(x/log(2))*(144*x^4*log(2) + 48*x^5))/(2 *log(2)*(x^3 - x*exp(5) + 4*x^7) + 8*x^3*exp((2*x)/log(2))*log(2) + 48*x^5 *exp(x/log(2))*log(2) - 32*x^4*exp((3*x)/(2*log(2)))*log(2) - 32*x^6*exp(x /(2*log(2)))*log(2)), x)
Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx=\mathrm {log}\left (4 e^{\frac {2 x}{\mathrm {log}\left (2\right )}} x^{2}+24 e^{\frac {x}{\mathrm {log}\left (2\right )}} x^{4}-16 e^{\frac {3 x}{2 \,\mathrm {log}\left (2\right )}} x^{3}-16 e^{\frac {x}{2 \,\mathrm {log}\left (2\right )}} x^{5}-e^{5}+4 x^{6}+x^{2}\right )-\mathrm {log}\left (x \right ) \] Input:
int(((8*x^2*log(2)+16*x^3)*exp(1/2*x/log(2))^4+(-64*x^3*log(2)-48*x^4)*exp (1/2*x/log(2))^3+(144*x^4*log(2)+48*x^5)*exp(1/2*x/log(2))^2+(-128*x^5*log (2)-16*x^6)*exp(1/2*x/log(2))+2*(exp(5)+20*x^6+x^2)*log(2))/(8*x^3*log(2)* exp(1/2*x/log(2))^4-32*x^4*log(2)*exp(1/2*x/log(2))^3+48*x^5*log(2)*exp(1/ 2*x/log(2))^2-32*x^6*log(2)*exp(1/2*x/log(2))+2*(-x*exp(5)+4*x^7+x^3)*log( 2)),x)
Output:
log(4*e**((2*x)/log(2))*x**2 + 24*e**(x/log(2))*x**4 - 16*e**((3*x)/(2*log (2)))*x**3 - 16*e**(x/(2*log(2)))*x**5 - e**5 + 4*x**6 + x**2) - log(x)