Integrand size = 75, antiderivative size = 20 \[ \int \frac {e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+128 x^7 \log ^8(4)}{e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+16 x^8 \log ^8(4)} \, dx=\log \left (e^x+\frac {16 x^8 \log ^8(4)}{(1+\log (2))^4}\right ) \] Output:
ln(4096*ln(2)^8*x^8/(1+ln(2))^4+exp(x))
Time = 1.49 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+128 x^7 \log ^8(4)}{e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+16 x^8 \log ^8(4)} \, dx=\log \left (e^x (1+\log (2))^4+16 x^8 \log ^8(4)\right ) \] Input:
Integrate[(E^x*(1 + 4*Log[2] + 6*Log[2]^2 + 4*Log[2]^3 + Log[2]^4) + 128*x ^7*Log[4]^8)/(E^x*(1 + 4*Log[2] + 6*Log[2]^2 + 4*Log[2]^3 + Log[2]^4) + 16 *x^8*Log[4]^8),x]
Output:
Log[E^x*(1 + Log[2])^4 + 16*x^8*Log[4]^8]
Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {7235}
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.
\(\displaystyle \int \frac {128 x^7 \log ^8(4)+e^x \left (1+\log ^4(2)+4 \log ^3(2)+6 \log ^2(2)+4 \log (2)\right )}{16 x^8 \log ^8(4)+e^x \left (1+\log ^4(2)+4 \log ^3(2)+6 \log ^2(2)+4 \log (2)\right )} \, dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (16 x^8 \log ^8(4)+e^x (1+\log (2))^4\right )\) |
Input:
Int[(E^x*(1 + 4*Log[2] + 6*Log[2]^2 + 4*Log[2]^3 + Log[2]^4) + 128*x^7*Log [4]^8)/(E^x*(1 + 4*Log[2] + 6*Log[2]^2 + 4*Log[2]^3 + Log[2]^4) + 16*x^8*L og[4]^8),x]
Output:
Log[E^x*(1 + Log[2])^4 + 16*x^8*Log[4]^8]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85
method | result | size |
derivativedivides | \(\ln \left (\left (\ln \left (2\right )^{4}+4 \ln \left (2\right )^{3}+6 \ln \left (2\right )^{2}+4 \ln \left (2\right )+1\right ) {\mathrm e}^{x}+4096 x^{8} \ln \left (2\right )^{8}\right )\) | \(37\) |
default | \(\ln \left (\left (\ln \left (2\right )^{4}+4 \ln \left (2\right )^{3}+6 \ln \left (2\right )^{2}+4 \ln \left (2\right )+1\right ) {\mathrm e}^{x}+4096 x^{8} \ln \left (2\right )^{8}\right )\) | \(37\) |
risch | \(\ln \left ({\mathrm e}^{x}+\frac {4096 x^{8} \ln \left (2\right )^{8}}{\ln \left (2\right )^{4}+4 \ln \left (2\right )^{3}+6 \ln \left (2\right )^{2}+4 \ln \left (2\right )+1}\right )\) | \(38\) |
norman | \(\ln \left (4096 x^{8} \ln \left (2\right )^{8}+{\mathrm e}^{x} \ln \left (2\right )^{4}+4 \ln \left (2\right )^{3} {\mathrm e}^{x}+6 \ln \left (2\right )^{2} {\mathrm e}^{x}+4 \,{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x}\right )\) | \(43\) |
parallelrisch | \(\ln \left (\frac {4096 x^{8} \ln \left (2\right )^{8}+{\mathrm e}^{x} \ln \left (2\right )^{4}+4 \ln \left (2\right )^{3} {\mathrm e}^{x}+6 \ln \left (2\right )^{2} {\mathrm e}^{x}+4 \,{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x}}{4096 \ln \left (2\right )^{8}}\right )\) | \(49\) |
Input:
int(((ln(2)^4+4*ln(2)^3+6*ln(2)^2+4*ln(2)+1)*exp(x)+32768*x^7*ln(2)^8)/((l n(2)^4+4*ln(2)^3+6*ln(2)^2+4*ln(2)+1)*exp(x)+4096*x^8*ln(2)^8),x,method=_R ETURNVERBOSE)
Output:
ln((ln(2)^4+4*ln(2)^3+6*ln(2)^2+4*ln(2)+1)*exp(x)+4096*x^8*ln(2)^8)
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+128 x^7 \log ^8(4)}{e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+16 x^8 \log ^8(4)} \, dx=\log \left (4096 \, x^{8} \log \left (2\right )^{8} + {\left (\log \left (2\right )^{4} + 4 \, \log \left (2\right )^{3} + 6 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) + 1\right )} e^{x}\right ) \] Input:
integrate(((log(2)^4+4*log(2)^3+6*log(2)^2+4*log(2)+1)*exp(x)+32768*x^7*lo g(2)^8)/((log(2)^4+4*log(2)^3+6*log(2)^2+4*log(2)+1)*exp(x)+4096*x^8*log(2 )^8),x, algorithm="fricas")
Output:
log(4096*x^8*log(2)^8 + (log(2)^4 + 4*log(2)^3 + 6*log(2)^2 + 4*log(2) + 1 )*e^x)
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+128 x^7 \log ^8(4)}{e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+16 x^8 \log ^8(4)} \, dx=\log {\left (\frac {4096 x^{8} \log {\left (2 \right )}^{8}}{\log {\left (2 \right )}^{4} + 1 + 4 \log {\left (2 \right )}^{3} + 4 \log {\left (2 \right )} + 6 \log {\left (2 \right )}^{2}} + e^{x} \right )} \] Input:
integrate(((ln(2)**4+4*ln(2)**3+6*ln(2)**2+4*ln(2)+1)*exp(x)+32768*x**7*ln (2)**8)/((ln(2)**4+4*ln(2)**3+6*ln(2)**2+4*ln(2)+1)*exp(x)+4096*x**8*ln(2) **8),x)
Output:
log(4096*x**8*log(2)**8/(log(2)**4 + 1 + 4*log(2)**3 + 4*log(2) + 6*log(2) **2) + exp(x))
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+128 x^7 \log ^8(4)}{e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+16 x^8 \log ^8(4)} \, dx=\log \left (4096 \, x^{8} \log \left (2\right )^{8} + {\left (\log \left (2\right )^{4} + 4 \, \log \left (2\right )^{3} + 6 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) + 1\right )} e^{x}\right ) \] Input:
integrate(((log(2)^4+4*log(2)^3+6*log(2)^2+4*log(2)+1)*exp(x)+32768*x^7*lo g(2)^8)/((log(2)^4+4*log(2)^3+6*log(2)^2+4*log(2)+1)*exp(x)+4096*x^8*log(2 )^8),x, algorithm="maxima")
Output:
log(4096*x^8*log(2)^8 + (log(2)^4 + 4*log(2)^3 + 6*log(2)^2 + 4*log(2) + 1 )*e^x)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10 \[ \int \frac {e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+128 x^7 \log ^8(4)}{e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+16 x^8 \log ^8(4)} \, dx=\log \left (4096 \, x^{8} \log \left (2\right )^{8} + e^{x} \log \left (2\right )^{4} + 4 \, e^{x} \log \left (2\right )^{3} + 6 \, e^{x} \log \left (2\right )^{2} + 4 \, e^{x} \log \left (2\right ) + e^{x}\right ) \] Input:
integrate(((log(2)^4+4*log(2)^3+6*log(2)^2+4*log(2)+1)*exp(x)+32768*x^7*lo g(2)^8)/((log(2)^4+4*log(2)^3+6*log(2)^2+4*log(2)+1)*exp(x)+4096*x^8*log(2 )^8),x, algorithm="giac")
Output:
log(4096*x^8*log(2)^8 + e^x*log(2)^4 + 4*e^x*log(2)^3 + 6*e^x*log(2)^2 + 4 *e^x*log(2) + e^x)
Time = 0.71 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10 \[ \int \frac {e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+128 x^7 \log ^8(4)}{e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+16 x^8 \log ^8(4)} \, dx=\ln \left ({\mathrm {e}}^x+4096\,x^8\,{\ln \left (2\right )}^8+6\,{\mathrm {e}}^x\,{\ln \left (2\right )}^2+4\,{\mathrm {e}}^x\,{\ln \left (2\right )}^3+{\mathrm {e}}^x\,{\ln \left (2\right )}^4+4\,{\mathrm {e}}^x\,\ln \left (2\right )\right ) \] Input:
int((32768*x^7*log(2)^8 + exp(x)*(4*log(2) + 6*log(2)^2 + 4*log(2)^3 + log (2)^4 + 1))/(4096*x^8*log(2)^8 + exp(x)*(4*log(2) + 6*log(2)^2 + 4*log(2)^ 3 + log(2)^4 + 1)),x)
Output:
log(exp(x) + 4096*x^8*log(2)^8 + 6*exp(x)*log(2)^2 + 4*exp(x)*log(2)^3 + e xp(x)*log(2)^4 + 4*exp(x)*log(2))
\[ \int \frac {e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+128 x^7 \log ^8(4)}{e^x \left (1+4 \log (2)+6 \log ^2(2)+4 \log ^3(2)+\log ^4(2)\right )+16 x^8 \log ^8(4)} \, dx=\left (\int \frac {e^{x}}{e^{x} \mathrm {log}\left (2\right )^{4}+4 e^{x} \mathrm {log}\left (2\right )^{3}+6 e^{x} \mathrm {log}\left (2\right )^{2}+4 e^{x} \mathrm {log}\left (2\right )+e^{x}+4096 \mathrm {log}\left (2\right )^{8} x^{8}}d x \right ) \mathrm {log}\left (2\right )^{4}+4 \left (\int \frac {e^{x}}{e^{x} \mathrm {log}\left (2\right )^{4}+4 e^{x} \mathrm {log}\left (2\right )^{3}+6 e^{x} \mathrm {log}\left (2\right )^{2}+4 e^{x} \mathrm {log}\left (2\right )+e^{x}+4096 \mathrm {log}\left (2\right )^{8} x^{8}}d x \right ) \mathrm {log}\left (2\right )^{3}+6 \left (\int \frac {e^{x}}{e^{x} \mathrm {log}\left (2\right )^{4}+4 e^{x} \mathrm {log}\left (2\right )^{3}+6 e^{x} \mathrm {log}\left (2\right )^{2}+4 e^{x} \mathrm {log}\left (2\right )+e^{x}+4096 \mathrm {log}\left (2\right )^{8} x^{8}}d x \right ) \mathrm {log}\left (2\right )^{2}+4 \left (\int \frac {e^{x}}{e^{x} \mathrm {log}\left (2\right )^{4}+4 e^{x} \mathrm {log}\left (2\right )^{3}+6 e^{x} \mathrm {log}\left (2\right )^{2}+4 e^{x} \mathrm {log}\left (2\right )+e^{x}+4096 \mathrm {log}\left (2\right )^{8} x^{8}}d x \right ) \mathrm {log}\left (2\right )+\int \frac {e^{x}}{e^{x} \mathrm {log}\left (2\right )^{4}+4 e^{x} \mathrm {log}\left (2\right )^{3}+6 e^{x} \mathrm {log}\left (2\right )^{2}+4 e^{x} \mathrm {log}\left (2\right )+e^{x}+4096 \mathrm {log}\left (2\right )^{8} x^{8}}d x +32768 \left (\int \frac {x^{7}}{e^{x} \mathrm {log}\left (2\right )^{4}+4 e^{x} \mathrm {log}\left (2\right )^{3}+6 e^{x} \mathrm {log}\left (2\right )^{2}+4 e^{x} \mathrm {log}\left (2\right )+e^{x}+4096 \mathrm {log}\left (2\right )^{8} x^{8}}d x \right ) \mathrm {log}\left (2\right )^{8} \] Input:
int(((log(2)^4+4*log(2)^3+6*log(2)^2+4*log(2)+1)*exp(x)+32768*x^7*log(2)^8 )/((log(2)^4+4*log(2)^3+6*log(2)^2+4*log(2)+1)*exp(x)+4096*x^8*log(2)^8),x )
Output:
int(e**x/(e**x*log(2)**4 + 4*e**x*log(2)**3 + 6*e**x*log(2)**2 + 4*e**x*lo g(2) + e**x + 4096*log(2)**8*x**8),x)*log(2)**4 + 4*int(e**x/(e**x*log(2)* *4 + 4*e**x*log(2)**3 + 6*e**x*log(2)**2 + 4*e**x*log(2) + e**x + 4096*log (2)**8*x**8),x)*log(2)**3 + 6*int(e**x/(e**x*log(2)**4 + 4*e**x*log(2)**3 + 6*e**x*log(2)**2 + 4*e**x*log(2) + e**x + 4096*log(2)**8*x**8),x)*log(2) **2 + 4*int(e**x/(e**x*log(2)**4 + 4*e**x*log(2)**3 + 6*e**x*log(2)**2 + 4 *e**x*log(2) + e**x + 4096*log(2)**8*x**8),x)*log(2) + int(e**x/(e**x*log( 2)**4 + 4*e**x*log(2)**3 + 6*e**x*log(2)**2 + 4*e**x*log(2) + e**x + 4096* log(2)**8*x**8),x) + 32768*int(x**7/(e**x*log(2)**4 + 4*e**x*log(2)**3 + 6 *e**x*log(2)**2 + 4*e**x*log(2) + e**x + 4096*log(2)**8*x**8),x)*log(2)**8