Integrand size = 94, antiderivative size = 25 \[ \int \frac {-e^x \log (5)+e^{5+e^5 (2+x)} \log (5)-\log (5) \log (16)}{1+e^{2 x}+e^{2 e^5 (2+x)}-2 x \log (16)+x^2 \log ^2(16)+e^{e^5 (2+x)} \left (2-2 e^x-2 x \log (16)\right )+e^x (-2+2 x \log (16))} \, dx=\frac {\log (5)}{-1+e^x-e^{e^5 (2+x)}+x \log (16)} \] Output:
ln(5)/(exp(x)-1-exp((2+x)*exp(5))+4*x*ln(2))
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-e^x \log (5)+e^{5+e^5 (2+x)} \log (5)-\log (5) \log (16)}{1+e^{2 x}+e^{2 e^5 (2+x)}-2 x \log (16)+x^2 \log ^2(16)+e^{e^5 (2+x)} \left (2-2 e^x-2 x \log (16)\right )+e^x (-2+2 x \log (16))} \, dx=\frac {\log (5)}{-1+e^x-e^{e^5 (2+x)}+x \log (16)} \] Input:
Integrate[(-(E^x*Log[5]) + E^(5 + E^5*(2 + x))*Log[5] - Log[5]*Log[16])/(1 + E^(2*x) + E^(2*E^5*(2 + x)) - 2*x*Log[16] + x^2*Log[16]^2 + E^(E^5*(2 + x))*(2 - 2*E^x - 2*x*Log[16]) + E^x*(-2 + 2*x*Log[16])),x]
Output:
Log[5]/(-1 + E^x - E^(E^5*(2 + x)) + x*Log[16])
Time = 0.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {7239, 27, 25, 7237}
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.
\(\displaystyle \int \frac {-e^x \log (5)+e^{e^5 (x+2)+5} \log (5)-\log (5) \log (16)}{x^2 \log ^2(16)+e^{2 x}+e^{2 e^5 (x+2)}-2 x \log (16)+e^{e^5 (x+2)} \left (-2 e^x-2 x \log (16)+2\right )+e^x (2 x \log (16)-2)+1} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\log (5) \left (-e^x+e^{e^5 (x+2)+5}-\log (16)\right )}{\left (-e^x+e^{e^5 (x+2)}+x (-\log (16))+1\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \log (5) \int -\frac {e^x-e^{e^5 (x+2)+5}+\log (16)}{\left (-\log (16) x-e^x+e^{e^5 (x+2)}+1\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\log (5) \int \frac {e^x-e^{e^5 (x+2)+5}+\log (16)}{\left (-\log (16) x-e^x+e^{e^5 (x+2)}+1\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle -\frac {\log (5)}{-e^x+e^{e^5 (x+2)}+x (-\log (16))+1}\) |
Input:
Int[(-(E^x*Log[5]) + E^(5 + E^5*(2 + x))*Log[5] - Log[5]*Log[16])/(1 + E^( 2*x) + E^(2*E^5*(2 + x)) - 2*x*Log[16] + x^2*Log[16]^2 + E^(E^5*(2 + x))*( 2 - 2*E^x - 2*x*Log[16]) + E^x*(-2 + 2*x*Log[16])),x]
Output:
-(Log[5]/(1 - E^x + E^(E^5*(2 + x)) - x*Log[16]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\frac {\ln \left (5\right )}{{\mathrm e}^{x}-1-{\mathrm e}^{\left (2+x \right ) {\mathrm e}^{5}}+4 x \ln \left (2\right )}\) | \(24\) |
risch | \(\frac {\ln \left (5\right )}{{\mathrm e}^{x}-1-{\mathrm e}^{\left (2+x \right ) {\mathrm e}^{5}}+4 x \ln \left (2\right )}\) | \(24\) |
parallelrisch | \(\frac {\ln \left (5\right )}{{\mathrm e}^{x}-1-{\mathrm e}^{\left (2+x \right ) {\mathrm e}^{5}}+4 x \ln \left (2\right )}\) | \(24\) |
Input:
int((exp(5)*ln(5)*exp((2+x)*exp(5))-exp(x)*ln(5)-4*ln(2)*ln(5))/(exp((2+x) *exp(5))^2+(-2*exp(x)-8*x*ln(2)+2)*exp((2+x)*exp(5))+exp(x)^2+(8*x*ln(2)-2 )*exp(x)+16*x^2*ln(2)^2-8*x*ln(2)+1),x,method=_RETURNVERBOSE)
Output:
ln(5)/(exp(x)-1-exp((2+x)*exp(5))+4*x*ln(2))
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-e^x \log (5)+e^{5+e^5 (2+x)} \log (5)-\log (5) \log (16)}{1+e^{2 x}+e^{2 e^5 (2+x)}-2 x \log (16)+x^2 \log ^2(16)+e^{e^5 (2+x)} \left (2-2 e^x-2 x \log (16)\right )+e^x (-2+2 x \log (16))} \, dx=\frac {e^{5} \log \left (5\right )}{4 \, x e^{5} \log \left (2\right ) - e^{5} - e^{\left ({\left (x + 2\right )} e^{5} + 5\right )} + e^{\left (x + 5\right )}} \] Input:
integrate((exp(5)*log(5)*exp((2+x)*exp(5))-exp(x)*log(5)-4*log(2)*log(5))/ (exp((2+x)*exp(5))^2+(-2*exp(x)-8*x*log(2)+2)*exp((2+x)*exp(5))+exp(x)^2+( 8*x*log(2)-2)*exp(x)+16*x^2*log(2)^2-8*x*log(2)+1),x, algorithm="fricas")
Output:
e^5*log(5)/(4*x*e^5*log(2) - e^5 - e^((x + 2)*e^5 + 5) + e^(x + 5))
Timed out. \[ \int \frac {-e^x \log (5)+e^{5+e^5 (2+x)} \log (5)-\log (5) \log (16)}{1+e^{2 x}+e^{2 e^5 (2+x)}-2 x \log (16)+x^2 \log ^2(16)+e^{e^5 (2+x)} \left (2-2 e^x-2 x \log (16)\right )+e^x (-2+2 x \log (16))} \, dx=\text {Timed out} \] Input:
integrate((exp(5)*ln(5)*exp((2+x)*exp(5))-exp(x)*ln(5)-4*ln(2)*ln(5))/(exp ((2+x)*exp(5))**2+(-2*exp(x)-8*x*ln(2)+2)*exp((2+x)*exp(5))+exp(x)**2+(8*x *ln(2)-2)*exp(x)+16*x**2*ln(2)**2-8*x*ln(2)+1),x)
Output:
Timed out
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-e^x \log (5)+e^{5+e^5 (2+x)} \log (5)-\log (5) \log (16)}{1+e^{2 x}+e^{2 e^5 (2+x)}-2 x \log (16)+x^2 \log ^2(16)+e^{e^5 (2+x)} \left (2-2 e^x-2 x \log (16)\right )+e^x (-2+2 x \log (16))} \, dx=\frac {\log \left (5\right )}{4 \, x \log \left (2\right ) - e^{\left (x e^{5} + 2 \, e^{5}\right )} + e^{x} - 1} \] Input:
integrate((exp(5)*log(5)*exp((2+x)*exp(5))-exp(x)*log(5)-4*log(2)*log(5))/ (exp((2+x)*exp(5))^2+(-2*exp(x)-8*x*log(2)+2)*exp((2+x)*exp(5))+exp(x)^2+( 8*x*log(2)-2)*exp(x)+16*x^2*log(2)^2-8*x*log(2)+1),x, algorithm="maxima")
Output:
log(5)/(4*x*log(2) - e^(x*e^5 + 2*e^5) + e^x - 1)
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-e^x \log (5)+e^{5+e^5 (2+x)} \log (5)-\log (5) \log (16)}{1+e^{2 x}+e^{2 e^5 (2+x)}-2 x \log (16)+x^2 \log ^2(16)+e^{e^5 (2+x)} \left (2-2 e^x-2 x \log (16)\right )+e^x (-2+2 x \log (16))} \, dx=\frac {\log \left (5\right )}{4 \, x \log \left (2\right ) - e^{\left (x e^{5} + 2 \, e^{5}\right )} + e^{x} - 1} \] Input:
integrate((exp(5)*log(5)*exp((2+x)*exp(5))-exp(x)*log(5)-4*log(2)*log(5))/ (exp((2+x)*exp(5))^2+(-2*exp(x)-8*x*log(2)+2)*exp((2+x)*exp(5))+exp(x)^2+( 8*x*log(2)-2)*exp(x)+16*x^2*log(2)^2-8*x*log(2)+1),x, algorithm="giac")
Output:
log(5)/(4*x*log(2) - e^(x*e^5 + 2*e^5) + e^x - 1)
Timed out. \[ \int \frac {-e^x \log (5)+e^{5+e^5 (2+x)} \log (5)-\log (5) \log (16)}{1+e^{2 x}+e^{2 e^5 (2+x)}-2 x \log (16)+x^2 \log ^2(16)+e^{e^5 (2+x)} \left (2-2 e^x-2 x \log (16)\right )+e^x (-2+2 x \log (16))} \, dx=\int -\frac {4\,\ln \left (2\right )\,\ln \left (5\right )+{\mathrm {e}}^x\,\ln \left (5\right )-{\mathrm {e}}^{{\mathrm {e}}^5\,\left (x+2\right )}\,{\mathrm {e}}^5\,\ln \left (5\right )}{{\mathrm {e}}^{2\,{\mathrm {e}}^5\,\left (x+2\right )}+{\mathrm {e}}^{2\,x}+16\,x^2\,{\ln \left (2\right )}^2-{\mathrm {e}}^{{\mathrm {e}}^5\,\left (x+2\right )}\,\left (2\,{\mathrm {e}}^x+8\,x\,\ln \left (2\right )-2\right )-8\,x\,\ln \left (2\right )+{\mathrm {e}}^x\,\left (8\,x\,\ln \left (2\right )-2\right )+1} \,d x \] Input:
int(-(4*log(2)*log(5) + exp(x)*log(5) - exp(exp(5)*(x + 2))*exp(5)*log(5)) /(exp(2*exp(5)*(x + 2)) + exp(2*x) + 16*x^2*log(2)^2 - exp(exp(5)*(x + 2)) *(2*exp(x) + 8*x*log(2) - 2) - 8*x*log(2) + exp(x)*(8*x*log(2) - 2) + 1),x )
Output:
int(-(4*log(2)*log(5) + exp(x)*log(5) - exp(exp(5)*(x + 2))*exp(5)*log(5)) /(exp(2*exp(5)*(x + 2)) + exp(2*x) + 16*x^2*log(2)^2 - exp(exp(5)*(x + 2)) *(2*exp(x) + 8*x*log(2) - 2) - 8*x*log(2) + exp(x)*(8*x*log(2) - 2) + 1), x)
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-e^x \log (5)+e^{5+e^5 (2+x)} \log (5)-\log (5) \log (16)}{1+e^{2 x}+e^{2 e^5 (2+x)}-2 x \log (16)+x^2 \log ^2(16)+e^{e^5 (2+x)} \left (2-2 e^x-2 x \log (16)\right )+e^x (-2+2 x \log (16))} \, dx=-\frac {\mathrm {log}\left (5\right )}{e^{e^{5} x +2 e^{5}}-e^{x}-4 \,\mathrm {log}\left (2\right ) x +1} \] Input:
int((exp(5)*log(5)*exp((2+x)*exp(5))-exp(x)*log(5)-4*log(2)*log(5))/(exp(( 2+x)*exp(5))^2+(-2*exp(x)-8*x*log(2)+2)*exp((2+x)*exp(5))+exp(x)^2+(8*x*lo g(2)-2)*exp(x)+16*x^2*log(2)^2-8*x*log(2)+1),x)
Output:
( - log(5))/(e**(e**5*x + 2*e**5) - e**x - 4*log(2)*x + 1)