Integrand size = 84, antiderivative size = 20 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (4 \left (4+\left (-5+5 e^{\frac {1}{\frac {1}{4}+x}}\right )^2\right )\right ) \] Output:
ln(4*(5*exp(1/(x+1/4))-5)^2+16)
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (29-50 e^{\frac {4}{1+4 x}}+25 e^{\frac {8}{1+4 x}}\right ) \] Input:
Integrate[(800*E^(4/(1 + 4*x)) - 800*E^(8/(1 + 4*x)))/(29 + 232*x + 464*x^ 2 + E^(4/(1 + 4*x))*(-50 - 400*x - 800*x^2) + E^(8/(1 + 4*x))*(25 + 200*x + 400*x^2)),x]
Output:
Log[29 - 50*E^(4/(1 + 4*x)) + 25*E^(8/(1 + 4*x))]
Time = 0.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {7292, 27, 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.
\(\displaystyle \int \frac {800 e^{\frac {4}{4 x+1}}-800 e^{\frac {8}{4 x+1}}}{464 x^2+e^{\frac {4}{4 x+1}} \left (-800 x^2-400 x-50\right )+e^{\frac {8}{4 x+1}} \left (400 x^2+200 x+25\right )+232 x+29} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {800 e^{\frac {4}{4 x+1}} \left (1-e^{\frac {4}{4 x+1}}\right )}{\left (-50 e^{\frac {4}{4 x+1}}+25 e^{\frac {8}{4 x+1}}+29\right ) (4 x+1)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 800 \int \frac {e^{\frac {4}{4 x+1}} \left (1-e^{\frac {4}{4 x+1}}\right )}{\left (29-50 e^{\frac {4}{4 x+1}}+25 e^{\frac {8}{4 x+1}}\right ) (4 x+1)^2}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (-50 e^{\frac {4}{4 x+1}}+25 e^{\frac {8}{4 x+1}}+29\right )\) |
Input:
Int[(800*E^(4/(1 + 4*x)) - 800*E^(8/(1 + 4*x)))/(29 + 232*x + 464*x^2 + E^ (4/(1 + 4*x))*(-50 - 400*x - 800*x^2) + E^(8/(1 + 4*x))*(25 + 200*x + 400* x^2)),x]
Output:
Log[29 - 50*E^(4/(1 + 4*x)) + 25*E^(8/(1 + 4*x))]
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_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30
method | result | size |
risch | \(\ln \left ({\mathrm e}^{\frac {8}{1+4 x}}-2 \,{\mathrm e}^{\frac {4}{1+4 x}}+\frac {29}{25}\right )\) | \(26\) |
parallelrisch | \(\ln \left ({\mathrm e}^{\frac {8}{1+4 x}}-2 \,{\mathrm e}^{\frac {4}{1+4 x}}+\frac {29}{25}\right )\) | \(28\) |
derivativedivides | \(\ln \left (25 \,{\mathrm e}^{\frac {8}{1+4 x}}-50 \,{\mathrm e}^{\frac {4}{1+4 x}}+29\right )\) | \(30\) |
default | \(\ln \left (25 \,{\mathrm e}^{\frac {8}{1+4 x}}-50 \,{\mathrm e}^{\frac {4}{1+4 x}}+29\right )\) | \(30\) |
norman | \(\ln \left (25 \,{\mathrm e}^{\frac {8}{1+4 x}}-50 \,{\mathrm e}^{\frac {4}{1+4 x}}+29\right )\) | \(30\) |
Input:
int((-800*exp(4/(1+4*x))^2+800*exp(4/(1+4*x)))/((400*x^2+200*x+25)*exp(4/( 1+4*x))^2+(-800*x^2-400*x-50)*exp(4/(1+4*x))+464*x^2+232*x+29),x,method=_R ETURNVERBOSE)
Output:
ln(exp(8/(1+4*x))-2*exp(4/(1+4*x))+29/25)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (25 \, e^{\left (\frac {8}{4 \, x + 1}\right )} - 50 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + 29\right ) \] Input:
integrate((-800*exp(4/(1+4*x))^2+800*exp(4/(1+4*x)))/((400*x^2+200*x+25)*e xp(4/(1+4*x))^2+(-800*x^2-400*x-50)*exp(4/(1+4*x))+464*x^2+232*x+29),x, al gorithm="fricas")
Output:
log(25*e^(8/(4*x + 1)) - 50*e^(4/(4*x + 1)) + 29)
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log {\left (e^{\frac {8}{4 x + 1}} - 2 e^{\frac {4}{4 x + 1}} + \frac {29}{25} \right )} \] Input:
integrate((-800*exp(4/(1+4*x))**2+800*exp(4/(1+4*x)))/((400*x**2+200*x+25) *exp(4/(1+4*x))**2+(-800*x**2-400*x-50)*exp(4/(1+4*x))+464*x**2+232*x+29), x)
Output:
log(exp(8/(4*x + 1)) - 2*exp(4/(4*x + 1)) + 29/25)
Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (e^{\left (\frac {8}{4 \, x + 1}\right )} - 2 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + \frac {29}{25}\right ) \] Input:
integrate((-800*exp(4/(1+4*x))^2+800*exp(4/(1+4*x)))/((400*x^2+200*x+25)*e xp(4/(1+4*x))^2+(-800*x^2-400*x-50)*exp(4/(1+4*x))+464*x^2+232*x+29),x, al gorithm="maxima")
Output:
log(e^(8/(4*x + 1)) - 2*e^(4/(4*x + 1)) + 29/25)
Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (25 \, e^{\left (\frac {8}{4 \, x + 1}\right )} - 50 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + 29\right ) \] Input:
integrate((-800*exp(4/(1+4*x))^2+800*exp(4/(1+4*x)))/((400*x^2+200*x+25)*e xp(4/(1+4*x))^2+(-800*x^2-400*x-50)*exp(4/(1+4*x))+464*x^2+232*x+29),x, al gorithm="giac")
Output:
log(25*e^(8/(4*x + 1)) - 50*e^(4/(4*x + 1)) + 29)
Time = 3.46 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\ln \left (25\,{\mathrm {e}}^{\frac {8}{4\,x+1}}-50\,{\mathrm {e}}^{\frac {4}{4\,x+1}}+29\right ) \] Input:
int((800*exp(4/(4*x + 1)) - 800*exp(8/(4*x + 1)))/(232*x + exp(8/(4*x + 1) )*(200*x + 400*x^2 + 25) - exp(4/(4*x + 1))*(400*x + 800*x^2 + 50) + 464*x ^2 + 29),x)
Output:
log(25*exp(8/(4*x + 1)) - 50*exp(4/(4*x + 1)) + 29)
Time = 0.23 (sec) , antiderivative size = 193, normalized size of antiderivative = 9.65 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\mathrm {log}\left (-e^{\frac {1}{4 x +1}} \sqrt {-\sqrt {\sqrt {29}+5}\, \sqrt {2}+2 \,29^{\frac {1}{4}}}\, 5^{\frac {1}{4}}+e^{\frac {2}{4 x +1}} \sqrt {5}+29^{\frac {1}{4}}\right )+\mathrm {log}\left (-e^{\frac {1}{4 x +1}} \sqrt {\sqrt {\sqrt {29}+5}\, \sqrt {2}+2 \,29^{\frac {1}{4}}}\, 5^{\frac {1}{4}}+e^{\frac {2}{4 x +1}} \sqrt {5}+29^{\frac {1}{4}}\right )+\mathrm {log}\left (e^{\frac {1}{4 x +1}} \sqrt {-\sqrt {\sqrt {29}+5}\, \sqrt {2}+2 \,29^{\frac {1}{4}}}\, 5^{\frac {1}{4}}+e^{\frac {2}{4 x +1}} \sqrt {5}+29^{\frac {1}{4}}\right )+\mathrm {log}\left (e^{\frac {1}{4 x +1}} \sqrt {\sqrt {\sqrt {29}+5}\, \sqrt {2}+2 \,29^{\frac {1}{4}}}\, 5^{\frac {1}{4}}+e^{\frac {2}{4 x +1}} \sqrt {5}+29^{\frac {1}{4}}\right ) \] Input:
int((-800*exp(4/(1+4*x))^2+800*exp(4/(1+4*x)))/((400*x^2+200*x+25)*exp(4/( 1+4*x))^2+(-800*x^2-400*x-50)*exp(4/(1+4*x))+464*x^2+232*x+29),x)
Output:
log( - e**(1/(4*x + 1))*sqrt( - sqrt(sqrt(29) + 5)*sqrt(2) + 2*29**(1/4))* 5**(1/4) + e**(2/(4*x + 1))*sqrt(5) + 29**(1/4)) + log( - e**(1/(4*x + 1)) *sqrt(sqrt(sqrt(29) + 5)*sqrt(2) + 2*29**(1/4))*5**(1/4) + e**(2/(4*x + 1) )*sqrt(5) + 29**(1/4)) + log(e**(1/(4*x + 1))*sqrt( - sqrt(sqrt(29) + 5)*s qrt(2) + 2*29**(1/4))*5**(1/4) + e**(2/(4*x + 1))*sqrt(5) + 29**(1/4)) + l og(e**(1/(4*x + 1))*sqrt(sqrt(sqrt(29) + 5)*sqrt(2) + 2*29**(1/4))*5**(1/4 ) + e**(2/(4*x + 1))*sqrt(5) + 29**(1/4))