\(\int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} (-50-400 x-800 x^2)+e^{\frac {8}{1+4 x}} (25+200 x+400 x^2)} \, dx\) [7719]

Optimal result

Integrand size = 84, antiderivative size = 20 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (4 \left (4+\left (-5+5 e^{\frac {1}{\frac {1}{4}+x}}\right )^2\right )\right ) \]

[Out]

ln(4*(5*exp(1/(x+1/4))-5)^2+16)

Rubi [A] (verified)

Time = 0.20 (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 = {6873, 12, 6816} \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (-50 e^{\frac {4}{4 x+1}}+25 e^{\frac {8}{4 x+1}}+29\right ) \]

[In]

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]

[Out]

Log[29 - 50*E^(4/(1 + 4*x)) + 25*E^(8/(1 + 4*x))]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps \begin{align*} \text {integral}& = \int \frac {800 e^{\frac {4}{1+4 x}} \left (1-e^{\frac {4}{1+4 x}}\right )}{\left (29-50 e^{\frac {4}{1+4 x}}+25 e^{\frac {8}{1+4 x}}\right ) (1+4 x)^2} \, dx \\ & = 800 \int \frac {e^{\frac {4}{1+4 x}} \left (1-e^{\frac {4}{1+4 x}}\right )}{\left (29-50 e^{\frac {4}{1+4 x}}+25 e^{\frac {8}{1+4 x}}\right ) (1+4 x)^2} \, dx \\ & = \log \left (29-50 e^{\frac {4}{1+4 x}}+25 e^{\frac {8}{1+4 x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (29-50 e^{\frac {4}{1+4 x}}+25 e^{\frac {8}{1+4 x}}\right ) \]

[In]

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 - 8
00*x^2) + E^(8/(1 + 4*x))*(25 + 200*x + 400*x^2)),x]

[Out]

Log[29 - 50*E^(4/(1 + 4*x)) + 25*E^(8/(1 + 4*x))]

Maple [A] (verified)

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	\(\ln \left (25 \,{\mathrm e}^{\frac {8}{1+4 x}}-50 \,{\mathrm e}^{\frac {4}{1+4 x}}+29\right )\)	\(30\)

[In]

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=_RETURNVERBOSE)

[Out]

ln(exp(8/(1+4*x))-2*exp(4/(1+4*x))+29/25)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (25 \, e^{\left (\frac {8}{4 \, x + 1}\right )} - 50 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + 29\right ) \]

[In]

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, algorithm="fricas")

[Out]

log(25*e^(8/(4*x + 1)) - 50*e^(4/(4*x + 1)) + 29)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log {\left (e^{\frac {8}{4 x + 1}} - 2 e^{\frac {4}{4 x + 1}} + \frac {29}{25} \right )} \]

[In]

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)

[Out]

log(exp(8/(4*x + 1)) - 2*exp(4/(4*x + 1)) + 29/25)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (e^{\left (\frac {8}{4 \, x + 1}\right )} - 2 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + \frac {29}{25}\right ) \]

[In]

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, algorithm="maxima")

[Out]

log(e^(8/(4*x + 1)) - 2*e^(4/(4*x + 1)) + 29/25)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\log \left (25 \, e^{\left (\frac {8}{4 \, x + 1}\right )} - 50 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + 29\right ) \]

[In]

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, algorithm="giac")

[Out]

log(25*e^(8/(4*x + 1)) - 50*e^(4/(4*x + 1)) + 29)

Mupad [B] (verification not implemented)

Time = 13.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {800 e^{\frac {4}{1+4 x}}-800 e^{\frac {8}{1+4 x}}}{29+232 x+464 x^2+e^{\frac {4}{1+4 x}} \left (-50-400 x-800 x^2\right )+e^{\frac {8}{1+4 x}} \left (25+200 x+400 x^2\right )} \, dx=\ln \left (25\,{\mathrm {e}}^{\frac {8}{4\,x+1}}-50\,{\mathrm {e}}^{\frac {4}{4\,x+1}}+29\right ) \]

[In]

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)

[Out]

log(25*exp(8/(4*x + 1)) - 50*exp(4/(4*x + 1)) + 29)