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

Optimal. Leaf size=20 \[ \log \left (4 \left (4+\left (-5+5 e^{\frac {1}{\frac {1}{4}+x}}\right )^2\right )\right ) \]

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Rubi [A]  time = 0.30, antiderivative size = 29, normalized size of antiderivative = 1.45, number of steps used = 3, number of rules used = 3, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6741, 12, 6684} \begin {gather*} \log \left (-50 e^{\frac {4}{4 x+1}}+25 e^{\frac {8}{4 x+1}}+29\right ) \end {gather*}

Antiderivative was successfully verified.

[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 6684

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

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 29, normalized size = 1.45 \begin {gather*} \log \left (29-50 e^{\frac {4}{1+4 x}}+25 e^{\frac {8}{1+4 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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))]

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fricas [A]  time = 0.54, size = 27, normalized size = 1.35 \begin {gather*} \log \left (25 \, e^{\left (\frac {8}{4 \, x + 1}\right )} - 50 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + 29\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-800*exp(4/(4*x+1))^2+800*exp(4/(4*x+1)))/((400*x^2+200*x+25)*exp(4/(4*x+1))^2+(-800*x^2-400*x-50)*
exp(4/(4*x+1))+464*x^2+232*x+29),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 27, normalized size = 1.35 \begin {gather*} \log \left (25 \, e^{\left (\frac {8}{4 \, x + 1}\right )} - 50 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + 29\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-800*exp(4/(4*x+1))^2+800*exp(4/(4*x+1)))/((400*x^2+200*x+25)*exp(4/(4*x+1))^2+(-800*x^2-400*x-50)*
exp(4/(4*x+1))+464*x^2+232*x+29),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.11, size = 30, normalized size = 1.50

method	result	size
derivativedivides	\(\ln \left (25 \,{\mathrm e}^{\frac {8}{4 x +1}}-50 \,{\mathrm e}^{\frac {4}{4 x +1}}+29\right )\)	\(30\)
default	\(\ln \left (25 \,{\mathrm e}^{\frac {8}{4 x +1}}-50 \,{\mathrm e}^{\frac {4}{4 x +1}}+29\right )\)	\(30\)
norman	\(\ln \left (25 \,{\mathrm e}^{\frac {8}{4 x +1}}-50 \,{\mathrm e}^{\frac {4}{4 x +1}}+29\right )\)	\(30\)
risch	\(\frac {2}{x +\frac {1}{4}}-\frac {8}{4 x +1}+\ln \left ({\mathrm e}^{\frac {8}{4 x +1}}-2 \,{\mathrm e}^{\frac {4}{4 x +1}}+\frac {29}{25}\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-800*exp(4/(4*x+1))^2+800*exp(4/(4*x+1)))/((400*x^2+200*x+25)*exp(4/(4*x+1))^2+(-800*x^2-400*x-50)*exp(4/
(4*x+1))+464*x^2+232*x+29),x,method=_RETURNVERBOSE)

[Out]

ln(25*exp(4/(4*x+1))^2-50*exp(4/(4*x+1))+29)

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maxima [A]  time = 0.41, size = 25, normalized size = 1.25 \begin {gather*} \log \left (e^{\left (\frac {8}{4 \, x + 1}\right )} - 2 \, e^{\left (\frac {4}{4 \, x + 1}\right )} + \frac {29}{25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-800*exp(4/(4*x+1))^2+800*exp(4/(4*x+1)))/((400*x^2+200*x+25)*exp(4/(4*x+1))^2+(-800*x^2-400*x-50)*
exp(4/(4*x+1))+464*x^2+232*x+29),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.49, size = 27, normalized size = 1.35 \begin {gather*} \ln \left (25\,{\mathrm {e}}^{\frac {8}{4\,x+1}}-50\,{\mathrm {e}}^{\frac {4}{4\,x+1}}+29\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.36, size = 22, normalized size = 1.10 \begin {gather*} \log {\left (e^{\frac {8}{4 x + 1}} - 2 e^{\frac {4}{4 x + 1}} + \frac {29}{25} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-800*exp(4/(4*x+1))**2+800*exp(4/(4*x+1)))/((400*x**2+200*x+25)*exp(4/(4*x+1))**2+(-800*x**2-400*x-
50)*exp(4/(4*x+1))+464*x**2+232*x+29),x)

[Out]

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

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