3.1.25 \(\int \frac {108}{(-160-68 x+11 x^2) \log (\frac {e^{e^4} (80-10 x)}{20+11 x})} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\log \left (\frac {e^{e^4} (8-x)}{2+\frac {11 x}{10}}\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 2504} \begin {gather*} \log \left (\log \left (\frac {10 e^{e^4} (8-x)}{11 x+20}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[108/((-160 - 68*x + 11*x^2)*Log[(E^E^4*(80 - 10*x))/(20 + 11*x)]),x]

[Out]

Log[Log[(10*E^E^4*(8 - x))/(20 + 11*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 2504

Int[(u_)/Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)], x_Symbol] :> With[{h
= Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]])/(p*r*(b*c - a*d)), x] /
; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=108 \int \frac {1}{\left (-160-68 x+11 x^2\right ) \log \left (\frac {e^{e^4} (80-10 x)}{20+11 x}\right )} \, dx\\ &=\log \left (\log \left (\frac {10 e^{e^4} (8-x)}{20+11 x}\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 19, normalized size = 0.86 \begin {gather*} \log \left (\log \left (-\frac {10 e^{e^4} (-8+x)}{20+11 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[108/((-160 - 68*x + 11*x^2)*Log[(E^E^4*(80 - 10*x))/(20 + 11*x)]),x]

[Out]

Log[Log[(-10*E^E^4*(-8 + x))/(20 + 11*x)]]

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 17, normalized size = 0.77 \begin {gather*} \log \left (\log \left (-\frac {10 \, {\left (x - 8\right )} e^{\left (e^{4}\right )}}{11 \, x + 20}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(108/(11*x^2-68*x-160)/log((-10*x+80)*exp(exp(4))/(11*x+20)),x, algorithm="fricas")

[Out]

log(log(-10*(x - 8)*e^(e^4)/(11*x + 20)))

________________________________________________________________________________________

giac [A]  time = 0.45, size = 17, normalized size = 0.77 \begin {gather*} \log \left (e^{4} + \log \left (-\frac {10 \, {\left (x - 8\right )}}{11 \, x + 20}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(108/(11*x^2-68*x-160)/log((-10*x+80)*exp(exp(4))/(11*x+20)),x, algorithm="giac")

[Out]

log(e^4 + log(-10*(x - 8)/(11*x + 20)))

________________________________________________________________________________________

maple [A]  time = 0.27, size = 19, normalized size = 0.86

method	result	size
norman	\(\ln \left (\ln \left (\frac {\left (-10 x +80\right ) {\mathrm e}^{{\mathrm e}^{4}}}{11 x +20}\right )\right )\)	\(19\)
risch	\(\ln \left (\ln \left (\frac {\left (-10 x +80\right ) {\mathrm e}^{{\mathrm e}^{4}}}{11 x +20}\right )\right )\)	\(19\)
derivativedivides	\(\ln \left (\ln \left (-\frac {10 \,{\mathrm e}^{{\mathrm e}^{4}}}{11}+\frac {1080 \,{\mathrm e}^{{\mathrm e}^{4}}}{11 \left (11 x +20\right )}\right )\right )\)	\(21\)
default	\(\ln \left (\ln \left (-\frac {10 \,{\mathrm e}^{{\mathrm e}^{4}}}{11}+\frac {1080 \,{\mathrm e}^{{\mathrm e}^{4}}}{11 \left (11 x +20\right )}\right )\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(108/(11*x^2-68*x-160)/ln((-10*x+80)*exp(exp(4))/(11*x+20)),x,method=_RETURNVERBOSE)

[Out]

ln(ln((-10*x+80)*exp(exp(4))/(11*x+20)))

________________________________________________________________________________________

maxima [C]  time = 0.62, size = 29, normalized size = 1.32 \begin {gather*} \log \left (-i \, \pi - e^{4} - \log \relax (5) - \log \relax (2) + \log \left (11 \, x + 20\right ) - \log \left (x - 8\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(108/(11*x^2-68*x-160)/log((-10*x+80)*exp(exp(4))/(11*x+20)),x, algorithm="maxima")

[Out]

log(-I*pi - e^4 - log(5) - log(2) + log(11*x + 20) - log(x - 8))

________________________________________________________________________________________

mupad [B]  time = 3.20, size = 19, normalized size = 0.86 \begin {gather*} \ln \left ({\mathrm {e}}^4+\ln \left (-\frac {10\,x-80}{11\,x+20}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-108/(log(-(exp(exp(4))*(10*x - 80))/(11*x + 20))*(68*x - 11*x^2 + 160)),x)

[Out]

log(exp(4) + log(-(10*x - 80)/(11*x + 20)))

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 17, normalized size = 0.77 \begin {gather*} \log {\left (\log {\left (\frac {\left (80 - 10 x\right ) e^{e^{4}}}{11 x + 20} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(108/(11*x**2-68*x-160)/ln((-10*x+80)*exp(exp(4))/(11*x+20)),x)

[Out]

log(log((80 - 10*x)*exp(exp(4))/(11*x + 20)))

________________________________________________________________________________________