3.31.71 \(\int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+(4 e^{10}-163 e^5 x+1660 x^2+(-8 e^5+163 x) \log (4)+4 \log ^2(4)) \log (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)})}{26560 x^2-13280 x^3+1660 x^4+e^{10} (64-32 x+4 x^2)+e^5 (-2608 x+1304 x^2-163 x^3)+(2608 x-1304 x^2+163 x^3+e^5 (-128+64 x-8 x^2)) \log (4)+(64-32 x+4 x^2) \log ^2(4)} \, dx\) [3071]

Optimal. Leaf size=31 \[ \frac {\log \left (\frac {1}{5} \left (-4+\frac {3 x}{e^5-20 x-\log (4)}\right )\right )}{4-x} \]

[Out]

ln(3/5*x/(exp(5)-2*ln(2)-20*x)-4/5)/(4-x)

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(249\) vs. \(2(31)=62\).
time = 1.16, antiderivative size = 249, normalized size of antiderivative = 8.03, number of steps used = 9, number of rules used = 7, integrand size = 181, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6, 6820, 6874, 84, 2554, 2351, 31} \begin {gather*} \frac {3 \left (e^5-\log (4)\right ) \log (4-x)}{\left (80-e^5+\log (4)\right ) \left (332-4 e^5+\log (256)\right )}-\frac {\left (3 e^5-\log (64)\right ) \log \left (\frac {4-x}{-20 x+e^5-\log (4)}\right )}{\left (80-e^5+\log (4)\right ) \left (332-4 e^5+\log (256)\right )}-\frac {60 \left (e^5-\log (4)\right ) \log \left (-20 x+e^5-\log (4)\right )}{\left (80-e^5+\log (4)\right ) \left (3 e^5-\log (64)\right )}+\frac {249 \left (e^5-\log (4)\right ) \log \left (-83 x+4 e^5-\log (256)\right )}{\left (3 e^5-\log (64)\right ) \left (332-4 e^5+\log (256)\right )}-\frac {\left (-83 x+4 e^5-\log (256)\right ) \log \left (-\frac {-83 x+4 e^5-\log (256)}{5 \left (-20 x+e^5-\log (4)\right )}\right )}{(4-x) \left (332-4 e^5+\log (256)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^5*(-12 + 3*x) + (12 - 3*x)*Log[4] + (4*E^10 - 163*E^5*x + 1660*x^2 + (-8*E^5 + 163*x)*Log[4] + 4*Log[4]
^2)*Log[(4*E^5 - 83*x - 4*Log[4])/(-5*E^5 + 100*x + 5*Log[4])])/(26560*x^2 - 13280*x^3 + 1660*x^4 + E^10*(64 -
32*x + 4*x^2) + E^5*(-2608*x + 1304*x^2 - 163*x^3) + (2608*x - 1304*x^2 + 163*x^3 + E^5*(-128 + 64*x - 8*x^2)
)*Log[4] + (64 - 32*x + 4*x^2)*Log[4]^2),x]

[Out]

(3*(E^5 - Log[4])*Log[4 - x])/((80 - E^5 + Log[4])*(332 - 4*E^5 + Log[256])) - ((3*E^5 - Log[64])*Log[(4 - x)/
(E^5 - 20*x - Log[4])])/((80 - E^5 + Log[4])*(332 - 4*E^5 + Log[256])) - (60*(E^5 - Log[4])*Log[E^5 - 20*x - L
og[4]])/((80 - E^5 + Log[4])*(3*E^5 - Log[64])) + (249*(E^5 - Log[4])*Log[4*E^5 - 83*x - Log[256]])/((3*E^5 -
Log[64])*(332 - 4*E^5 + Log[256])) - ((4*E^5 - 83*x - Log[256])*Log[-1/5*(4*E^5 - 83*x - Log[256])/(E^5 - 20*x
- Log[4])])/((4 - x)*(332 - 4*E^5 + Log[256]))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2554

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.), x_Symbol] :> Dist[b*c - a*d, Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^
(m + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGt
Q[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rule 6820

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

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+\left (4 e^{10}-163 e^5 x+1660 x^2+\left (-8 e^5+163 x\right ) \log (4)+4 \log ^2(4)\right ) \log \left (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)}\right )}{26560 x^2-13280 x^3+1660 x^4+e^5 \left (-2608 x+1304 x^2-163 x^3\right )+\left (2608 x-1304 x^2+163 x^3+e^5 \left (-128+64 x-8 x^2\right )\right ) \log (4)+\left (64-32 x+4 x^2\right ) \left (e^{10}+\log ^2(4)\right )} \, dx\\ &=\int \frac {\frac {3 (-4+x) \left (e^5-\log (4)\right )}{4 e^{10}+1660 x^2+163 x \log (4)+4 \log ^2(4)-e^5 (163 x+8 \log (4))}+\log \left (\frac {-4 e^5+83 x+\log (256)}{5 \left (e^5-20 x-\log (4)\right )}\right )}{(4-x)^2} \, dx\\ &=\int \left (\frac {3 \left (e^5-\log (4)\right )}{(-4+x) \left (-e^5+20 x+\log (4)\right ) \left (-4 e^5+83 x+\log (256)\right )}+\frac {\log \left (\frac {-4 e^5+83 x+\log (256)}{5 \left (e^5-20 x-\log (4)\right )}\right )}{(-4+x)^2}\right ) \, dx\\ &=\left (3 \left (e^5-\log (4)\right )\right ) \int \frac {1}{(-4+x) \left (-e^5+20 x+\log (4)\right ) \left (-4 e^5+83 x+\log (256)\right )} \, dx+\int \frac {\log \left (\frac {-4 e^5+83 x+\log (256)}{5 \left (e^5-20 x-\log (4)\right )}\right )}{(-4+x)^2} \, dx\\ &=-\frac {\left (e^5-20 x-\log (4)\right ) \log \left (-\frac {4 e^5-83 x-\log (256)}{5 \left (e^5-20 x-\log (4)\right )}\right )}{(4-x) \left (80-e^5+\log (4)\right )}+\left (3 \left (e^5-\log (4)\right )\right ) \int \left (-\frac {400}{\left (-80+e^5-\log (4)\right ) \left (e^5-20 x-\log (4)\right ) \left (3 e^5-\log (64)\right )}+\frac {1}{(-4+x) \left (-80+e^5-\log (4)\right ) \left (-332+4 e^5-\log (256)\right )}+\frac {6889}{\left (3 e^5-\log (64)\right ) \left (-332+4 e^5-\log (256)\right ) \left (4 e^5-83 x-\log (256)\right )}\right ) \, dx-\frac {\left (3 e^5-\log (64)\right ) \int \frac {1}{(-4+x) \left (-4 e^5+83 x+\log (256)\right )} \, dx}{80-e^5+\log (4)}\\ &=\frac {3 \left (e^5-\log (4)\right ) \log (4-x)}{\left (80-e^5+\log (4)\right ) \left (332-4 e^5+\log (256)\right )}-\frac {60 \left (e^5-\log (4)\right ) \log \left (e^5-20 x-\log (4)\right )}{\left (80-e^5+\log (4)\right ) \left (3 e^5-\log (64)\right )}+\frac {249 \left (e^5-\log (4)\right ) \log \left (4 e^5-83 x-\log (256)\right )}{\left (3 e^5-\log (64)\right ) \left (332-4 e^5+\log (256)\right )}-\frac {\left (e^5-20 x-\log (4)\right ) \log \left (-\frac {4 e^5-83 x-\log (256)}{5 \left (e^5-20 x-\log (4)\right )}\right )}{(4-x) \left (80-e^5+\log (4)\right )}-\frac {\left (3 e^5-\log (64)\right ) \int \frac {1}{-4+x} \, dx}{\left (80-e^5+\log (4)\right ) \left (332-4 e^5+\log (256)\right )}+\frac {\left (83 \left (3 e^5-\log (64)\right )\right ) \int \frac {1}{-4 e^5+83 x+\log (256)} \, dx}{\left (80-e^5+\log (4)\right ) \left (332-4 e^5+\log (256)\right )}\\ &=\frac {3 \left (e^5-\log (4)\right ) \log (4-x)}{\left (80-e^5+\log (4)\right ) \left (332-4 e^5+\log (256)\right )}-\frac {\left (3 e^5-\log (64)\right ) \log (4-x)}{\left (80-e^5+\log (4)\right ) \left (332-4 e^5+\log (256)\right )}-\frac {60 \left (e^5-\log (4)\right ) \log \left (e^5-20 x-\log (4)\right )}{\left (80-e^5+\log (4)\right ) \left (3 e^5-\log (64)\right )}+\frac {249 \left (e^5-\log (4)\right ) \log \left (4 e^5-83 x-\log (256)\right )}{\left (3 e^5-\log (64)\right ) \left (332-4 e^5+\log (256)\right )}+\frac {\left (3 e^5-\log (64)\right ) \log \left (4 e^5-83 x-\log (256)\right )}{\left (80-e^5+\log (4)\right ) \left (332-4 e^5+\log (256)\right )}-\frac {\left (e^5-20 x-\log (4)\right ) \log \left (-\frac {4 e^5-83 x-\log (256)}{5 \left (e^5-20 x-\log (4)\right )}\right )}{(4-x) \left (80-e^5+\log (4)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.09, size = 35, normalized size = 1.13 \begin {gather*} -\frac {\log \left (\frac {-4 e^5+83 x+\log (256)}{5 e^5-100 x-\log (1024)}\right )}{-4+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(-12 + 3*x) + (12 - 3*x)*Log[4] + (4*E^10 - 163*E^5*x + 1660*x^2 + (-8*E^5 + 163*x)*Log[4] + 4*
Log[4]^2)*Log[(4*E^5 - 83*x - 4*Log[4])/(-5*E^5 + 100*x + 5*Log[4])])/(26560*x^2 - 13280*x^3 + 1660*x^4 + E^10
*(64 - 32*x + 4*x^2) + E^5*(-2608*x + 1304*x^2 - 163*x^3) + (2608*x - 1304*x^2 + 163*x^3 + E^5*(-128 + 64*x -
8*x^2))*Log[4] + (64 - 32*x + 4*x^2)*Log[4]^2),x]

[Out]

-(Log[(-4*E^5 + 83*x + Log[256])/(5*E^5 - 100*x - Log[1024])]/(-4 + x))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.88, size = 1439, normalized size = 46.42

method	result	size
norman	\(-\frac {\ln \left (\frac {-8 \ln \left (2\right )+4 \,{\mathrm e}^{5}-83 x}{10 \ln \left (2\right )-5 \,{\mathrm e}^{5}+100 x}\right )}{x -4}\)	\(36\)
risch	\(-\frac {\ln \left (\frac {-8 \ln \left (2\right )+4 \,{\mathrm e}^{5}-83 x}{10 \ln \left (2\right )-5 \,{\mathrm e}^{5}+100 x}\right )}{x -4}\)	\(36\)
derivativedivides	\(\text {Expression too large to display}\)	\(1439\)
default	\(\text {Expression too large to display}\)	\(1439\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*ln(2)^2+2*(-8*exp(5)+163*x)*ln(2)+4*exp(5)^2-163*x*exp(5)+1660*x^2)*ln((-8*ln(2)+4*exp(5)-83*x)/(10*l
n(2)-5*exp(5)+100*x))+2*(-3*x+12)*ln(2)+(3*x-12)*exp(5))/(4*(4*x^2-32*x+64)*ln(2)^2+2*((-8*x^2+64*x-128)*exp(5
)+163*x^3-1304*x^2+2608*x)*ln(2)+(4*x^2-32*x+64)*exp(5)^2+(-163*x^3+1304*x^2-2608*x)*exp(5)+1660*x^4-13280*x^3
+26560*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/20*(-300*exp(5)+600*ln(2))*(-1/120*(ln(1/4*(20*exp(5)*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*l
n(2)+20*x))-20*ln(2)^2*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))+16*exp(5)*ln(2)+800*exp(5)*(
-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-16*ln(2)^2-1600*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp
(5))/(-exp(5)+2*ln(2)+20*x))-5*exp(10)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))+652*exp(5)-1
304*ln(2)-4*exp(10)-12*(exp(5)^2-exp(10))^(1/2)-32000*(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))/(4*exp
(5)*ln(2)-4*ln(2)^2+163*exp(5)-326*ln(2)-exp(10)-3*(exp(5)^2-exp(10))^(1/2)-6640))-ln(1/4*(20*exp(5)*ln(2)*(-8
3/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-20*ln(2)^2*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)
+2*ln(2)+20*x))+16*exp(5)*ln(2)+800*exp(5)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-16*ln(2)
^2-1600*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-5*exp(10)*(-83/100+(3/50*ln(2)-3/100*
exp(5))/(-exp(5)+2*ln(2)+20*x))+652*exp(5)-1304*ln(2)-4*exp(10)+12*(exp(5)^2-exp(10))^(1/2)-32000*(3/50*ln(2)-
3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))/(4*exp(5)*ln(2)-4*ln(2)^2+163*exp(5)-326*ln(2)-exp(10)+3*(exp(5)^2-exp(1
0))^(1/2)-6640)))*ln(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))/(exp(5)^2-exp(10))^(1/2)-1/120/
(exp(5)^2-exp(10))^(1/2)*dilog(1/4*(20*exp(5)*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))
-20*ln(2)^2*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))+16*exp(5)*ln(2)+800*exp(5)*(-83/100+(3/
50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-16*ln(2)^2-1600*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(
5)+2*ln(2)+20*x))-5*exp(10)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))+652*exp(5)-1304*ln(2)-4
*exp(10)-12*(exp(5)^2-exp(10))^(1/2)-32000*(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))/(4*exp(5)*ln(2)-4
*ln(2)^2+163*exp(5)-326*ln(2)-exp(10)-3*(exp(5)^2-exp(10))^(1/2)-6640))+1/120/(exp(5)^2-exp(10))^(1/2)*dilog(1
/4*(20*exp(5)*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-20*ln(2)^2*(-83/100+(3/50*ln(2)
-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))+16*exp(5)*ln(2)+800*exp(5)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+
2*ln(2)+20*x))-16*ln(2)^2-1600*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-5*exp(10)*(-83
/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))+652*exp(5)-1304*ln(2)-4*exp(10)+12*(exp(5)^2-exp(10))^(
1/2)-32000*(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))/(4*exp(5)*ln(2)-4*ln(2)^2+163*exp(5)-326*ln(2)-ex
p(10)+3*(exp(5)^2-exp(10))^(1/2)-6640))-83/15/(exp(5)-2*ln(2))/(4*exp(5)-8*ln(2)-332)*ln(-83/100+(3/50*ln(2)-3
/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-2/(exp(5)-2*ln(2))/(4*exp(5)-8*ln(2)-332)/(5*exp(5)-10*ln(2)-400)*ln(5*ex
p(5)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))-10*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-
exp(5)+2*ln(2)+20*x))+4*exp(5)-8*ln(2)-400*(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))*ln(2)+1/(exp(5)-2
*ln(2))/(4*exp(5)-8*ln(2)-332)/(5*exp(5)-10*ln(2)-400)*ln(5*exp(5)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)
+2*ln(2)+20*x))-10*ln(2)*(-83/100+(3/50*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))+4*exp(5)-8*ln(2)-400*(3/50
*ln(2)-3/100*exp(5))/(-exp(5)+2*ln(2)+20*x))*exp(5))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1260 vs. \(2 (27) = 54\).
time = 0.64, size = 1260, normalized size = 40.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*log(2)^2+2*(-8*exp(5)+163*x)*log(2)+4*exp(5)^2-163*x*exp(5)+1660*x^2)*log((-8*log(2)+4*exp(5)-8
3*x)/(10*log(2)-5*exp(5)+100*x))+2*(-3*x+12)*log(2)+(3*x-12)*exp(5))/(4*(4*x^2-32*x+64)*log(2)^2+2*((-8*x^2+64
*x-128)*exp(5)+163*x^3-1304*x^2+2608*x)*log(2)+(4*x^2-32*x+64)*exp(5)^2+(-163*x^3+1304*x^2-2608*x)*exp(5)+1660
*x^4-13280*x^3+26560*x^2),x, algorithm="maxima")

[Out]

1/4*(3*(4*e^5*log(2) - 4*log(2)^2 - e^10 + 6640)*log(x - 4)/(16*(2*e^5 - 163)*log(2)^3 - 16*log(2)^4 - 12*(2*e
^10 - 326*e^5 + 13283)*log(2)^2 + 4*(2*e^15 - 489*e^10 + 39849*e^5 - 1082320)*log(2) - e^20 + 326*e^15 - 39849
*e^10 + 2164640*e^5 - 44089600) + 83*log(83*x - 4*e^5 + 8*log(2))/(4*(e^5 - 83)*log(2) - 4*log(2)^2 - e^10 + 1
66*e^5 - 6889) - 80*log(20*x - e^5 + 2*log(2))/(4*(e^5 - 80)*log(2) - 4*log(2)^2 - e^10 + 160*e^5 - 6400) + 12
/((2*(2*e^5 - 163)*log(2) - 4*log(2)^2 - e^10 + 163*e^5 - 6640)*x - 8*(2*e^5 - 163)*log(2) + 16*log(2)^2 + 4*e
^10 - 652*e^5 + 26560))*e^5 + 1/4*(3*(163*e^5 - 326*log(2) - 13280)*log(x - 4)/(16*(2*e^5 - 163)*log(2)^3 - 16
*log(2)^4 - 12*(2*e^10 - 326*e^5 + 13283)*log(2)^2 + 4*(2*e^15 - 489*e^10 + 39849*e^5 - 1082320)*log(2) - e^20
+ 326*e^15 - 39849*e^10 + 2164640*e^5 - 44089600) + 6889*log(83*x - 4*e^5 + 8*log(2))/(4*(3*e^5 - 166)*log(2)
^2 - 8*log(2)^3 - 2*(3*e^10 - 332*e^5 + 6889)*log(2) + e^15 - 166*e^10 + 6889*e^5) - 6400*log(20*x - e^5 + 2*l
og(2))/(4*(3*e^5 - 160)*log(2)^2 - 8*log(2)^3 - 2*(3*e^10 - 320*e^5 + 6400)*log(2) + e^15 - 160*e^10 + 6400*e^
5) - 12/((2*(2*e^5 - 163)*log(2) - 4*log(2)^2 - e^10 + 163*e^5 - 6640)*x - 8*(2*e^5 - 163)*log(2) + 16*log(2)^
2 + 4*e^10 - 652*e^5 + 26560))*e^5 - 1/2*(3*(4*e^5*log(2) - 4*log(2)^2 - e^10 + 6640)*log(x - 4)/(16*(2*e^5 -
163)*log(2)^3 - 16*log(2)^4 - 12*(2*e^10 - 326*e^5 + 13283)*log(2)^2 + 4*(2*e^15 - 489*e^10 + 39849*e^5 - 1082
320)*log(2) - e^20 + 326*e^15 - 39849*e^10 + 2164640*e^5 - 44089600) + 83*log(83*x - 4*e^5 + 8*log(2))/(4*(e^5
- 83)*log(2) - 4*log(2)^2 - e^10 + 166*e^5 - 6889) - 80*log(20*x - e^5 + 2*log(2))/(4*(e^5 - 80)*log(2) - 4*l
og(2)^2 - e^10 + 160*e^5 - 6400) + 12/((2*(2*e^5 - 163)*log(2) - 4*log(2)^2 - e^10 + 163*e^5 - 6640)*x - 8*(2*
e^5 - 163)*log(2) + 16*log(2)^2 + 4*e^10 - 652*e^5 + 26560))*log(2) - 1/2*(3*(163*e^5 - 326*log(2) - 13280)*lo
g(x - 4)/(16*(2*e^5 - 163)*log(2)^3 - 16*log(2)^4 - 12*(2*e^10 - 326*e^5 + 13283)*log(2)^2 + 4*(2*e^15 - 489*e
^10 + 39849*e^5 - 1082320)*log(2) - e^20 + 326*e^15 - 39849*e^10 + 2164640*e^5 - 44089600) + 6889*log(83*x - 4
*e^5 + 8*log(2))/(4*(3*e^5 - 166)*log(2)^2 - 8*log(2)^3 - 2*(3*e^10 - 332*e^5 + 6889)*log(2) + e^15 - 166*e^10
+ 6889*e^5) - 6400*log(20*x - e^5 + 2*log(2))/(4*(3*e^5 - 160)*log(2)^2 - 8*log(2)^3 - 2*(3*e^10 - 320*e^5 +
6400)*log(2) + e^15 - 160*e^10 + 6400*e^5) - 12/((2*(2*e^5 - 163)*log(2) - 4*log(2)^2 - e^10 + 163*e^5 - 6640)
*x - 8*(2*e^5 - 163)*log(2) + 16*log(2)^2 + 4*e^10 - 652*e^5 + 26560))*log(2) + 3/4*(e^5 - 2*log(2))*log(x - 4
)/((4*log(2) + 163)*e^5 - 4*log(2)^2 - e^10 - 326*log(2) - 6640) - 1/4*(16*log(5)*log(2)^2 - 4*(4*log(5)*log(2
) + 163*log(5))*e^5 + 4*e^10*log(5) + 1304*log(5)*log(2) - 4*(20*x*(e^5 - 2*log(2) - 83) + (4*log(2) + 83)*e^5
- 4*log(2)^2 - e^10 - 166*log(2))*log(20*x - e^5 + 2*log(2)) + (83*x*(e^5 - 2*log(2) - 80) + 16*(log(2) + 20)
*e^5 - 16*log(2)^2 - 4*e^10 - 640*log(2))*log(-83*x + 4*e^5 - 8*log(2)) + 26560*log(5))/(((4*log(2) + 163)*e^5
- 4*log(2)^2 - e^10 - 326*log(2) - 6640)*x - 4*(4*log(2) + 163)*e^5 + 16*log(2)^2 + 4*e^10 + 1304*log(2) + 26
560)

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Fricas [A]
time = 0.38, size = 36, normalized size = 1.16 \begin {gather*} -\frac {\log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right )}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*log(2)^2+2*(-8*exp(5)+163*x)*log(2)+4*exp(5)^2-163*x*exp(5)+1660*x^2)*log((-8*log(2)+4*exp(5)-8
3*x)/(10*log(2)-5*exp(5)+100*x))+2*(-3*x+12)*log(2)+(3*x-12)*exp(5))/(4*(4*x^2-32*x+64)*log(2)^2+2*((-8*x^2+64
*x-128)*exp(5)+163*x^3-1304*x^2+2608*x)*log(2)+(4*x^2-32*x+64)*exp(5)^2+(-163*x^3+1304*x^2-2608*x)*exp(5)+1660
*x^4-13280*x^3+26560*x^2),x, algorithm="fricas")

[Out]

-log(-1/5*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 + 2*log(2)))/(x - 4)

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Sympy [A]
time = 0.18, size = 32, normalized size = 1.03 \begin {gather*} - \frac {\log {\left (\frac {- 83 x - 8 \log {\left (2 \right )} + 4 e^{5}}{100 x - 5 e^{5} + 10 \log {\left (2 \right )}} \right )}}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*ln(2)**2+2*(-8*exp(5)+163*x)*ln(2)+4*exp(5)**2-163*x*exp(5)+1660*x**2)*ln((-8*ln(2)+4*exp(5)-83
*x)/(10*ln(2)-5*exp(5)+100*x))+2*(-3*x+12)*ln(2)+(3*x-12)*exp(5))/(4*(4*x**2-32*x+64)*ln(2)**2+2*((-8*x**2+64*
x-128)*exp(5)+163*x**3-1304*x**2+2608*x)*ln(2)+(4*x**2-32*x+64)*exp(5)**2+(-163*x**3+1304*x**2-2608*x)*exp(5)+
1660*x**4-13280*x**3+26560*x**2),x)

[Out]

-log((-83*x - 8*log(2) + 4*exp(5))/(100*x - 5*exp(5) + 10*log(2)))/(x - 4)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (27) = 54\).
time = 0.79, size = 295, normalized size = 9.52 \begin {gather*} -\frac {\frac {20 \, {\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )} e^{5} \log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right )}{20 \, x - e^{5} + 2 \, \log \left (2\right )} - 83 \, e^{5} \log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right ) - \frac {40 \, {\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )} \log \left (2\right ) \log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right )}{20 \, x - e^{5} + 2 \, \log \left (2\right )} + 166 \, \log \left (2\right ) \log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right )}{{\left (\frac {{\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )} e^{5}}{20 \, x - e^{5} + 2 \, \log \left (2\right )} - \frac {2 \, {\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )} \log \left (2\right )}{20 \, x - e^{5} + 2 \, \log \left (2\right )} - \frac {80 \, {\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )}}{20 \, x - e^{5} + 2 \, \log \left (2\right )} - 4 \, e^{5} + 8 \, \log \left (2\right ) + 332\right )} {\left (e^{5} - 2 \, \log \left (2\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*log(2)^2+2*(-8*exp(5)+163*x)*log(2)+4*exp(5)^2-163*x*exp(5)+1660*x^2)*log((-8*log(2)+4*exp(5)-8
3*x)/(10*log(2)-5*exp(5)+100*x))+2*(-3*x+12)*log(2)+(3*x-12)*exp(5))/(4*(4*x^2-32*x+64)*log(2)^2+2*((-8*x^2+64
*x-128)*exp(5)+163*x^3-1304*x^2+2608*x)*log(2)+(4*x^2-32*x+64)*exp(5)^2+(-163*x^3+1304*x^2-2608*x)*exp(5)+1660
*x^4-13280*x^3+26560*x^2),x, algorithm="giac")

[Out]

-(20*(83*x - 4*e^5 + 8*log(2))*e^5*log(-1/5*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 + 2*log(2)))/(20*x - e^5 + 2
*log(2)) - 83*e^5*log(-1/5*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 + 2*log(2))) - 40*(83*x - 4*e^5 + 8*log(2))*l
og(2)*log(-1/5*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 + 2*log(2)))/(20*x - e^5 + 2*log(2)) + 166*log(2)*log(-1/
5*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 + 2*log(2))))/(((83*x - 4*e^5 + 8*log(2))*e^5/(20*x - e^5 + 2*log(2))
- 2*(83*x - 4*e^5 + 8*log(2))*log(2)/(20*x - e^5 + 2*log(2)) - 80*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 + 2*lo
g(2)) - 4*e^5 + 8*log(2) + 332)*(e^5 - 2*log(2)))

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Mupad [B]
time = 2.63, size = 36, normalized size = 1.16 \begin {gather*} -\frac {\ln \left (-\frac {83\,x-4\,{\mathrm {e}}^5+8\,\ln \left (2\right )}{5\,\left (20\,x-{\mathrm {e}}^5+2\,\ln \left (2\right )\right )}\right )}{x-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(83*x - 4*exp(5) + 8*log(2))/(100*x - 5*exp(5) + 10*log(2)))*(4*exp(10) - 163*x*exp(5) + 2*log(2)*(1
63*x - 8*exp(5)) + 16*log(2)^2 + 1660*x^2) - 2*log(2)*(3*x - 12) + exp(5)*(3*x - 12))/(exp(10)*(4*x^2 - 32*x +
64) - exp(5)*(2608*x - 1304*x^2 + 163*x^3) + 4*log(2)^2*(4*x^2 - 32*x + 64) + 2*log(2)*(2608*x - exp(5)*(8*x^
2 - 64*x + 128) - 1304*x^2 + 163*x^3) + 26560*x^2 - 13280*x^3 + 1660*x^4),x)

[Out]

-log(-(83*x - 4*exp(5) + 8*log(2))/(5*(20*x - exp(5) + 2*log(2))))/(x - 4)

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