3.36.80 \(\int \frac {220 x^3+(27500 x-33000 e^4 x+9900 e^8 x+1090 x^2) \log (3)+(-1250+1500 e^4-450 e^8-50 x) \log ^2(3)+(2 x^3+(250 x-300 e^4 x+90 e^8 x+10 x^2) \log (3)) \log (25-30 e^4+9 e^8+x)}{625 x^3-750 e^4 x^3+225 e^8 x^3+25 x^4} \, dx\) [3580]

Optimal. Leaf size=26 \[ \left (-22+\frac {\log (3)}{x}-\frac {1}{5} \log \left (\left (5-3 e^4\right )^2+x\right )\right )^2 \]

[Out]

(ln(3)/x-22-1/5*ln((5-3*exp(4))^2+x))^2

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(26)=52\).
time = 1.42, antiderivative size = 121, normalized size of antiderivative = 4.65, number of steps used = 17, number of rules used = 13, integrand size = 125, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {6, 1607, 6820, 12, 6874, 1626, 2465, 2437, 2338, 2442, 36, 29, 31} \begin {gather*} \frac {\log ^2(3)}{x^2}+\frac {1}{25} \log ^2\left (x+\left (5-3 e^4\right )^2\right )+\frac {2 \left (550-660 e^4+198 e^8+\log (3)\right ) \log \left (x+\left (5-3 e^4\right )^2\right )}{5 \left (5-3 e^4\right )^2}-\frac {2 \log (3) \log \left (x+\left (5-3 e^4\right )^2\right )}{5 x}-\frac {2 \log (3) \log \left (x+\left (5-3 e^4\right )^2\right )}{5 \left (5-3 e^4\right )^2}-\frac {44 \log (3)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(220*x^3 + (27500*x - 33000*E^4*x + 9900*E^8*x + 1090*x^2)*Log[3] + (-1250 + 1500*E^4 - 450*E^8 - 50*x)*Lo
g[3]^2 + (2*x^3 + (250*x - 300*E^4*x + 90*E^8*x + 10*x^2)*Log[3])*Log[25 - 30*E^4 + 9*E^8 + x])/(625*x^3 - 750
*E^4*x^3 + 225*E^8*x^3 + 25*x^4),x]

[Out]

(-44*Log[3])/x + Log[3]^2/x^2 - (2*Log[3]*Log[(5 - 3*E^4)^2 + x])/(5*(5 - 3*E^4)^2) - (2*Log[3]*Log[(5 - 3*E^4
)^2 + x])/(5*x) + (2*(550 - 660*E^4 + 198*E^8 + Log[3])*Log[(5 - 3*E^4)^2 + x])/(5*(5 - 3*E^4)^2) + Log[(5 - 3
*E^4)^2 + x]^2/25

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 12

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1626

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2437

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

Rule 2442

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

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 {220 x^3+\left (27500 x-33000 e^4 x+9900 e^8 x+1090 x^2\right ) \log (3)+\left (-1250+1500 e^4-450 e^8-50 x\right ) \log ^2(3)+\left (2 x^3+\left (250 x-300 e^4 x+90 e^8 x+10 x^2\right ) \log (3)\right ) \log \left (25-30 e^4+9 e^8+x\right )}{225 e^8 x^3+\left (625-750 e^4\right ) x^3+25 x^4} \, dx\\ &=\int \frac {220 x^3+\left (27500 x-33000 e^4 x+9900 e^8 x+1090 x^2\right ) \log (3)+\left (-1250+1500 e^4-450 e^8-50 x\right ) \log ^2(3)+\left (2 x^3+\left (250 x-300 e^4 x+90 e^8 x+10 x^2\right ) \log (3)\right ) \log \left (25-30 e^4+9 e^8+x\right )}{\left (625-750 e^4+225 e^8\right ) x^3+25 x^4} \, dx\\ &=\int \frac {220 x^3+\left (27500 x-33000 e^4 x+9900 e^8 x+1090 x^2\right ) \log (3)+\left (-1250+1500 e^4-450 e^8-50 x\right ) \log ^2(3)+\left (2 x^3+\left (250 x-300 e^4 x+90 e^8 x+10 x^2\right ) \log (3)\right ) \log \left (25-30 e^4+9 e^8+x\right )}{x^3 \left (625-750 e^4+225 e^8+25 x\right )} \, dx\\ &=\int \frac {2 \left (x^2+5 \left (5-3 e^4\right )^2 \log (3)+5 x \log (3)\right ) \left (110 x-5 \log (3)+x \log \left (25-30 e^4+9 e^8+x\right )\right )}{25 x^3 \left (25-30 e^4+9 e^8+x\right )} \, dx\\ &=\frac {2}{25} \int \frac {\left (x^2+5 \left (5-3 e^4\right )^2 \log (3)+5 x \log (3)\right ) \left (110 x-5 \log (3)+x \log \left (25-30 e^4+9 e^8+x\right )\right )}{x^3 \left (25-30 e^4+9 e^8+x\right )} \, dx\\ &=\frac {2}{25} \int \left (\frac {5 (22 x-\log (3)) \left (x^2+5 \left (5-3 e^4\right )^2 \log (3)+5 x \log (3)\right )}{x^3 \left (25-30 e^4+9 e^8+x\right )}+\frac {\left (x^2+5 \left (5-3 e^4\right )^2 \log (3)+5 x \log (3)\right ) \log \left (25-30 e^4+9 e^8+x\right )}{x^2 \left (25-30 e^4+9 e^8+x\right )}\right ) \, dx\\ &=\frac {2}{25} \int \frac {\left (x^2+5 \left (5-3 e^4\right )^2 \log (3)+5 x \log (3)\right ) \log \left (25-30 e^4+9 e^8+x\right )}{x^2 \left (25-30 e^4+9 e^8+x\right )} \, dx+\frac {2}{5} \int \frac {(22 x-\log (3)) \left (x^2+5 \left (5-3 e^4\right )^2 \log (3)+5 x \log (3)\right )}{x^3 \left (25-30 e^4+9 e^8+x\right )} \, dx\\ &=\frac {2}{25} \int \left (\frac {\log \left (25-30 e^4+9 e^8+x\right )}{25-30 e^4+9 e^8+x}+\frac {5 \log (3) \log \left (25-30 e^4+9 e^8+x\right )}{x^2}\right ) \, dx+\frac {2}{5} \int \left (\frac {110 \log (3)}{x^2}-\frac {\log (3)}{\left (-5+3 e^4\right )^2 x}-\frac {5 \log ^2(3)}{x^3}+\frac {550-660 e^4+198 e^8+\log (3)}{\left (-5+3 e^4\right )^2 \left (25-30 e^4+9 e^8+x\right )}\right ) \, dx\\ &=-\frac {44 \log (3)}{x}+\frac {\log ^2(3)}{x^2}-\frac {2 \log (3) \log (x)}{5 \left (5-3 e^4\right )^2}+\frac {2 \left (550-660 e^4+198 e^8+\log (3)\right ) \log \left (\left (5-3 e^4\right )^2+x\right )}{5 \left (5-3 e^4\right )^2}+\frac {2}{25} \int \frac {\log \left (25-30 e^4+9 e^8+x\right )}{25-30 e^4+9 e^8+x} \, dx+\frac {1}{5} (2 \log (3)) \int \frac {\log \left (25-30 e^4+9 e^8+x\right )}{x^2} \, dx\\ &=-\frac {44 \log (3)}{x}+\frac {\log ^2(3)}{x^2}-\frac {2 \log (3) \log (x)}{5 \left (5-3 e^4\right )^2}-\frac {2 \log (3) \log \left (\left (5-3 e^4\right )^2+x\right )}{5 x}+\frac {2 \left (550-660 e^4+198 e^8+\log (3)\right ) \log \left (\left (5-3 e^4\right )^2+x\right )}{5 \left (5-3 e^4\right )^2}+\frac {2}{25} \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,25-30 e^4+9 e^8+x\right )+\frac {1}{5} (2 \log (3)) \int \frac {1}{x \left (25-30 e^4+9 e^8+x\right )} \, dx\\ &=-\frac {44 \log (3)}{x}+\frac {\log ^2(3)}{x^2}-\frac {2 \log (3) \log (x)}{5 \left (5-3 e^4\right )^2}-\frac {2 \log (3) \log \left (\left (5-3 e^4\right )^2+x\right )}{5 x}+\frac {2 \left (550-660 e^4+198 e^8+\log (3)\right ) \log \left (\left (5-3 e^4\right )^2+x\right )}{5 \left (5-3 e^4\right )^2}+\frac {1}{25} \log ^2\left (\left (5-3 e^4\right )^2+x\right )+\frac {(2 \log (3)) \int \frac {1}{x} \, dx}{5 \left (5-3 e^4\right )^2}-\frac {(2 \log (3)) \int \frac {1}{25-30 e^4+9 e^8+x} \, dx}{5 \left (5-3 e^4\right )^2}\\ &=-\frac {44 \log (3)}{x}+\frac {\log ^2(3)}{x^2}-\frac {2 \log (3) \log \left (\left (5-3 e^4\right )^2+x\right )}{5 \left (5-3 e^4\right )^2}-\frac {2 \log (3) \log \left (\left (5-3 e^4\right )^2+x\right )}{5 x}+\frac {2 \left (550-660 e^4+198 e^8+\log (3)\right ) \log \left (\left (5-3 e^4\right )^2+x\right )}{5 \left (5-3 e^4\right )^2}+\frac {1}{25} \log ^2\left (\left (5-3 e^4\right )^2+x\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(26)=52\).
time = 0.13, size = 80, normalized size = 3.08 \begin {gather*} \frac {2}{25} \left (-\frac {550 \log (3)}{x}+\frac {25 \log ^2(3)}{2 x^2}+110 \log \left (25-30 e^4+9 e^8+x\right )-\frac {5 \log (3) \log \left (25-30 e^4+9 e^8+x\right )}{x}+\frac {1}{2} \log ^2\left (25-30 e^4+9 e^8+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(220*x^3 + (27500*x - 33000*E^4*x + 9900*E^8*x + 1090*x^2)*Log[3] + (-1250 + 1500*E^4 - 450*E^8 - 50
*x)*Log[3]^2 + (2*x^3 + (250*x - 300*E^4*x + 90*E^8*x + 10*x^2)*Log[3])*Log[25 - 30*E^4 + 9*E^8 + x])/(625*x^3
- 750*E^4*x^3 + 225*E^8*x^3 + 25*x^4),x]

[Out]

(2*((-550*Log[3])/x + (25*Log[3]^2)/(2*x^2) + 110*Log[25 - 30*E^4 + 9*E^8 + x] - (5*Log[3]*Log[25 - 30*E^4 + 9
*E^8 + x])/x + Log[25 - 30*E^4 + 9*E^8 + x]^2/2))/25

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(210\) vs. \(2(23)=46\).
time = 1.10, size = 211, normalized size = 8.12

method	result	size
risch	\(\frac {\ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )^{2}}{25}-\frac {2 \ln \left (3\right ) \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )}{5 x}+\frac {44 x^{2} \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )+5 \ln \left (3\right )^{2}-220 x \ln \left (3\right )}{5 x^{2}}\)	\(71\)
norman	\(\frac {\ln \left (3\right )^{2}+\frac {44 x^{2} \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )}{5}-44 x \ln \left (3\right )+\frac {x^{2} \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )^{2}}{25}-\frac {2 \ln \left (3\right ) \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right ) x}{5}}{x^{2}}\)	\(74\)
derivativedivides	\(\frac {2 \ln \left (3\right ) \ln \left (-x \right )}{5 \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+25\right )}-\frac {2 \ln \left (3\right ) \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right ) \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )}{5 \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+25\right ) x}-\frac {44 \ln \left (3\right )}{x}-\frac {2 \ln \left (3\right ) \ln \left (x \right )}{5 \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+25\right )}+\frac {2 \ln \left (3\right ) \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )}{5 \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+25\right )}+\frac {44 \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )}{5}+\frac {\ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )^{2}}{25}+\frac {\ln \left (3\right )^{2}}{x^{2}}\)	\(211\)
default	\(\frac {2 \ln \left (3\right ) \ln \left (-x \right )}{5 \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+25\right )}-\frac {2 \ln \left (3\right ) \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right ) \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )}{5 \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+25\right ) x}-\frac {44 \ln \left (3\right )}{x}-\frac {2 \ln \left (3\right ) \ln \left (x \right )}{5 \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+25\right )}+\frac {2 \ln \left (3\right ) \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )}{5 \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+25\right )}+\frac {44 \ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )}{5}+\frac {\ln \left (9 \,{\mathrm e}^{8}-30 \,{\mathrm e}^{4}+x +25\right )^{2}}{25}+\frac {\ln \left (3\right )^{2}}{x^{2}}\)	\(211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((90*x*exp(4)^2-300*x*exp(4)+10*x^2+250*x)*ln(3)+2*x^3)*ln(9*exp(4)^2-30*exp(4)+x+25)+(-450*exp(4)^2+1500
*exp(4)-50*x-1250)*ln(3)^2+(9900*x*exp(4)^2-33000*x*exp(4)+1090*x^2+27500*x)*ln(3)+220*x^3)/(225*x^3*exp(4)^2-
750*x^3*exp(4)+25*x^4+625*x^3),x,method=_RETURNVERBOSE)

[Out]

2/5*ln(3)/(9*exp(8)-30*exp(4)+25)*ln(9*exp(8)-9*exp(4)^2-x)+2/5*ln(3)*ln(9*exp(4)^2-30*exp(4)+x+25)*(9*exp(4)^
2-30*exp(4)+x+25)/(9*exp(8)-30*exp(4)+25)/(9*exp(8)-9*exp(4)^2-x)-44*ln(3)/(-9*exp(8)+9*exp(4)^2+x)-2/5*ln(3)/
(9*exp(8)-30*exp(4)+25)*ln(-9*exp(8)+9*exp(4)^2+x)+2/5*ln(3)/(9*exp(8)-30*exp(4)+25)*ln(9*exp(4)^2-30*exp(4)+x
+25)+44/5*ln(9*exp(4)^2-30*exp(4)+x+25)+1/25*ln(9*exp(4)^2-30*exp(4)+x+25)^2+ln(3)^2/x^2

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 841 vs. \(2 (26) = 52\).
time = 0.53, size = 841, normalized size = 32.35 \begin {gather*} 9 \, {\left (\frac {2 \, \log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{729 \, e^{24} - 7290 \, e^{20} + 30375 \, e^{16} - 67500 \, e^{12} + 84375 \, e^{8} - 56250 \, e^{4} + 15625} - \frac {2 \, \log \left (x\right )}{729 \, e^{24} - 7290 \, e^{20} + 30375 \, e^{16} - 67500 \, e^{12} + 84375 \, e^{8} - 56250 \, e^{4} + 15625} - \frac {2 \, x - 9 \, e^{8} + 30 \, e^{4} - 25}{x^{2} {\left (81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625\right )}}\right )} e^{8} \log \left (3\right )^{2} - 30 \, {\left (\frac {2 \, \log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{729 \, e^{24} - 7290 \, e^{20} + 30375 \, e^{16} - 67500 \, e^{12} + 84375 \, e^{8} - 56250 \, e^{4} + 15625} - \frac {2 \, \log \left (x\right )}{729 \, e^{24} - 7290 \, e^{20} + 30375 \, e^{16} - 67500 \, e^{12} + 84375 \, e^{8} - 56250 \, e^{4} + 15625} - \frac {2 \, x - 9 \, e^{8} + 30 \, e^{4} - 25}{x^{2} {\left (81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625\right )}}\right )} e^{4} \log \left (3\right )^{2} + 396 \, {\left (\frac {\log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625} - \frac {\log \left (x\right )}{81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625} - \frac {1}{x {\left (9 \, e^{8} - 30 \, e^{4} + 25\right )}}\right )} e^{8} \log \left (3\right ) - 1320 \, {\left (\frac {\log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625} - \frac {\log \left (x\right )}{81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625} - \frac {1}{x {\left (9 \, e^{8} - 30 \, e^{4} + 25\right )}}\right )} e^{4} \log \left (3\right ) + 25 \, {\left (\frac {2 \, \log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{729 \, e^{24} - 7290 \, e^{20} + 30375 \, e^{16} - 67500 \, e^{12} + 84375 \, e^{8} - 56250 \, e^{4} + 15625} - \frac {2 \, \log \left (x\right )}{729 \, e^{24} - 7290 \, e^{20} + 30375 \, e^{16} - 67500 \, e^{12} + 84375 \, e^{8} - 56250 \, e^{4} + 15625} - \frac {2 \, x - 9 \, e^{8} + 30 \, e^{4} - 25}{x^{2} {\left (81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625\right )}}\right )} \log \left (3\right )^{2} - 2 \, {\left (\frac {\log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625} - \frac {\log \left (x\right )}{81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625} - \frac {1}{x {\left (9 \, e^{8} - 30 \, e^{4} + 25\right )}}\right )} \log \left (3\right )^{2} + 1100 \, {\left (\frac {\log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625} - \frac {\log \left (x\right )}{81 \, e^{16} - 540 \, e^{12} + 1350 \, e^{8} - 1500 \, e^{4} + 625} - \frac {1}{x {\left (9 \, e^{8} - 30 \, e^{4} + 25\right )}}\right )} \log \left (3\right ) - \frac {218}{5} \, {\left (\frac {\log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{9 \, e^{8} - 30 \, e^{4} + 25} - \frac {\log \left (x\right )}{9 \, e^{8} - 30 \, e^{4} + 25}\right )} \log \left (3\right ) + \frac {2 \, \log \left (3\right ) \log \left (x\right )}{5 \, {\left (9 \, e^{8} - 30 \, e^{4} + 25\right )}} + \frac {x {\left (9 \, e^{8} - 30 \, e^{4} + 25\right )} \log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )^{2} - 10 \, {\left (x \log \left (3\right ) + 9 \, e^{8} \log \left (3\right ) - 30 \, e^{4} \log \left (3\right ) + 25 \, \log \left (3\right )\right )} \log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{25 \, x {\left (9 \, e^{8} - 30 \, e^{4} + 25\right )}} + \frac {44}{5} \, \log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((90*x*exp(4)^2-300*x*exp(4)+10*x^2+250*x)*log(3)+2*x^3)*log(9*exp(4)^2-30*exp(4)+x+25)+(-450*exp(4
)^2+1500*exp(4)-50*x-1250)*log(3)^2+(9900*x*exp(4)^2-33000*x*exp(4)+1090*x^2+27500*x)*log(3)+220*x^3)/(225*x^3
*exp(4)^2-750*x^3*exp(4)+25*x^4+625*x^3),x, algorithm="maxima")

[Out]

9*(2*log(x + 9*e^8 - 30*e^4 + 25)/(729*e^24 - 7290*e^20 + 30375*e^16 - 67500*e^12 + 84375*e^8 - 56250*e^4 + 15
625) - 2*log(x)/(729*e^24 - 7290*e^20 + 30375*e^16 - 67500*e^12 + 84375*e^8 - 56250*e^4 + 15625) - (2*x - 9*e^
8 + 30*e^4 - 25)/(x^2*(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 + 625)))*e^8*log(3)^2 - 30*(2*log(x + 9*e^8 -
30*e^4 + 25)/(729*e^24 - 7290*e^20 + 30375*e^16 - 67500*e^12 + 84375*e^8 - 56250*e^4 + 15625) - 2*log(x)/(729*
e^24 - 7290*e^20 + 30375*e^16 - 67500*e^12 + 84375*e^8 - 56250*e^4 + 15625) - (2*x - 9*e^8 + 30*e^4 - 25)/(x^2
*(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 + 625)))*e^4*log(3)^2 + 396*(log(x + 9*e^8 - 30*e^4 + 25)/(81*e^16
- 540*e^12 + 1350*e^8 - 1500*e^4 + 625) - log(x)/(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 + 625) - 1/(x*(9*e^
8 - 30*e^4 + 25)))*e^8*log(3) - 1320*(log(x + 9*e^8 - 30*e^4 + 25)/(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 +
625) - log(x)/(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 + 625) - 1/(x*(9*e^8 - 30*e^4 + 25)))*e^4*log(3) + 25
*(2*log(x + 9*e^8 - 30*e^4 + 25)/(729*e^24 - 7290*e^20 + 30375*e^16 - 67500*e^12 + 84375*e^8 - 56250*e^4 + 156
25) - 2*log(x)/(729*e^24 - 7290*e^20 + 30375*e^16 - 67500*e^12 + 84375*e^8 - 56250*e^4 + 15625) - (2*x - 9*e^8
+ 30*e^4 - 25)/(x^2*(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 + 625)))*log(3)^2 - 2*(log(x + 9*e^8 - 30*e^4 +
25)/(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 + 625) - log(x)/(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 + 625
) - 1/(x*(9*e^8 - 30*e^4 + 25)))*log(3)^2 + 1100*(log(x + 9*e^8 - 30*e^4 + 25)/(81*e^16 - 540*e^12 + 1350*e^8
- 1500*e^4 + 625) - log(x)/(81*e^16 - 540*e^12 + 1350*e^8 - 1500*e^4 + 625) - 1/(x*(9*e^8 - 30*e^4 + 25)))*log
(3) - 218/5*(log(x + 9*e^8 - 30*e^4 + 25)/(9*e^8 - 30*e^4 + 25) - log(x)/(9*e^8 - 30*e^4 + 25))*log(3) + 2/5*l
og(3)*log(x)/(9*e^8 - 30*e^4 + 25) + 1/25*(x*(9*e^8 - 30*e^4 + 25)*log(x + 9*e^8 - 30*e^4 + 25)^2 - 10*(x*log(
3) + 9*e^8*log(3) - 30*e^4*log(3) + 25*log(3))*log(x + 9*e^8 - 30*e^4 + 25))/(x*(9*e^8 - 30*e^4 + 25)) + 44/5*
log(x + 9*e^8 - 30*e^4 + 25)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
time = 0.39, size = 60, normalized size = 2.31 \begin {gather*} \frac {x^{2} \log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )^{2} - 1100 \, x \log \left (3\right ) + 25 \, \log \left (3\right )^{2} + 10 \, {\left (22 \, x^{2} - x \log \left (3\right )\right )} \log \left (x + 9 \, e^{8} - 30 \, e^{4} + 25\right )}{25 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((90*x*exp(4)^2-300*x*exp(4)+10*x^2+250*x)*log(3)+2*x^3)*log(9*exp(4)^2-30*exp(4)+x+25)+(-450*exp(4
)^2+1500*exp(4)-50*x-1250)*log(3)^2+(9900*x*exp(4)^2-33000*x*exp(4)+1090*x^2+27500*x)*log(3)+220*x^3)/(225*x^3
*exp(4)^2-750*x^3*exp(4)+25*x^4+625*x^3),x, algorithm="fricas")

[Out]

1/25*(x^2*log(x + 9*e^8 - 30*e^4 + 25)^2 - 1100*x*log(3) + 25*log(3)^2 + 10*(22*x^2 - x*log(3))*log(x + 9*e^8
- 30*e^4 + 25))/x^2

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
time = 1.96, size = 75, normalized size = 2.88 \begin {gather*} \frac {\log {\left (x - 30 e^{4} + 25 + 9 e^{8} \right )}^{2}}{25} + \frac {44 \log {\left (x - 30 e^{4} + 25 + 9 e^{8} \right )}}{5} - \frac {2 \log {\left (3 \right )} \log {\left (x - 30 e^{4} + 25 + 9 e^{8} \right )}}{5 x} + \frac {- 44 x \log {\left (3 \right )} + \log {\left (3 \right )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((90*x*exp(4)**2-300*x*exp(4)+10*x**2+250*x)*ln(3)+2*x**3)*ln(9*exp(4)**2-30*exp(4)+x+25)+(-450*exp
(4)**2+1500*exp(4)-50*x-1250)*ln(3)**2+(9900*x*exp(4)**2-33000*x*exp(4)+1090*x**2+27500*x)*ln(3)+220*x**3)/(22
5*x**3*exp(4)**2-750*x**3*exp(4)+25*x**4+625*x**3),x)

[Out]

log(x - 30*exp(4) + 25 + 9*exp(8))**2/25 + 44*log(x - 30*exp(4) + 25 + 9*exp(8))/5 - 2*log(3)*log(x - 30*exp(4
) + 25 + 9*exp(8))/(5*x) + (-44*x*log(3) + log(3)**2)/x**2

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((90*x*exp(4)^2-300*x*exp(4)+10*x^2+250*x)*log(3)+2*x^3)*log(9*exp(4)^2-30*exp(4)+x+25)+(-450*exp(4
)^2+1500*exp(4)-50*x-1250)*log(3)^2+(9900*x*exp(4)^2-33000*x*exp(4)+1090*x^2+27500*x)*log(3)+220*x^3)/(225*x^3
*exp(4)^2-750*x^3*exp(4)+25*x^4+625*x^3),x, algorithm="giac")

[Out]

integrate(2/25*(110*x^3 - 25*(x + 9*e^8 - 30*e^4 + 25)*log(3)^2 + 5*(109*x^2 + 990*x*e^8 - 3300*x*e^4 + 2750*x
)*log(3) + (x^3 + 5*(x^2 + 9*x*e^8 - 30*x*e^4 + 25*x)*log(3))*log(x + 9*e^8 - 30*e^4 + 25))/(x^4 + 9*x^3*e^8 -
30*x^3*e^4 + 25*x^3), x)

________________________________________________________________________________________

Mupad [B]
time = 3.09, size = 88, normalized size = 3.38 \begin {gather*} \frac {44\,\ln \left (x+{\left (3\,{\mathrm {e}}^4-5\right )}^2\right )}{5}-\frac {220\,x\,\ln \left (3\right )-5\,{\ln \left (3\right )}^2}{5\,x^2}+\frac {{\ln \left (x-30\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8+25\right )}^2}{25}-\frac {\ln \left (x-30\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8+25\right )\,\left (\frac {12\,{\mathrm {e}}^4}{5}-\frac {18\,{\mathrm {e}}^8}{25}+\frac {2\,\ln \left (3\right )}{5}+\frac {2\,{\left (3\,{\mathrm {e}}^4-5\right )}^2}{25}-2\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x - 30*exp(4) + 9*exp(8) + 25)*(log(3)*(250*x - 300*x*exp(4) + 90*x*exp(8) + 10*x^2) + 2*x^3) + log(3
)*(27500*x - 33000*x*exp(4) + 9900*x*exp(8) + 1090*x^2) + 220*x^3 - log(3)^2*(50*x - 1500*exp(4) + 450*exp(8)
+ 1250))/(225*x^3*exp(8) - 750*x^3*exp(4) + 625*x^3 + 25*x^4),x)

[Out]

(44*log(x + (3*exp(4) - 5)^2))/5 - (220*x*log(3) - 5*log(3)^2)/(5*x^2) + log(x - 30*exp(4) + 9*exp(8) + 25)^2/
25 - (log(x - 30*exp(4) + 9*exp(8) + 25)*((12*exp(4))/5 - (18*exp(8))/25 + (2*log(3))/5 + (2*(3*exp(4) - 5)^2)
/25 - 2))/x

________________________________________________________________________________________