\(\int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+(-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}) \log (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10})}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx\) [1581]

Optimal result

Integrand size = 144, antiderivative size = 28 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=\left (-x+\log \left (2 \left (e^{e^3}+x^2 \left (x+900 x^2\right )^4\right )\right )\right )^2 \] Output:

(ln(2*exp(exp(3))+2*(900*x^2+x)^4*x^2)-x)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=\left (x-\log \left (2 \left (e^{e^3}+x^6 (1+900 x)^4\right )\right )\right )^2 \] Input:

Integrate[(2*E^E^3*x - 12*x^6 - 50398*x^7 - 77752800*x^8 - 52478280000*x^9 
 - 13116168000000*x^10 + 1312200000000*x^11 + (-2*E^E^3 + 12*x^5 + 50398*x 
^6 + 77752800*x^7 + 52478280000*x^8 + 13116168000000*x^9 - 1312200000000*x 
^10)*Log[2*E^E^3 + 2*x^6 + 7200*x^7 + 9720000*x^8 + 5832000000*x^9 + 13122 
00000000*x^10])/(E^E^3 + x^6 + 3600*x^7 + 4860000*x^8 + 2916000000*x^9 + 6 
56100000000*x^10),x]
 

Output:

(x - Log[2*(E^E^3 + x^6*(1 + 900*x)^4)])^2
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {7239, 27, 7237}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1312200000000 x^{11}-13116168000000 x^{10}-52478280000 x^9-77752800 x^8-50398 x^7-12 x^6+\left (-1312200000000 x^{10}[phone] x^9+52478280000 x^8+77752800 x^7+50398 x^6+12 x^5-2 e^{e^3}\right ) \log \left (1312200000000 x^{10}[phone] x^9+9720000 x^8+7200 x^7+2 x^6+2 e^{e^3}\right )+2 e^{e^3} x}{656100000000 x^{10}[phone] x^9+4860000 x^8+3600 x^7+x^6+e^{e^3}} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left ((900 x+1)^3 \left (900 x^2-8999 x-6\right ) x^5+e^{e^3}\right ) \left (x-\log \left (2 \left ((900 x+1)^4 x^6+e^{e^3}\right )\right )\right )}{(900 x+1)^4 x^6+e^{e^3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {\left (e^{e^3}-x^5 (900 x+1)^3 \left (-900 x^2+8999 x+6\right )\right ) \left (x-\log \left (2 \left ((900 x+1)^4 x^6+e^{e^3}\right )\right )\right )}{(900 x+1)^4 x^6+e^{e^3}}dx\)

\(\Big \downarrow \) 7237

\(\displaystyle \left (x-\log \left (2 \left ((900 x+1)^4 x^6+e^{e^3}\right )\right )\right )^2\)

Input:

Int[(2*E^E^3*x - 12*x^6 - 50398*x^7 - 77752800*x^8 - 52478280000*x^9 - 131 
16168000000*x^10 + 1312200000000*x^11 + (-2*E^E^3 + 12*x^5 + 50398*x^6 + 7 
7752800*x^7 + 52478280000*x^8 + 13116168000000*x^9 - 1312200000000*x^10)*L 
og[2*E^E^3 + 2*x^6 + 7200*x^7 + 9720000*x^8 + 5832000000*x^9 + 13122000000 
00*x^10])/(E^E^3 + x^6 + 3600*x^7 + 4860000*x^8 + 2916000000*x^9 + 6561000 
00000*x^10),x]
 

Output:

(x - Log[2*(E^E^3 + x^6*(1 + 900*x)^4)])^2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7239

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(27)=54\).

Time = 61.46 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.64

method	result	size
norman	\(x^{2}+\ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )^{2}-2 x \ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )\)	\(74\)
risch	\(x^{2}+\ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )^{2}-2 x \ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )\)	\(74\)
parallelrisch	\(x^{2}-2 x \ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )+\ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )^{2}-1312200000000 \,{\mathrm e}^{{\mathrm e}^{3}}\)	\(79\)
default	\(\text {Expression too large to display}\)	\(809\)
parts	\(\text {Expression too large to display}\)	\(809\)

Input:

int(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+52478280000*x^8 
[phone]*x^7+50398*x^6+12*x^5)*ln(2*exp(exp(3))[phone]*x^10+583200 
0000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))[phone]*x^11-13 
116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp( 
3))[phone]*x^10+2916000000*x^9+4860000*x^8+3600*x^7+x^6),x,method=_R 
ETURNVERBOSE)
 

Output:

x^2+ln(2*exp(exp(3))[phone]*x^10+5832000000*x^9+9720000*x^8+7200*x^ 
7+2*x^6)^2-2*x*ln(2*exp(exp(3))[phone]*x^10+5832000000*x^9+9720000* 
x^8+7200*x^7+2*x^6)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (27) = 54\).

Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=x^{2} - 2 \, x \log \left (1312200000000 \, x^{10} + 5832000000 \, x^{9} + 9720000 \, x^{8} + 7200 \, x^{7} + 2 \, x^{6} + 2 \, e^{\left (e^{3}\right )}\right ) + \log \left (1312200000000 \, x^{10} + 5832000000 \, x^{9} + 9720000 \, x^{8} + 7200 \, x^{7} + 2 \, x^{6} + 2 \, e^{\left (e^{3}\right )}\right )^{2} \] Input:

integrate(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+524782800 
00*x^8+77752800*x^7+50398*x^6+12*x^5)*log(2*exp(exp(3))[phone]*x^10 
[phone]*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))[phone]* 
x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(e 
xp(exp(3))[phone]*x^10+2916000000*x^9+4860000*x^8+3600*x^7+x^6),x, a 
lgorithm="fricas")
 

Output:

x^2 - 2*x*log(1312200000000*x^10 + 5832000000*x^9 + 9720000*x^8 + 7200*x^7 
 + 2*x^6 + 2*e^(e^3)) + log(1312200000000*x^10 + 5832000000*x^9 + 9720000* 
x^8 + 7200*x^7 + 2*x^6 + 2*e^(e^3))^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (24) = 48\).

Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=x^{2} - 2 x \log {\left (1312200000000 x^{10} + 5832000000 x^{9} + 9720000 x^{8} + 7200 x^{7} + 2 x^{6} + 2 e^{e^{3}} \right )} + \log {\left (1312200000000 x^{10} + 5832000000 x^{9} + 9720000 x^{8} + 7200 x^{7} + 2 x^{6} + 2 e^{e^{3}} \right )}^{2} \] Input:

integrate(((-2*exp(exp(3))-1312200000000*x**10+13116168000000*x**9+5247828 
0000*x**8+77752800*x**7+50398*x**6+12*x**5)*ln(2*exp(exp(3))[phone] 
*x**10+5832000000*x**9+9720000*x**8+7200*x**7+2*x**6)+2*x*exp(exp(3))+1312 
200000000*x**11-13116168000000*x**10-52478280000*x**9-77752800*x**8-50398* 
x**7-12*x**6)/(exp(exp(3))[phone]*x**10+2916000000*x**9+4860000*x**8 
+3600*x**7+x**6),x)
 

Output:

x**2 - 2*x*log(1312200000000*x**10 + 5832000000*x**9 + 9720000*x**8 + 7200 
*x**7 + 2*x**6 + 2*exp(exp(3))) + log(1312200000000*x**10 + 5832000000*x** 
9 + 9720000*x**8 + 7200*x**7 + 2*x**6 + 2*exp(exp(3)))**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (27) = 54\).

Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=x^{2} - 2 \, x \log \left (2\right ) - 2 \, {\left (x - \log \left (2\right )\right )} \log \left (656100000000 \, x^{10} + 2916000000 \, x^{9} + 4860000 \, x^{8} + 3600 \, x^{7} + x^{6} + e^{\left (e^{3}\right )}\right ) + \log \left (656100000000 \, x^{10} + 2916000000 \, x^{9} + 4860000 \, x^{8} + 3600 \, x^{7} + x^{6} + e^{\left (e^{3}\right )}\right )^{2} \] Input:

integrate(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+524782800 
00*x^8+77752800*x^7+50398*x^6+12*x^5)*log(2*exp(exp(3))[phone]*x^10 
[phone]*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))[phone]* 
x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(e 
xp(exp(3))[phone]*x^10+2916000000*x^9+4860000*x^8+3600*x^7+x^6),x, a 
lgorithm="maxima")
 

Output:

x^2 - 2*x*log(2) - 2*(x - log(2))*log(656100000000*x^10 + 2916000000*x^9 + 
 4860000*x^8 + 3600*x^7 + x^6 + e^(e^3)) + log(656100000000*x^10 + 2916000 
000*x^9 + 4860000*x^8 + 3600*x^7 + x^6 + e^(e^3))^2
 

Giac [F(-2)]

Exception generated. \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+524782800 
00*x^8+77752800*x^7+50398*x^6+12*x^5)*log(2*exp(exp(3))[phone]*x^10 
[phone]*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))[phone]* 
x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(e 
xp(exp(3))[phone]*x^10+2916000000*x^9+4860000*x^8+3600*x^7+x^6),x, a 
lgorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[1.0,0 
.0,infinity,infinity,infinity,infinity,infinity,infinity,infinity,infinity 
,infinity
 

Mupad [B] (verification not implemented)

Time = 3.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx={\left (x-\ln \left (1312200000000\,x^{10}[phone]\,x^9+9720000\,x^8+7200\,x^7+2\,x^6+2\,{\mathrm {e}}^{{\mathrm {e}}^3}\right )\right )}^2 \] Input:

int(-(12*x^6 - log(2*exp(exp(3)) + 2*x^6 + 7200*x^7 + 9720000*x^8 + 583200 
0000*x^9 + 1312200000000*x^10)*(12*x^5 - 2*exp(exp(3)) + 50398*x^6 + 77752 
800*x^7 + 52478280000*x^8 + 13116168000000*x^9 - 1312200000000*x^10) - 2*x 
*exp(exp(3)) + 50398*x^7 + 77752800*x^8 + 52478280000*x^9 + 13116168000000 
*x^10 - 1312200000000*x^11)/(exp(exp(3)) + x^6 + 3600*x^7 + 4860000*x^8 + 
2916000000*x^9 + 656100000000*x^10),x)
 

Output:

(x - log(2*exp(exp(3)) + 2*x^6 + 7200*x^7 + 9720000*x^8 + 5832000000*x^9 + 
 1312200000000*x^10))^2
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.18 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=\frac {\mathrm {log}\left (e^{e^{3}}[phone] x^{10}[phone] x^{9}+4860000 x^{8}+3600 x^{7}+x^{6}\right )}{1125}+\mathrm {log}\left (2 e^{e^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )^{2}-2 \,\mathrm {log}\left (2 e^{e^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right ) x -\frac {\mathrm {log}\left (2 e^{e^{3}}[phone] x^{10}[phone] x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )}{1125}+x^{2} \] Input:

int(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+52478280000*x^8 
[phone]*x^7+50398*x^6+12*x^5)*log(2*exp(exp(3))[phone]*x^10+58320 
00000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))[phone]*x^11-1 
3116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp 
(3))[phone]*x^10+2916000000*x^9+4860000*x^8+3600*x^7+x^6),x)
 

Output:

(log(e**(e**3) + 656100000000*x**10 + 2916000000*x**9 + 4860000*x**8 + 360 
0*x**7 + x**6) + 1125*log(2*e**(e**3) + 1312200000000*x**10 + 5832000000*x 
**9 + 9720000*x**8 + 7200*x**7 + 2*x**6)**2 - 2250*log(2*e**(e**3) + 13122 
00000000*x**10 + 5832000000*x**9 + 9720000*x**8 + 7200*x**7 + 2*x**6)*x - 
log(2*e**(e**3) + 1312200000000*x**10 + 5832000000*x**9 + 9720000*x**8 + 7 
200*x**7 + 2*x**6) + 1125*x**2)/1125