\(\int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+(-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}) \log (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10})}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx\) [7581]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.25 (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 = {6820, 12, 6818} \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=\left (x-\log \left (2 \left ((900 x+1)^4 x^6+e^{e^3}\right )\right )\right )^2 \]

[In]

Int[(2*E^E^3*x - 12*x^6 - 50398*x^7 - 77752800*x^8 - 52478280000*x^9 - 13116168000000*x^10 + 1312200000000*x^1
1 + (-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),x]

[Out]

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

Rule 12

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

Rule 6818

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

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (e^{e^3}+x^5 (1+900 x)^3 \left (-6-8999 x+900 x^2\right )\right ) \left (x-\log \left (2 \left (e^{e^3}+x^6 (1+900 x)^4\right )\right )\right )}{e^{e^3}+x^6 (1+900 x)^4} \, dx \\ & = 2 \int \frac {\left (e^{e^3}+x^5 (1+900 x)^3 \left (-6-8999 x+900 x^2\right )\right ) \left (x-\log \left (2 \left (e^{e^3}+x^6 (1+900 x)^4\right )\right )\right )}{e^{e^3}+x^6 (1+900 x)^4} \, dx \\ & = \left (x-\log \left (2 \left (e^{e^3}+x^6 (1+900 x)^4\right )\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=\left (x-\log \left (2 \left (e^{e^3}+x^6 (1+900 x)^4\right )\right )\right )^2 \]

[In]

Integrate[(2*E^E^3*x - 12*x^6 - 50398*x^7 - 77752800*x^8 - 52478280000*x^9 - 13116168000000*x^10 + 13122000000
00*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 + 360
0*x^7 + 4860000*x^8 + 2916000000*x^9 + 656100000000*x^10),x]

[Out]

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

Maple [B] (verified)

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

Time = 2.51 (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	\(\text {Expression too large to display}\)	\(809\)

[In]

int(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+52478280000*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))[phone]*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,method=_RETURNVERBOSE)

[Out]

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))+1312
200000000*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 = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=x^{2} - 2 \, x \log \left (1312200000000 \, x^{10} + 5832000000 \, x^{9} + 9720000 \, x^{8} + 7200 \, x^{7} + 2 \, x^{6} + 2 \, e^{\left (e^{3}\right )}\right ) + \log \left (1312200000000 \, x^{10} + 5832000000 \, x^{9} + 9720000 \, x^{8} + 7200 \, x^{7} + 2 \, x^{6} + 2 \, e^{\left (e^{3}\right )}\right )^{2} \]

[In]

integrate(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+52478280000*x^8+77752800*x^7+50398*x^6+12*x^5
)*log(2*exp(exp(3))[phone]*x^10+5832000000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))[phone]
0*x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp(3))[phone]*x^10+29160
00000*x^9+4860000*x^8+3600*x^7+x^6),x, algorithm="fricas")

[Out]

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(13122000
00000*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 = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=x^{2} - 2 x \log {\left (1312200000000 x^{10} + 5832000000 x^{9} + 9720000 x^{8} + 7200 x^{7} + 2 x^{6} + 2 e^{e^{3}} \right )} + \log {\left (1312200000000 x^{10} + 5832000000 x^{9} + 9720000 x^{8} + 7200 x^{7} + 2 x^{6} + 2 e^{e^{3}} \right )}^{2} \]

[In]

integrate(((-2*exp(exp(3))-1312200000000*x**10+13116168000000*x**9+52478280000*x**8+77752800*x**7+50398*x**6+1
2*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))+13
12200000000*x**11-13116168000000*x**10-52478280000*x**9-77752800*x**8-50398*x**7-12*x**6)/(exp(exp(3))+6561000
00000*x**10+2916000000*x**9+4860000*x**8+3600*x**7+x**6),x)

[Out]

x**2 - 2*x*log(1312200000000*x**10 + 5832000000*x**9 + 9720000*x**8 + 7200*x**7 + 2*x**6 + 2*exp(exp(3))) + lo
g(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 = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx=x^{2} - 2 \, x \log \left (2\right ) - 2 \, {\left (x - \log \left (2\right )\right )} \log \left (656100000000 \, x^{10} + 2916000000 \, x^{9} + 4860000 \, x^{8} + 3600 \, x^{7} + x^{6} + e^{\left (e^{3}\right )}\right ) + \log \left (656100000000 \, x^{10} + 2916000000 \, x^{9} + 4860000 \, x^{8} + 3600 \, x^{7} + x^{6} + e^{\left (e^{3}\right )}\right )^{2} \]

[In]

integrate(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+52478280000*x^8+77752800*x^7+50398*x^6+12*x^5
)*log(2*exp(exp(3))[phone]*x^10+5832000000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))[phone]
0*x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp(3))[phone]*x^10+29160
00000*x^9+4860000*x^8+3600*x^7+x^6),x, algorithm="maxima")

[Out]

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 + 2916000000*x^9 + 4860000*x^8 + 3600*x^7 + x^6 + e^(e^3))^2

Giac [F(-2)]

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

[In]

integrate(((-2*exp(exp(3))-1312200000000*x^10+13116168000000*x^9+52478280000*x^8+77752800*x^7+50398*x^6+12*x^5
)*log(2*exp(exp(3))[phone]*x^10+5832000000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))[phone]
0*x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp(3))[phone]*x^10+29160
00000*x^9+4860000*x^8+3600*x^7+x^6),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Francis algorithm failure for[1.0,0.0,infinity,infinity,infinity,infinity,infinity,infinity,infinity,infini
ty,infinity

Mupad [B] (verification not implemented)

Time = 15.57 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}[phone] x^{11}+\left (-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}\right ) \log \left (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10}\right )}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx={\left (x-\ln \left (1312200000000\,x^{10}[phone]\,x^9+9720000\,x^8+7200\,x^7+2\,x^6+2\,{\mathrm {e}}^{{\mathrm {e}}^3}\right )\right )}^2 \]

[In]

int(-(12*x^6 - log(2*exp(exp(3)) + 2*x^6 + 7200*x^7 + 9720000*x^8 + 5832000000*x^9 + 1312200000000*x^10)*(12*x
^5 - 2*exp(exp(3)) + 50398*x^6 + 77752800*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(e
xp(3)) + x^6 + 3600*x^7 + 4860000*x^8 + 2916000000*x^9 + 656100000000*x^10),x)

[Out]

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