3.76.81 \(\int \frac {2 e^{e^3} x-12 x^6-50398 x^7-77752800 x^8-52478280000 x^9-13116168000000 x^{10}+1312200000000 x^{11}+(-2 e^{e^3}+12 x^5+50398 x^6+77752800 x^7+52478280000 x^8+13116168000000 x^9-1312200000000 x^{10}) \log (2 e^{e^3}+2 x^6+7200 x^7+9720000 x^8+5832000000 x^9+1312200000000 x^{10})}{e^{e^3}+x^6+3600 x^7+4860000 x^8+2916000000 x^9+656100000000 x^{10}} \, dx\)
Optimal. Leaf size=28 \[ \left (-x+\log \left (2 \left (e^{e^3}+x^2 \left (x+900 x^2\right )^4\right )\right )\right )^2 \]
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Rubi [A] time = 0.37, antiderivative size = 26, normalized size of antiderivative = 0.93,
number of steps used = 3, number of rules used = 3, integrand size = 144, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used
= {6688, 12, 6686} \begin {gather*} \left (x-\log \left (2 \left ((900 x+1)^4 x^6+e^{e^3}\right )\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
[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 6686
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 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rubi steps
\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 26, normalized size = 0.93 \begin {gather*} \left (x-\log \left (2 \left (e^{e^3}+x^6 (1+900 x)^4\right )\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
[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
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fricas [B] time = 0.54, size = 73, normalized size = 2.61 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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))+1312200000000*x^10+5832000000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))+131220000000
0*x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp(3))+656100000000*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
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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))+1312200000000*x^10+5832000000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))+131220000000
0*x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp(3))+656100000000*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
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maple [B] time = 0.58, size = 74, normalized size = 2.64
|
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| method |
result |
size |
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| norman |
\(x^{2}+\ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}+1312200000000 x^{10}+5832000000 x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )^{2}-2 x \ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}+1312200000000 x^{10}+5832000000 x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )\) |
\(74\) |
| risch |
\(x^{2}+\ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}+1312200000000 x^{10}+5832000000 x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )^{2}-2 x \ln \left (2 \,{\mathrm e}^{{\mathrm e}^{3}}+1312200000000 x^{10}+5832000000 x^{9}+9720000 x^{8}+7200 x^{7}+2 x^{6}\right )\) |
\(74\) |
| default |
error in gcdex: invalid arguments\ |
N/A |
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Verification of antiderivative is not currently implemented for this CAS.
[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))+1312200000000*x^10+5832000000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))+1312200000000*x^11-
13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp(3))+656100000000*x^10+2916000000*x
^9+4860000*x^8+3600*x^7+x^6),x,method=_RETURNVERBOSE)
[Out]
x^2+ln(2*exp(exp(3))+1312200000000*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)
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maxima [B] time = 0.51, size = 75, normalized size = 2.68 \begin {gather*} x^{2} - 2 \, x \log \relax (2) - 2 \, {\left (x - \log \relax (2)\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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))+1312200000000*x^10+5832000000*x^9+9720000*x^8+7200*x^7+2*x^6)+2*x*exp(exp(3))+131220000000
0*x^11-13116168000000*x^10-52478280000*x^9-77752800*x^8-50398*x^7-12*x^6)/(exp(exp(3))+656100000000*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
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mupad [B] time = 5.26, size = 38, normalized size = 1.36 \begin {gather*} {\left (x-\ln \left (1312200000000\,x^{10}+5832000000\,x^9+9720000\,x^8+7200\,x^7+2\,x^6+2\,{\mathrm {e}}^{{\mathrm {e}}^3}\right )\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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sympy [B] time = 0.33, size = 75, normalized size = 2.68 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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))+1312200000000*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
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