3.19.31 \(\int \frac {-x+18 e^{2 x} x+e^x (-54 x-18 e^4 x+18 x \log (5))+(-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x (54+18 e^4-18 \log (5))+(54+18 e^4) \log (5)-9 \log ^2(5)) \log (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x (54+18 e^4-18 \log (5))+(54+18 e^4) \log (5)-9 \log ^2(5))}{80 x^2+54 e^4 x^2+9 e^8 x^2+9 e^{2 x} x^2-x^3+(-54 x^2-18 e^4 x^2) \log (5)+9 x^2 \log ^2(5)+e^x (-54 x^2-18 e^4 x^2+18 x^2 \log (5))+(4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+(-2700 x-900 e^4 x) \log (5)+450 x \log ^2(5)+e^x (-2700 x-900 e^4 x+900 x \log (5))) \log (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x (54+18 e^4-18 \log (5))+(54+18 e^4) \log (5)-9 \log ^2(5))+(50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+(-33750-11250 e^4) \log (5)+5625 \log ^2(5)+e^x (-33750-11250 e^4+11250 \log (5))) \log ^2(-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x (54+18 e^4-18 \log (5))+(54+18 e^4) \log (5)-9 \log ^2(5))} \, dx\)
Optimal. Leaf size=28 \[ \frac {1}{25+\frac {x}{\log \left (1+x-9 \left (-3-e^4+e^x+\log (5)\right )^2\right )}} \]
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Rubi [B] time = 2.05, antiderivative size = 58, normalized size of antiderivative = 2.07,
number of steps used = 6, number of rules used = 4, integrand size = 455, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used
= {6, 6688, 6711, 32} \begin {gather*} \frac {1}{\frac {x}{\log \left (x-9 e^{2 x}+18 e^x \left (3+e^4-\log (5)\right )-9 e^8-54 e^4-80-9 \log ^2(5)+18 \left (3+e^4\right ) \log (5)\right )}+25} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-x + 18*E^(2*x)*x + E^x*(-54*x - 18*E^4*x + 18*x*Log[5]) + (-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(5
4 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2)*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(5
4 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2])/(80*x^2 + 54*E^4*x^2 + 9*E^8*x^2 + 9*E^(2*x)*x^2
- x^3 + (-54*x^2 - 18*E^4*x^2)*Log[5] + 9*x^2*Log[5]^2 + E^x*(-54*x^2 - 18*E^4*x^2 + 18*x^2*Log[5]) + (4000*x
+ 2700*E^4*x + 450*E^8*x + 450*E^(2*x)*x - 50*x^2 + (-2700*x - 900*E^4*x)*Log[5] + 450*x*Log[5]^2 + E^x*(-270
0*x - 900*E^4*x + 900*x*Log[5]))*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(54 + 18*E^4 - 18*Log[5]) + (5
4 + 18*E^4)*Log[5] - 9*Log[5]^2] + (50000 + 33750*E^4 + 5625*E^8 + 5625*E^(2*x) - 625*x + (-33750 - 11250*E^4)
*Log[5] + 5625*Log[5]^2 + E^x*(-33750 - 11250*E^4 + 11250*Log[5]))*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x +
E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2]^2),x]
[Out]
(25 + x/Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + 18*E^x*(3 + E^4 - Log[5]) + 18*(3 + E^4)*Log[5] - 9*Log[5]^
2])^(-1)
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 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6711
Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{9 e^8 x^2+9 e^{2 x} x^2+\left (80+54 e^4\right ) x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx\\ &=\int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{9 e^{2 x} x^2+\left (80+54 e^4+9 e^8\right ) x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+9 x^2 \log ^2(5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx\\ &=\int \frac {-x+18 e^{2 x} x+e^x \left (-54 x-18 e^4 x+18 x \log (5)\right )+\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )}{9 e^{2 x} x^2-x^3+\left (-54 x^2-18 e^4 x^2\right ) \log (5)+e^x \left (-54 x^2-18 e^4 x^2+18 x^2 \log (5)\right )+x^2 \left (80+54 e^4+9 e^8+9 \log ^2(5)\right )+\left (4000 x+2700 e^4 x+450 e^8 x+450 e^{2 x} x-50 x^2+\left (-2700 x-900 e^4 x\right ) \log (5)+450 x \log ^2(5)+e^x \left (-2700 x-900 e^4 x+900 x \log (5)\right )\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )+\left (50000+33750 e^4+5625 e^8+5625 e^{2 x}-625 x+\left (-33750-11250 e^4\right ) \log (5)+5625 \log ^2(5)+e^x \left (-33750-11250 e^4+11250 \log (5)\right )\right ) \log ^2\left (-80-54 e^4-9 e^8-9 e^{2 x}+x+e^x \left (54+18 e^4-18 \log (5)\right )+\left (54+18 e^4\right ) \log (5)-9 \log ^2(5)\right )} \, dx\\ &=\int \frac {x \left (-1+18 e^{2 x}-18 e^{4+x}+18 e^x (-3+\log (5))\right )+\left (-80-9 e^8-9 e^{2 x}+18 e^{4+x}+x+18 e^4 (-3+\log (5))-18 e^x (-3+\log (5))+54 \log (5)-9 \log ^2(5)\right ) \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+18 e^x \left (3+e^4-\log (5)\right )+18 \left (3+e^4\right ) \log (5)-9 \log ^2(5)\right )}{\left (9 e^{2 x}-x-18 e^{4+x} \left (1+\frac {3-\log (5)}{e^4}\right )+80 \left (1+\frac {9}{80} \left (e^8-2 e^4 (-3+\log (5))+(-6+\log (5)) \log (5)\right )\right )\right ) \left (x+25 \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+18 e^x \left (3+e^4-\log (5)\right )+18 \left (3+e^4\right ) \log (5)-9 \log ^2(5)\right )\right )^2} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{(25+x)^2} \, dx,x,\frac {x}{\log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+18 e^x \left (3+e^4-\log (5)\right )+18 \left (3+e^4\right ) \log (5)-9 \log ^2(5)\right )}\right )\\ &=\frac {1}{25+\frac {x}{\log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+18 e^x \left (3+e^4-\log (5)\right )+18 \left (3+e^4\right ) \log (5)-9 \log ^2(5)\right )}}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.22, size = 61, normalized size = 2.18 \begin {gather*} -\frac {x}{25 \left (x+25 \log \left (-80-54 e^4-9 e^8-9 e^{2 x}+x+18 e^x \left (3+e^4-\log (5)\right )+18 \left (3+e^4\right ) \log (5)-9 \log ^2(5)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-x + 18*E^(2*x)*x + E^x*(-54*x - 18*E^4*x + 18*x*Log[5]) + (-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x +
E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2)*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x +
E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2])/(80*x^2 + 54*E^4*x^2 + 9*E^8*x^2 + 9*E^(2*
x)*x^2 - x^3 + (-54*x^2 - 18*E^4*x^2)*Log[5] + 9*x^2*Log[5]^2 + E^x*(-54*x^2 - 18*E^4*x^2 + 18*x^2*Log[5]) + (
4000*x + 2700*E^4*x + 450*E^8*x + 450*E^(2*x)*x - 50*x^2 + (-2700*x - 900*E^4*x)*Log[5] + 450*x*Log[5]^2 + E^x
*(-2700*x - 900*E^4*x + 900*x*Log[5]))*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + E^x*(54 + 18*E^4 - 18*Log[5]
) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2] + (50000 + 33750*E^4 + 5625*E^8 + 5625*E^(2*x) - 625*x + (-33750 - 1125
0*E^4)*Log[5] + 5625*Log[5]^2 + E^x*(-33750 - 11250*E^4 + 11250*Log[5]))*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x)
+ x + E^x*(54 + 18*E^4 - 18*Log[5]) + (54 + 18*E^4)*Log[5] - 9*Log[5]^2]^2),x]
[Out]
-1/25*x/(x + 25*Log[-80 - 54*E^4 - 9*E^8 - 9*E^(2*x) + x + 18*E^x*(3 + E^4 - Log[5]) + 18*(3 + E^4)*Log[5] - 9
*Log[5]^2])
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fricas [A] time = 0.91, size = 53, normalized size = 1.89 \begin {gather*} -\frac {x}{25 \, {\left (x + 25 \, \log \left (18 \, {\left (e^{4} - \log \relax (5) + 3\right )} e^{x} + 18 \, {\left (e^{4} + 3\right )} \log \relax (5) - 9 \, \log \relax (5)^{2} + x - 9 \, e^{8} - 54 \, e^{4} - 9 \, e^{\left (2 \, x\right )} - 80\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)
+x-80)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+
x-80)+18*x*exp(x)^2+(18*x*log(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2+(11250*log(5)-11250*exp(4)-33750)
*exp(x)+5625*log(5)^2+(-11250*exp(4)-33750)*log(5)+5625*exp(4)^2+33750*exp(4)-625*x+50000)*log(-9*exp(x)^2+(-1
8*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)^2+(450*x*exp(x)^2+(9
00*x*log(5)-900*x*exp(4)-2700*x)*exp(x)+450*x*log(5)^2+(-900*x*exp(4)-2700*x)*log(5)+450*x*exp(4)^2+2700*x*exp
(4)-50*x^2+4000*x)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^
2-54*exp(4)+x-80)+9*exp(x)^2*x^2+(18*x^2*log(5)-18*x^2*exp(4)-54*x^2)*exp(x)+9*x^2*log(5)^2+(-18*x^2*exp(4)-54
*x^2)*log(5)+9*x^2*exp(4)^2+54*x^2*exp(4)-x^3+80*x^2),x, algorithm="fricas")
[Out]
-1/25*x/(x + 25*log(18*(e^4 - log(5) + 3)*e^x + 18*(e^4 + 3)*log(5) - 9*log(5)^2 + x - 9*e^8 - 54*e^4 - 9*e^(2
*x) - 80))
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giac [B] time = 4.23, size = 59, normalized size = 2.11 \begin {gather*} -\frac {x}{25 \, {\left (x + 25 \, \log \left (18 \, e^{4} \log \relax (5) - 18 \, e^{x} \log \relax (5) - 9 \, \log \relax (5)^{2} + x - 9 \, e^{8} - 54 \, e^{4} - 9 \, e^{\left (2 \, x\right )} + 18 \, e^{\left (x + 4\right )} + 54 \, e^{x} + 54 \, \log \relax (5) - 80\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)
+x-80)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+
x-80)+18*x*exp(x)^2+(18*x*log(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2+(11250*log(5)-11250*exp(4)-33750)
*exp(x)+5625*log(5)^2+(-11250*exp(4)-33750)*log(5)+5625*exp(4)^2+33750*exp(4)-625*x+50000)*log(-9*exp(x)^2+(-1
8*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)^2+(450*x*exp(x)^2+(9
00*x*log(5)-900*x*exp(4)-2700*x)*exp(x)+450*x*log(5)^2+(-900*x*exp(4)-2700*x)*log(5)+450*x*exp(4)^2+2700*x*exp
(4)-50*x^2+4000*x)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^
2-54*exp(4)+x-80)+9*exp(x)^2*x^2+(18*x^2*log(5)-18*x^2*exp(4)-54*x^2)*exp(x)+9*x^2*log(5)^2+(-18*x^2*exp(4)-54
*x^2)*log(5)+9*x^2*exp(4)^2+54*x^2*exp(4)-x^3+80*x^2),x, algorithm="giac")
[Out]
-1/25*x/(x + 25*log(18*e^4*log(5) - 18*e^x*log(5) - 9*log(5)^2 + x - 9*e^8 - 54*e^4 - 9*e^(2*x) + 18*e^(x + 4)
+ 54*e^x + 54*log(5) - 80))
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maple [A] time = 0.10, size = 56, normalized size = 2.00
|
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| method |
result |
size |
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| risch |
\(-\frac {x}{25 \left (x +25 \ln \left (-9 \,{\mathrm e}^{2 x}+\left (-18 \ln \relax (5)+18 \,{\mathrm e}^{4}+54\right ) {\mathrm e}^{x}-9 \ln \relax (5)^{2}+\left (18 \,{\mathrm e}^{4}+54\right ) \ln \relax (5)-9 \,{\mathrm e}^{8}-54 \,{\mathrm e}^{4}+x -80\right )\right )}\) |
\(56\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((-9*exp(x)^2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)^2+(18*exp(4)+54)*ln(5)-9*exp(4)^2-54*exp(4)+x-80)*ln
(-9*exp(x)^2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)^2+(18*exp(4)+54)*ln(5)-9*exp(4)^2-54*exp(4)+x-80)+18*x*ex
p(x)^2+(18*x*ln(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2+(11250*ln(5)-11250*exp(4)-33750)*exp(x)+5625*ln
(5)^2+(-11250*exp(4)-33750)*ln(5)+5625*exp(4)^2+33750*exp(4)-625*x+50000)*ln(-9*exp(x)^2+(-18*ln(5)+18*exp(4)+
54)*exp(x)-9*ln(5)^2+(18*exp(4)+54)*ln(5)-9*exp(4)^2-54*exp(4)+x-80)^2+(450*x*exp(x)^2+(900*x*ln(5)-900*x*exp(
4)-2700*x)*exp(x)+450*x*ln(5)^2+(-900*x*exp(4)-2700*x)*ln(5)+450*x*exp(4)^2+2700*x*exp(4)-50*x^2+4000*x)*ln(-9
*exp(x)^2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)^2+(18*exp(4)+54)*ln(5)-9*exp(4)^2-54*exp(4)+x-80)+9*exp(x)^2
*x^2+(18*x^2*ln(5)-18*x^2*exp(4)-54*x^2)*exp(x)+9*x^2*ln(5)^2+(-18*x^2*exp(4)-54*x^2)*ln(5)+9*x^2*exp(4)^2+54*
x^2*exp(4)-x^3+80*x^2),x,method=_RETURNVERBOSE)
[Out]
-1/25*x/(x+25*ln(-9*exp(2*x)+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)^2+(18*exp(4)+54)*ln(5)-9*exp(8)-54*exp(4)
+x-80))
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maxima [A] time = 11.84, size = 53, normalized size = 1.89 \begin {gather*} -\frac {x}{25 \, {\left (x + 25 \, \log \left (18 \, {\left (e^{4} - \log \relax (5) + 3\right )} e^{x} + 18 \, {\left (e^{4} + 3\right )} \log \relax (5) - 9 \, \log \relax (5)^{2} + x - 9 \, e^{8} - 54 \, e^{4} - 9 \, e^{\left (2 \, x\right )} - 80\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)
+x-80)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+
x-80)+18*x*exp(x)^2+(18*x*log(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)^2+(11250*log(5)-11250*exp(4)-33750)
*exp(x)+5625*log(5)^2+(-11250*exp(4)-33750)*log(5)+5625*exp(4)^2+33750*exp(4)-625*x+50000)*log(-9*exp(x)^2+(-1
8*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^2-54*exp(4)+x-80)^2+(450*x*exp(x)^2+(9
00*x*log(5)-900*x*exp(4)-2700*x)*exp(x)+450*x*log(5)^2+(-900*x*exp(4)-2700*x)*log(5)+450*x*exp(4)^2+2700*x*exp
(4)-50*x^2+4000*x)*log(-9*exp(x)^2+(-18*log(5)+18*exp(4)+54)*exp(x)-9*log(5)^2+(18*exp(4)+54)*log(5)-9*exp(4)^
2-54*exp(4)+x-80)+9*exp(x)^2*x^2+(18*x^2*log(5)-18*x^2*exp(4)-54*x^2)*exp(x)+9*x^2*log(5)^2+(-18*x^2*exp(4)-54
*x^2)*log(5)+9*x^2*exp(4)^2+54*x^2*exp(4)-x^3+80*x^2),x, algorithm="maxima")
[Out]
-1/25*x/(x + 25*log(18*(e^4 - log(5) + 3)*e^x + 18*(e^4 + 3)*log(5) - 9*log(5)^2 + x - 9*e^8 - 54*e^4 - 9*e^(2
*x) - 80))
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {x-18\,x\,{\mathrm {e}}^{2\,x}+\ln \left (x-9\,{\mathrm {e}}^{2\,x}-54\,{\mathrm {e}}^4-9\,{\mathrm {e}}^8+{\mathrm {e}}^x\,\left (18\,{\mathrm {e}}^4-18\,\ln \relax (5)+54\right )-9\,{\ln \relax (5)}^2+\ln \relax (5)\,\left (18\,{\mathrm {e}}^4+54\right )-80\right )\,\left (9\,{\mathrm {e}}^{2\,x}-x+54\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8-{\mathrm {e}}^x\,\left (18\,{\mathrm {e}}^4-18\,\ln \relax (5)+54\right )+9\,{\ln \relax (5)}^2-\ln \relax (5)\,\left (18\,{\mathrm {e}}^4+54\right )+80\right )+{\mathrm {e}}^x\,\left (54\,x+18\,x\,{\mathrm {e}}^4-18\,x\,\ln \relax (5)\right )}{9\,x^2\,{\ln \relax (5)}^2-{\mathrm {e}}^x\,\left (18\,x^2\,{\mathrm {e}}^4-18\,x^2\,\ln \relax (5)+54\,x^2\right )+{\ln \left (x-9\,{\mathrm {e}}^{2\,x}-54\,{\mathrm {e}}^4-9\,{\mathrm {e}}^8+{\mathrm {e}}^x\,\left (18\,{\mathrm {e}}^4-18\,\ln \relax (5)+54\right )-9\,{\ln \relax (5)}^2+\ln \relax (5)\,\left (18\,{\mathrm {e}}^4+54\right )-80\right )}^2\,\left (5625\,{\mathrm {e}}^{2\,x}-625\,x+33750\,{\mathrm {e}}^4+5625\,{\mathrm {e}}^8-{\mathrm {e}}^x\,\left (11250\,{\mathrm {e}}^4-11250\,\ln \relax (5)+33750\right )+5625\,{\ln \relax (5)}^2-\ln \relax (5)\,\left (11250\,{\mathrm {e}}^4+33750\right )+50000\right )+9\,x^2\,{\mathrm {e}}^{2\,x}+54\,x^2\,{\mathrm {e}}^4+9\,x^2\,{\mathrm {e}}^8+80\,x^2-x^3+\ln \left (x-9\,{\mathrm {e}}^{2\,x}-54\,{\mathrm {e}}^4-9\,{\mathrm {e}}^8+{\mathrm {e}}^x\,\left (18\,{\mathrm {e}}^4-18\,\ln \relax (5)+54\right )-9\,{\ln \relax (5)}^2+\ln \relax (5)\,\left (18\,{\mathrm {e}}^4+54\right )-80\right )\,\left (4000\,x+450\,x\,{\mathrm {e}}^{2\,x}+2700\,x\,{\mathrm {e}}^4+450\,x\,{\mathrm {e}}^8-\ln \relax (5)\,\left (2700\,x+900\,x\,{\mathrm {e}}^4\right )+450\,x\,{\ln \relax (5)}^2-50\,x^2-{\mathrm {e}}^x\,\left (2700\,x+900\,x\,{\mathrm {e}}^4-900\,x\,\ln \relax (5)\right )\right )-\ln \relax (5)\,\left (18\,x^2\,{\mathrm {e}}^4+54\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(x - 18*x*exp(2*x) + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp(x)*(18*exp(4) - 18*log(5) + 54) - 9*
log(5)^2 + log(5)*(18*exp(4) + 54) - 80)*(9*exp(2*x) - x + 54*exp(4) + 9*exp(8) - exp(x)*(18*exp(4) - 18*log(5
) + 54) + 9*log(5)^2 - log(5)*(18*exp(4) + 54) + 80) + exp(x)*(54*x + 18*x*exp(4) - 18*x*log(5)))/(9*x^2*log(5
)^2 - exp(x)*(18*x^2*exp(4) - 18*x^2*log(5) + 54*x^2) + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp(x)*(18
*exp(4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 80)^2*(5625*exp(2*x) - 625*x + 33750*exp(4)
+ 5625*exp(8) - exp(x)*(11250*exp(4) - 11250*log(5) + 33750) + 5625*log(5)^2 - log(5)*(11250*exp(4) + 33750)
+ 50000) + 9*x^2*exp(2*x) + 54*x^2*exp(4) + 9*x^2*exp(8) + 80*x^2 - x^3 + log(x - 9*exp(2*x) - 54*exp(4) - 9*e
xp(8) + exp(x)*(18*exp(4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 80)*(4000*x + 450*x*exp(2
*x) + 2700*x*exp(4) + 450*x*exp(8) - log(5)*(2700*x + 900*x*exp(4)) + 450*x*log(5)^2 - 50*x^2 - exp(x)*(2700*x
+ 900*x*exp(4) - 900*x*log(5))) - log(5)*(18*x^2*exp(4) + 54*x^2)),x)
[Out]
int(-(x - 18*x*exp(2*x) + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp(x)*(18*exp(4) - 18*log(5) + 54) - 9*
log(5)^2 + log(5)*(18*exp(4) + 54) - 80)*(9*exp(2*x) - x + 54*exp(4) + 9*exp(8) - exp(x)*(18*exp(4) - 18*log(5
) + 54) + 9*log(5)^2 - log(5)*(18*exp(4) + 54) + 80) + exp(x)*(54*x + 18*x*exp(4) - 18*x*log(5)))/(9*x^2*log(5
)^2 - exp(x)*(18*x^2*exp(4) - 18*x^2*log(5) + 54*x^2) + log(x - 9*exp(2*x) - 54*exp(4) - 9*exp(8) + exp(x)*(18
*exp(4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 80)^2*(5625*exp(2*x) - 625*x + 33750*exp(4)
+ 5625*exp(8) - exp(x)*(11250*exp(4) - 11250*log(5) + 33750) + 5625*log(5)^2 - log(5)*(11250*exp(4) + 33750)
+ 50000) + 9*x^2*exp(2*x) + 54*x^2*exp(4) + 9*x^2*exp(8) + 80*x^2 - x^3 + log(x - 9*exp(2*x) - 54*exp(4) - 9*e
xp(8) + exp(x)*(18*exp(4) - 18*log(5) + 54) - 9*log(5)^2 + log(5)*(18*exp(4) + 54) - 80)*(4000*x + 450*x*exp(2
*x) + 2700*x*exp(4) + 450*x*exp(8) - log(5)*(2700*x + 900*x*exp(4)) + 450*x*log(5)^2 - 50*x^2 - exp(x)*(2700*x
+ 900*x*exp(4) - 900*x*log(5))) - log(5)*(18*x^2*exp(4) + 54*x^2)), x)
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sympy [B] time = 0.87, size = 61, normalized size = 2.18 \begin {gather*} - \frac {x}{25 x + 625 \log {\left (x - 9 e^{2 x} + \left (- 18 \log {\relax (5 )} + 54 + 18 e^{4}\right ) e^{x} - 9 e^{8} - 54 e^{4} - 80 - 9 \log {\relax (5 )}^{2} + \left (54 + 18 e^{4}\right ) \log {\relax (5 )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-9*exp(x)**2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)**2+(18*exp(4)+54)*ln(5)-9*exp(4)**2-54*exp(4)
+x-80)*ln(-9*exp(x)**2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)**2+(18*exp(4)+54)*ln(5)-9*exp(4)**2-54*exp(4)+x
-80)+18*x*exp(x)**2+(18*x*ln(5)-18*x*exp(4)-54*x)*exp(x)-x)/((5625*exp(x)**2+(11250*ln(5)-11250*exp(4)-33750)*
exp(x)+5625*ln(5)**2+(-11250*exp(4)-33750)*ln(5)+5625*exp(4)**2+33750*exp(4)-625*x+50000)*ln(-9*exp(x)**2+(-18
*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)**2+(18*exp(4)+54)*ln(5)-9*exp(4)**2-54*exp(4)+x-80)**2+(450*x*exp(x)**2+(9
00*x*ln(5)-900*x*exp(4)-2700*x)*exp(x)+450*x*ln(5)**2+(-900*x*exp(4)-2700*x)*ln(5)+450*x*exp(4)**2+2700*x*exp(
4)-50*x**2+4000*x)*ln(-9*exp(x)**2+(-18*ln(5)+18*exp(4)+54)*exp(x)-9*ln(5)**2+(18*exp(4)+54)*ln(5)-9*exp(4)**2
-54*exp(4)+x-80)+9*exp(x)**2*x**2+(18*x**2*ln(5)-18*x**2*exp(4)-54*x**2)*exp(x)+9*x**2*ln(5)**2+(-18*x**2*exp(
4)-54*x**2)*ln(5)+9*x**2*exp(4)**2+54*x**2*exp(4)-x**3+80*x**2),x)
[Out]
-x/(25*x + 625*log(x - 9*exp(2*x) + (-18*log(5) + 54 + 18*exp(4))*exp(x) - 9*exp(8) - 54*exp(4) - 80 - 9*log(5
)**2 + (54 + 18*exp(4))*log(5)))
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