3.66.41 \(\int \frac {-x^2+(e^x (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7)+e^{2 x} (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9)) \log (x)+(-x^2+e^x (40 x+500 x^3+200 x^4+20 x^5)) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{(x^2+e^x (-20 x-250 x^3-100 x^4-10 x^5)+e^{2 x} (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8)) \log (x)+(10 x+e^x (-100-1250 x^2-500 x^3-50 x^4)) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx\)
Optimal. Leaf size=33 \[ \frac {x^2}{-5+\frac {x}{e^x \left (2+x^2 (5+x)^2\right )-\log (\log (x))}} \]
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Rubi [F] time = 138.30, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-x^2 + (E^x*(2*x^2 + 2*x^3 + 75*x^4 + 65*x^5 + 15*x^6 + x^7) + E^(2*x)*(-40*x - 1000*x^3 - 400*x^4 - 6290
*x^5 - 5000*x^6 - 1500*x^7 - 200*x^8 - 10*x^9))*Log[x] + (-x^2 + E^x*(40*x + 500*x^3 + 200*x^4 + 20*x^5))*Log[
x]*Log[Log[x]] - 10*x*Log[x]*Log[Log[x]]^2)/((x^2 + E^x*(-20*x - 250*x^3 - 100*x^4 - 10*x^5) + E^(2*x)*(100 +
2500*x^2 + 1000*x^3 + 15725*x^4 + 12500*x^5 + 3750*x^6 + 500*x^7 + 25*x^8))*Log[x] + (10*x + E^x*(-100 - 1250*
x^2 - 500*x^3 - 50*x^4))*Log[x]*Log[Log[x]] + 25*Log[x]*Log[Log[x]]^2),x]
[Out]
-1/5*x^2 + 10*Defer[Int][(10*E^x - x + 125*E^x*x^2 + 50*E^x*x^3 + 5*E^x*x^4 - 5*Log[Log[x]])^(-1), x] - 2*Defe
r[Int][x/(10*E^x - x + 125*E^x*x^2 + 50*E^x*x^3 + 5*E^x*x^4 - 5*Log[Log[x]]), x] + Defer[Int][x^2/(10*E^x - x
+ 125*E^x*x^2 + 50*E^x*x^3 + 5*E^x*x^4 - 5*Log[Log[x]]), x]/5 + Defer[Int][x^3/(10*E^x - x + 125*E^x*x^2 + 50*
E^x*x^3 + 5*E^x*x^4 - 5*Log[Log[x]]), x]/5 - 20*Defer[Int][1/((2 + 25*x^2 + 10*x^3 + x^4)*(10*E^x - x + 125*E^
x*x^2 + 50*E^x*x^3 + 5*E^x*x^4 - 5*Log[Log[x]])), x] + 4*Defer[Int][x/((2 + 25*x^2 + 10*x^3 + x^4)*(10*E^x - x
+ 125*E^x*x^2 + 50*E^x*x^3 + 5*E^x*x^4 - 5*Log[Log[x]])), x] - (1258*Defer[Int][x^2/((2 + 25*x^2 + 10*x^3 + x
^4)*(10*E^x - x + 125*E^x*x^2 + 50*E^x*x^3 + 5*E^x*x^4 - 5*Log[Log[x]])), x])/5 - 50*Defer[Int][x^3/((2 + 25*x
^2 + 10*x^3 + x^4)*(10*E^x - x + 125*E^x*x^2 + 50*E^x*x^3 + 5*E^x*x^4 - 5*Log[Log[x]])), x] + 50*Defer[Int][Lo
g[Log[x]]/(-10*E^x + x - 125*E^x*x^2 - 50*E^x*x^3 - 5*E^x*x^4 + 5*Log[Log[x]])^2, x] - 50*Defer[Int][(x - 5*E^
x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^(-2), x] + 10*Defer[Int][x/(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^
4) + 5*Log[Log[x]])^2, x] - 2*Defer[Int][x^2/(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2, x] + (
3*Defer[Int][x^3/(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2, x])/5 + Defer[Int][x^4/(x - 5*E^x*
(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2, x]/5 + 100*Defer[Int][1/((2 + 25*x^2 + 10*x^3 + x^4)*(x - 5*E^
x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2), x] - 20*Defer[Int][x/((2 + 25*x^2 + 10*x^3 + x^4)*(x - 5*E^
x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2), x] + 1254*Defer[Int][x^2/((2 + 25*x^2 + 10*x^3 + x^4)*(x -
5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2), x] + (1242*Defer[Int][x^3/((2 + 25*x^2 + 10*x^3 + x^4)*
(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2), x])/5 - Defer[Int][x^2/(Log[x]*(x - 5*E^x*(2 + 25*
x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2), x] - 10*Defer[Int][(x*Log[Log[x]])/(x - 5*E^x*(2 + 25*x^2 + 10*x^3 +
x^4) + 5*Log[Log[x]])^2, x] + 4*Defer[Int][(x^2*Log[Log[x]])/(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Lo
g[x]])^2, x] + Defer[Int][(x^3*Log[Log[x]])/(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2, x] - 10
0*Defer[Int][Log[Log[x]]/((2 + 25*x^2 + 10*x^3 + x^4)*(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^
2), x] + 20*Defer[Int][(x*Log[Log[x]])/((2 + 25*x^2 + 10*x^3 + x^4)*(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5
*Log[Log[x]])^2), x] - 1258*Defer[Int][(x^2*Log[Log[x]])/((2 + 25*x^2 + 10*x^3 + x^4)*(x - 5*E^x*(2 + 25*x^2 +
10*x^3 + x^4) + 5*Log[Log[x]])^2), x] - 250*Defer[Int][(x^3*Log[Log[x]])/((2 + 25*x^2 + 10*x^3 + x^4)*(x - 5*
E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])^2), x]
Rubi steps
Aborted
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Mathematica [A] time = 0.28, size = 41, normalized size = 1.24 \begin {gather*} -\frac {1}{5} x^2 \left (1-\frac {x}{x-5 e^x \left (2+25 x^2+10 x^3+x^4\right )+5 \log (\log (x))}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-x^2 + (E^x*(2*x^2 + 2*x^3 + 75*x^4 + 65*x^5 + 15*x^6 + x^7) + E^(2*x)*(-40*x - 1000*x^3 - 400*x^4
- 6290*x^5 - 5000*x^6 - 1500*x^7 - 200*x^8 - 10*x^9))*Log[x] + (-x^2 + E^x*(40*x + 500*x^3 + 200*x^4 + 20*x^5)
)*Log[x]*Log[Log[x]] - 10*x*Log[x]*Log[Log[x]]^2)/((x^2 + E^x*(-20*x - 250*x^3 - 100*x^4 - 10*x^5) + E^(2*x)*(
100 + 2500*x^2 + 1000*x^3 + 15725*x^4 + 12500*x^5 + 3750*x^6 + 500*x^7 + 25*x^8))*Log[x] + (10*x + E^x*(-100 -
1250*x^2 - 500*x^3 - 50*x^4))*Log[x]*Log[Log[x]] + 25*Log[x]*Log[Log[x]]^2),x]
[Out]
-1/5*(x^2*(1 - x/(x - 5*E^x*(2 + 25*x^2 + 10*x^3 + x^4) + 5*Log[Log[x]])))
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fricas [A] time = 0.76, size = 62, normalized size = 1.88 \begin {gather*} \frac {x^{2} \log \left (\log \relax (x)\right ) - {\left (x^{6} + 10 \, x^{5} + 25 \, x^{4} + 2 \, x^{2}\right )} e^{x}}{5 \, {\left (x^{4} + 10 \, x^{3} + 25 \, x^{2} + 2\right )} e^{x} - x - 5 \, \log \left (\log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-10*x*log(x)*log(log(x))^2+((20*x^5+200*x^4+500*x^3+40*x)*exp(x)-x^2)*log(x)*log(log(x))+((-10*x^9-
200*x^8-1500*x^7-5000*x^6-6290*x^5-400*x^4-1000*x^3-40*x)*exp(x)^2+(x^7+15*x^6+65*x^5+75*x^4+2*x^3+2*x^2)*exp(
x))*log(x)-x^2)/(25*log(x)*log(log(x))^2+((-50*x^4-500*x^3-1250*x^2-100)*exp(x)+10*x)*log(x)*log(log(x))+((25*
x^8+500*x^7+3750*x^6+12500*x^5+15725*x^4+1000*x^3+2500*x^2+100)*exp(x)^2+(-10*x^5-100*x^4-250*x^3-20*x)*exp(x)
+x^2)*log(x)),x, algorithm="fricas")
[Out]
(x^2*log(log(x)) - (x^6 + 10*x^5 + 25*x^4 + 2*x^2)*e^x)/(5*(x^4 + 10*x^3 + 25*x^2 + 2)*e^x - x - 5*log(log(x))
)
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giac [B] time = 0.51, size = 74, normalized size = 2.24 \begin {gather*} -\frac {x^{6} e^{x} + 10 \, x^{5} e^{x} + 25 \, x^{4} e^{x} + 2 \, x^{2} e^{x} - x^{2} \log \left (\log \relax (x)\right )}{5 \, x^{4} e^{x} + 50 \, x^{3} e^{x} + 125 \, x^{2} e^{x} - x + 10 \, e^{x} - 5 \, \log \left (\log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-10*x*log(x)*log(log(x))^2+((20*x^5+200*x^4+500*x^3+40*x)*exp(x)-x^2)*log(x)*log(log(x))+((-10*x^9-
200*x^8-1500*x^7-5000*x^6-6290*x^5-400*x^4-1000*x^3-40*x)*exp(x)^2+(x^7+15*x^6+65*x^5+75*x^4+2*x^3+2*x^2)*exp(
x))*log(x)-x^2)/(25*log(x)*log(log(x))^2+((-50*x^4-500*x^3-1250*x^2-100)*exp(x)+10*x)*log(x)*log(log(x))+((25*
x^8+500*x^7+3750*x^6+12500*x^5+15725*x^4+1000*x^3+2500*x^2+100)*exp(x)^2+(-10*x^5-100*x^4-250*x^3-20*x)*exp(x)
+x^2)*log(x)),x, algorithm="giac")
[Out]
-(x^6*e^x + 10*x^5*e^x + 25*x^4*e^x + 2*x^2*e^x - x^2*log(log(x)))/(5*x^4*e^x + 50*x^3*e^x + 125*x^2*e^x - x +
10*e^x - 5*log(log(x)))
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maple [A] time = 0.06, size = 48, normalized size = 1.45
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| method |
result |
size |
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| risch |
\(-\frac {x^{2}}{5}-\frac {x^{3}}{5 \left (5 \,{\mathrm e}^{x} x^{4}+50 \,{\mathrm e}^{x} x^{3}+125 \,{\mathrm e}^{x} x^{2}-x +10 \,{\mathrm e}^{x}-5 \ln \left (\ln \relax (x )\right )\right )}\) |
\(48\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-10*x*ln(x)*ln(ln(x))^2+((20*x^5+200*x^4+500*x^3+40*x)*exp(x)-x^2)*ln(x)*ln(ln(x))+((-10*x^9-200*x^8-1500
*x^7-5000*x^6-6290*x^5-400*x^4-1000*x^3-40*x)*exp(x)^2+(x^7+15*x^6+65*x^5+75*x^4+2*x^3+2*x^2)*exp(x))*ln(x)-x^
2)/(25*ln(x)*ln(ln(x))^2+((-50*x^4-500*x^3-1250*x^2-100)*exp(x)+10*x)*ln(x)*ln(ln(x))+((25*x^8+500*x^7+3750*x^
6+12500*x^5+15725*x^4+1000*x^3+2500*x^2+100)*exp(x)^2+(-10*x^5-100*x^4-250*x^3-20*x)*exp(x)+x^2)*ln(x)),x,meth
od=_RETURNVERBOSE)
[Out]
-1/5*x^2-1/5*x^3/(5*exp(x)*x^4+50*exp(x)*x^3+125*exp(x)*x^2-x+10*exp(x)-5*ln(ln(x)))
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maxima [A] time = 0.62, size = 62, normalized size = 1.88 \begin {gather*} \frac {x^{2} \log \left (\log \relax (x)\right ) - {\left (x^{6} + 10 \, x^{5} + 25 \, x^{4} + 2 \, x^{2}\right )} e^{x}}{5 \, {\left (x^{4} + 10 \, x^{3} + 25 \, x^{2} + 2\right )} e^{x} - x - 5 \, \log \left (\log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-10*x*log(x)*log(log(x))^2+((20*x^5+200*x^4+500*x^3+40*x)*exp(x)-x^2)*log(x)*log(log(x))+((-10*x^9-
200*x^8-1500*x^7-5000*x^6-6290*x^5-400*x^4-1000*x^3-40*x)*exp(x)^2+(x^7+15*x^6+65*x^5+75*x^4+2*x^3+2*x^2)*exp(
x))*log(x)-x^2)/(25*log(x)*log(log(x))^2+((-50*x^4-500*x^3-1250*x^2-100)*exp(x)+10*x)*log(x)*log(log(x))+((25*
x^8+500*x^7+3750*x^6+12500*x^5+15725*x^4+1000*x^3+2500*x^2+100)*exp(x)^2+(-10*x^5-100*x^4-250*x^3-20*x)*exp(x)
+x^2)*log(x)),x, algorithm="maxima")
[Out]
(x^2*log(log(x)) - (x^6 + 10*x^5 + 25*x^4 + 2*x^2)*e^x)/(5*(x^4 + 10*x^3 + 25*x^2 + 2)*e^x - x - 5*log(log(x))
)
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x}\,\left (10\,x^9+200\,x^8+1500\,x^7+5000\,x^6+6290\,x^5+400\,x^4+1000\,x^3+40\,x\right )-{\mathrm {e}}^x\,\left (x^7+15\,x^6+65\,x^5+75\,x^4+2\,x^3+2\,x^2\right )\right )+x^2-\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (20\,x^5+200\,x^4+500\,x^3+40\,x\right )-x^2\right )+10\,x\,{\ln \left (\ln \relax (x)\right )}^2\,\ln \relax (x)}{25\,\ln \relax (x)\,{\ln \left (\ln \relax (x)\right )}^2+\ln \relax (x)\,\left (10\,x-{\mathrm {e}}^x\,\left (50\,x^4+500\,x^3+1250\,x^2+100\right )\right )\,\ln \left (\ln \relax (x)\right )+\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x}\,\left (25\,x^8+500\,x^7+3750\,x^6+12500\,x^5+15725\,x^4+1000\,x^3+2500\,x^2+100\right )-{\mathrm {e}}^x\,\left (10\,x^5+100\,x^4+250\,x^3+20\,x\right )+x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(log(x)*(exp(2*x)*(40*x + 1000*x^3 + 400*x^4 + 6290*x^5 + 5000*x^6 + 1500*x^7 + 200*x^8 + 10*x^9) - exp(x
)*(2*x^2 + 2*x^3 + 75*x^4 + 65*x^5 + 15*x^6 + x^7)) + x^2 - log(log(x))*log(x)*(exp(x)*(40*x + 500*x^3 + 200*x
^4 + 20*x^5) - x^2) + 10*x*log(log(x))^2*log(x))/(log(x)*(exp(2*x)*(2500*x^2 + 1000*x^3 + 15725*x^4 + 12500*x^
5 + 3750*x^6 + 500*x^7 + 25*x^8 + 100) - exp(x)*(20*x + 250*x^3 + 100*x^4 + 10*x^5) + x^2) + 25*log(log(x))^2*
log(x) + log(log(x))*log(x)*(10*x - exp(x)*(1250*x^2 + 500*x^3 + 50*x^4 + 100))),x)
[Out]
int(-(log(x)*(exp(2*x)*(40*x + 1000*x^3 + 400*x^4 + 6290*x^5 + 5000*x^6 + 1500*x^7 + 200*x^8 + 10*x^9) - exp(x
)*(2*x^2 + 2*x^3 + 75*x^4 + 65*x^5 + 15*x^6 + x^7)) + x^2 - log(log(x))*log(x)*(exp(x)*(40*x + 500*x^3 + 200*x
^4 + 20*x^5) - x^2) + 10*x*log(log(x))^2*log(x))/(log(x)*(exp(2*x)*(2500*x^2 + 1000*x^3 + 15725*x^4 + 12500*x^
5 + 3750*x^6 + 500*x^7 + 25*x^8 + 100) - exp(x)*(20*x + 250*x^3 + 100*x^4 + 10*x^5) + x^2) + 25*log(log(x))^2*
log(x) + log(log(x))*log(x)*(10*x - exp(x)*(1250*x^2 + 500*x^3 + 50*x^4 + 100))), x)
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sympy [A] time = 0.95, size = 39, normalized size = 1.18 \begin {gather*} - \frac {x^{3}}{- 5 x + \left (25 x^{4} + 250 x^{3} + 625 x^{2} + 50\right ) e^{x} - 25 \log {\left (\log {\relax (x )} \right )}} - \frac {x^{2}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-10*x*ln(x)*ln(ln(x))**2+((20*x**5+200*x**4+500*x**3+40*x)*exp(x)-x**2)*ln(x)*ln(ln(x))+((-10*x**9-
200*x**8-1500*x**7-5000*x**6-6290*x**5-400*x**4-1000*x**3-40*x)*exp(x)**2+(x**7+15*x**6+65*x**5+75*x**4+2*x**3
+2*x**2)*exp(x))*ln(x)-x**2)/(25*ln(x)*ln(ln(x))**2+((-50*x**4-500*x**3-1250*x**2-100)*exp(x)+10*x)*ln(x)*ln(l
n(x))+((25*x**8+500*x**7+3750*x**6+12500*x**5+15725*x**4+1000*x**3+2500*x**2+100)*exp(x)**2+(-10*x**5-100*x**4
-250*x**3-20*x)*exp(x)+x**2)*ln(x)),x)
[Out]
-x**3/(-5*x + (25*x**4 + 250*x**3 + 625*x**2 + 50)*exp(x) - 25*log(log(x))) - x**2/5
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