3.18.83 \(\int \frac {500 x+200 x \log (3)+20 x \log ^2(3)+e^{e^x+x} (-500 x-200 x \log (3)-20 x \log ^2(3))+(-1500+250 x+(-600+100 x) \log (3)+(-60+10 x) \log ^2(3)+e^{e^x} (-250-100 \log (3)-10 \log ^2(3))) \log (6+e^{e^x}-x)}{(6 x^2+e^{e^x} x^2-x^3) \log ^3(6+e^{e^x}-x)} \, dx\)
Optimal. Leaf size=24 \[ \frac {10 (5+\log (3))^2}{x \log ^2\left (6+e^{e^x}-x\right )} \]
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Rubi [F] time = 3.47, 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 {500 x+200 x \log (3)+20 x \log ^2(3)+e^{e^x+x} \left (-500 x-200 x \log (3)-20 x \log ^2(3)\right )+\left (-1500+250 x+(-600+100 x) \log (3)+(-60+10 x) \log ^2(3)+e^{e^x} \left (-250-100 \log (3)-10 \log ^2(3)\right )\right ) \log \left (6+e^{e^x}-x\right )}{\left (6 x^2+e^{e^x} x^2-x^3\right ) \log ^3\left (6+e^{e^x}-x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(500*x + 200*x*Log[3] + 20*x*Log[3]^2 + E^(E^x + x)*(-500*x - 200*x*Log[3] - 20*x*Log[3]^2) + (-1500 + 250
*x + (-600 + 100*x)*Log[3] + (-60 + 10*x)*Log[3]^2 + E^E^x*(-250 - 100*Log[3] - 10*Log[3]^2))*Log[6 + E^E^x -
x])/((6*x^2 + E^E^x*x^2 - x^3)*Log[6 + E^E^x - x]^3),x]
[Out]
20*(5 + Log[3])^2*Defer[Int][1/((6 + E^E^x - x)*x*Log[6 + E^E^x - x]^3), x] - 20*(5 + Log[3])^2*Defer[Int][E^(
E^x + x)/((6 + E^E^x - x)*x*Log[6 + E^E^x - x]^3), x] - 10*(5 + Log[3])^2*Defer[Int][1/(x^2*Log[6 + E^E^x - x]
^2), x]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 x \log ^2(3)+x (500+200 \log (3))+e^{e^x+x} \left (-500 x-200 x \log (3)-20 x \log ^2(3)\right )+\left (-1500+250 x+(-600+100 x) \log (3)+(-60+10 x) \log ^2(3)+e^{e^x} \left (-250-100 \log (3)-10 \log ^2(3)\right )\right ) \log \left (6+e^{e^x}-x\right )}{\left (6 x^2+e^{e^x} x^2-x^3\right ) \log ^3\left (6+e^{e^x}-x\right )} \, dx\\ &=\int \frac {x \left (500+200 \log (3)+20 \log ^2(3)\right )+e^{e^x+x} \left (-500 x-200 x \log (3)-20 x \log ^2(3)\right )+\left (-1500+250 x+(-600+100 x) \log (3)+(-60+10 x) \log ^2(3)+e^{e^x} \left (-250-100 \log (3)-10 \log ^2(3)\right )\right ) \log \left (6+e^{e^x}-x\right )}{\left (6 x^2+e^{e^x} x^2-x^3\right ) \log ^3\left (6+e^{e^x}-x\right )} \, dx\\ &=\int \frac {10 (5+\log (3))^2 \left (-2 \left (-1+e^{e^x+x}\right ) x-\left (6+e^{e^x}-x\right ) \log \left (6+e^{e^x}-x\right )\right )}{\left (6+e^{e^x}-x\right ) x^2 \log ^3\left (6+e^{e^x}-x\right )} \, dx\\ &=\left (10 (5+\log (3))^2\right ) \int \frac {-2 \left (-1+e^{e^x+x}\right ) x-\left (6+e^{e^x}-x\right ) \log \left (6+e^{e^x}-x\right )}{\left (6+e^{e^x}-x\right ) x^2 \log ^3\left (6+e^{e^x}-x\right )} \, dx\\ &=\left (10 (5+\log (3))^2\right ) \int \left (-\frac {2 e^{e^x+x}}{\left (6+e^{e^x}-x\right ) x \log ^3\left (6+e^{e^x}-x\right )}-\frac {-2 x+6 \log \left (6+e^{e^x}-x\right )+e^{e^x} \log \left (6+e^{e^x}-x\right )-x \log \left (6+e^{e^x}-x\right )}{\left (6+e^{e^x}-x\right ) x^2 \log ^3\left (6+e^{e^x}-x\right )}\right ) \, dx\\ &=-\left (\left (10 (5+\log (3))^2\right ) \int \frac {-2 x+6 \log \left (6+e^{e^x}-x\right )+e^{e^x} \log \left (6+e^{e^x}-x\right )-x \log \left (6+e^{e^x}-x\right )}{\left (6+e^{e^x}-x\right ) x^2 \log ^3\left (6+e^{e^x}-x\right )} \, dx\right )-\left (20 (5+\log (3))^2\right ) \int \frac {e^{e^x+x}}{\left (6+e^{e^x}-x\right ) x \log ^3\left (6+e^{e^x}-x\right )} \, dx\\ &=-\left (\left (10 (5+\log (3))^2\right ) \int \frac {-\frac {2 x}{6+e^{e^x}-x}+\log \left (6+e^{e^x}-x\right )}{x^2 \log ^3\left (6+e^{e^x}-x\right )} \, dx\right )-\left (20 (5+\log (3))^2\right ) \int \frac {e^{e^x+x}}{\left (6+e^{e^x}-x\right ) x \log ^3\left (6+e^{e^x}-x\right )} \, dx\\ &=-\left (\left (10 (5+\log (3))^2\right ) \int \left (-\frac {2}{\left (6+e^{e^x}-x\right ) x \log ^3\left (6+e^{e^x}-x\right )}+\frac {1}{x^2 \log ^2\left (6+e^{e^x}-x\right )}\right ) \, dx\right )-\left (20 (5+\log (3))^2\right ) \int \frac {e^{e^x+x}}{\left (6+e^{e^x}-x\right ) x \log ^3\left (6+e^{e^x}-x\right )} \, dx\\ &=-\left (\left (10 (5+\log (3))^2\right ) \int \frac {1}{x^2 \log ^2\left (6+e^{e^x}-x\right )} \, dx\right )+\left (20 (5+\log (3))^2\right ) \int \frac {1}{\left (6+e^{e^x}-x\right ) x \log ^3\left (6+e^{e^x}-x\right )} \, dx-\left (20 (5+\log (3))^2\right ) \int \frac {e^{e^x+x}}{\left (6+e^{e^x}-x\right ) x \log ^3\left (6+e^{e^x}-x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.45, size = 24, normalized size = 1.00 \begin {gather*} \frac {10 (5+\log (3))^2}{x \log ^2\left (6+e^{e^x}-x\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(500*x + 200*x*Log[3] + 20*x*Log[3]^2 + E^(E^x + x)*(-500*x - 200*x*Log[3] - 20*x*Log[3]^2) + (-1500
+ 250*x + (-600 + 100*x)*Log[3] + (-60 + 10*x)*Log[3]^2 + E^E^x*(-250 - 100*Log[3] - 10*Log[3]^2))*Log[6 + E^
E^x - x])/((6*x^2 + E^E^x*x^2 - x^3)*Log[6 + E^E^x - x]^3),x]
[Out]
(10*(5 + Log[3])^2)/(x*Log[6 + E^E^x - x]^2)
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fricas [A] time = 0.69, size = 38, normalized size = 1.58 \begin {gather*} \frac {10 \, {\left (\log \relax (3)^{2} + 10 \, \log \relax (3) + 25\right )}}{x \log \left (-{\left ({\left (x - 6\right )} e^{x} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-10*log(3)^2-100*log(3)-250)*exp(exp(x))+(10*x-60)*log(3)^2+(100*x-600)*log(3)+250*x-1500)*log(ex
p(exp(x))-x+6)+(-20*x*log(3)^2-200*x*log(3)-500*x)*exp(x)*exp(exp(x))+20*x*log(3)^2+200*x*log(3)+500*x)/(exp(e
xp(x))*x^2-x^3+6*x^2)/log(exp(exp(x))-x+6)^3,x, algorithm="fricas")
[Out]
10*(log(3)^2 + 10*log(3) + 25)/(x*log(-((x - 6)*e^x - e^(x + e^x))*e^(-x))^2)
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-10*log(3)^2-100*log(3)-250)*exp(exp(x))+(10*x-60)*log(3)^2+(100*x-600)*log(3)+250*x-1500)*log(ex
p(exp(x))-x+6)+(-20*x*log(3)^2-200*x*log(3)-500*x)*exp(x)*exp(exp(x))+20*x*log(3)^2+200*x*log(3)+500*x)/(exp(e
xp(x))*x^2-x^3+6*x^2)/log(exp(exp(x))-x+6)^3,x, algorithm="giac")
[Out]
Timed out
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maple [A] time = 0.07, size = 27, normalized size = 1.12
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\(\frac {10 \ln \relax (3)^{2}+100 \ln \relax (3)+250}{\ln \left ({\mathrm e}^{{\mathrm e}^{x}}-x +6\right )^{2} x}\) |
\(27\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((-10*ln(3)^2-100*ln(3)-250)*exp(exp(x))+(10*x-60)*ln(3)^2+(100*x-600)*ln(3)+250*x-1500)*ln(exp(exp(x))-x
+6)+(-20*x*ln(3)^2-200*x*ln(3)-500*x)*exp(x)*exp(exp(x))+20*x*ln(3)^2+200*x*ln(3)+500*x)/(exp(exp(x))*x^2-x^3+
6*x^2)/ln(exp(exp(x))-x+6)^3,x,method=_RETURNVERBOSE)
[Out]
10*(ln(3)^2+10*ln(3)+25)/x/ln(exp(exp(x))-x+6)^2
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maxima [A] time = 0.99, size = 26, normalized size = 1.08 \begin {gather*} \frac {10 \, {\left (\log \relax (3)^{2} + 10 \, \log \relax (3) + 25\right )}}{x \log \left (-x + e^{\left (e^{x}\right )} + 6\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-10*log(3)^2-100*log(3)-250)*exp(exp(x))+(10*x-60)*log(3)^2+(100*x-600)*log(3)+250*x-1500)*log(ex
p(exp(x))-x+6)+(-20*x*log(3)^2-200*x*log(3)-500*x)*exp(x)*exp(exp(x))+20*x*log(3)^2+200*x*log(3)+500*x)/(exp(e
xp(x))*x^2-x^3+6*x^2)/log(exp(exp(x))-x+6)^3,x, algorithm="maxima")
[Out]
10*(log(3)^2 + 10*log(3) + 25)/(x*log(-x + e^(e^x) + 6)^2)
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mupad [B] time = 4.01, size = 1482, normalized size = 61.75 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((500*x + 200*x*log(3) + 20*x*log(3)^2 + log(exp(exp(x)) - x + 6)*(250*x + log(3)*(100*x - 600) + log(3)^2*
(10*x - 60) - exp(exp(x))*(100*log(3) + 10*log(3)^2 + 250) - 1500) - exp(exp(x))*exp(x)*(500*x + 200*x*log(3)
+ 20*x*log(3)^2))/(log(exp(exp(x)) - x + 6)^3*(x^2*exp(exp(x)) + 6*x^2 - x^3)),x)
[Out]
((10*(log(3) + 5)^2)/x + (5*log(exp(exp(x)) - x + 6)*(log(3) + 5)^2*(exp(exp(x)) - x + 6))/(x^2*(exp(x + exp(x
)) - 1)))/log(exp(exp(x)) - x + 6)^2 - ((5*(log(3) + 5)^2*(exp(exp(x)) - x + 6))/(x^2*(exp(x + exp(x)) - 1)) +
(5*log(exp(exp(x)) - x + 6)*(log(3) + 5)^2*(exp(exp(x)) - x + 6)*(x + 12*exp(x + exp(x)) - 2*exp(exp(x)) + 2*
exp(x + 2*exp(x)) + x*exp(x + 2*exp(x)) + 6*x*exp(2*x + exp(x)) - x^2*exp(x + exp(x)) - x^2*exp(2*x + exp(x))
+ 6*x*exp(x + exp(x)) - 12))/(x^3*(exp(x + exp(x)) - 1)^3))/log(exp(exp(x)) - x + 6) + (exp(-2*x)*(100*log(3)
+ x*(50*log(3) + 5*log(3)^2 + 125) + 10*log(3)^2 + 250))/x^3 + (5*exp(-x)*(100*x + 3*x^2*log(3)^2 + 600*x*exp(
2*x) + 325*x^2*exp(x) - 50*x^3*exp(x) + 40*x*log(3) + 400*x^2*exp(2*x) + 150*x^2*exp(3*x) - 75*x^3*exp(2*x) -
25*x^3*exp(3*x) + 4*x*log(3)^2 + 30*x^2*log(3) + 700*x*exp(x) + 75*x^2 + 280*x*exp(x)*log(3) + 16*x^2*exp(2*x)
*log(3)^2 + 6*x^2*exp(3*x)*log(3)^2 - 3*x^3*exp(2*x)*log(3)^2 - x^3*exp(3*x)*log(3)^2 + 240*x*exp(2*x)*log(3)
+ 28*x*exp(x)*log(3)^2 + 130*x^2*exp(x)*log(3) - 20*x^3*exp(x)*log(3) + 24*x*exp(2*x)*log(3)^2 + 160*x^2*exp(2
*x)*log(3) + 60*x^2*exp(3*x)*log(3) - 30*x^3*exp(2*x)*log(3) - 10*x^3*exp(3*x)*log(3) + 13*x^2*exp(x)*log(3)^2
- 2*x^3*exp(x)*log(3)^2))/(x^4*(exp(2*x) + exp(x))*(exp(x + exp(x)) - 1)) + (5*exp(-x)*(50*x + 3*x^2*log(3)^2
+ 2400*x*exp(2*x) + 1800*x*exp(3*x) + 650*x^2*exp(x) - 100*x^3*exp(x) + 20*x*log(3) + 1475*x^2*exp(2*x) + 180
0*x^2*exp(3*x) - 450*x^3*exp(2*x) + 900*x^2*exp(4*x) - 650*x^3*exp(3*x) + 25*x^4*exp(2*x) - 300*x^3*exp(4*x) +
50*x^4*exp(3*x) + 25*x^4*exp(4*x) + 2*x*log(3)^2 + 30*x^2*log(3) + 650*x*exp(x) + 75*x^2 + 260*x*exp(x)*log(3
) + 59*x^2*exp(2*x)*log(3)^2 + 72*x^2*exp(3*x)*log(3)^2 - 18*x^3*exp(2*x)*log(3)^2 + 36*x^2*exp(4*x)*log(3)^2
- 26*x^3*exp(3*x)*log(3)^2 + x^4*exp(2*x)*log(3)^2 - 12*x^3*exp(4*x)*log(3)^2 + 2*x^4*exp(3*x)*log(3)^2 + x^4*
exp(4*x)*log(3)^2 + 960*x*exp(2*x)*log(3) + 720*x*exp(3*x)*log(3) + 26*x*exp(x)*log(3)^2 + 260*x^2*exp(x)*log(
3) - 40*x^3*exp(x)*log(3) + 96*x*exp(2*x)*log(3)^2 + 590*x^2*exp(2*x)*log(3) + 72*x*exp(3*x)*log(3)^2 + 720*x^
2*exp(3*x)*log(3) - 180*x^3*exp(2*x)*log(3) + 360*x^2*exp(4*x)*log(3) - 260*x^3*exp(3*x)*log(3) + 10*x^4*exp(2
*x)*log(3) - 120*x^3*exp(4*x)*log(3) + 20*x^4*exp(3*x)*log(3) + 10*x^4*exp(4*x)*log(3) + 26*x^2*exp(x)*log(3)^
2 - 4*x^3*exp(x)*log(3)^2))/(x^4*(exp(2*x) + exp(x))*(exp(2*x + 2*exp(x)) - 2*exp(x + exp(x)) + 1)) + (5*exp(-
2*x)*(25*x^2*exp(x) + 350*x^2*exp(2*x) + 1525*x^2*exp(3*x) - 50*x^3*exp(2*x) + 2100*x^2*exp(4*x) - 400*x^3*exp
(3*x) + 900*x^2*exp(5*x) - 650*x^3*exp(4*x) + 25*x^4*exp(3*x) - 300*x^3*exp(5*x) + 50*x^4*exp(4*x) + 25*x^4*ex
p(5*x) + 14*x^2*exp(2*x)*log(3)^2 + 61*x^2*exp(3*x)*log(3)^2 - 2*x^3*exp(2*x)*log(3)^2 + 84*x^2*exp(4*x)*log(3
)^2 - 16*x^3*exp(3*x)*log(3)^2 + 36*x^2*exp(5*x)*log(3)^2 - 26*x^3*exp(4*x)*log(3)^2 + x^4*exp(3*x)*log(3)^2 -
12*x^3*exp(5*x)*log(3)^2 + 2*x^4*exp(4*x)*log(3)^2 + x^4*exp(5*x)*log(3)^2 + 10*x^2*exp(x)*log(3) + 140*x^2*e
xp(2*x)*log(3) + 610*x^2*exp(3*x)*log(3) - 20*x^3*exp(2*x)*log(3) + 840*x^2*exp(4*x)*log(3) - 160*x^3*exp(3*x)
*log(3) + 360*x^2*exp(5*x)*log(3) - 260*x^3*exp(4*x)*log(3) + 10*x^4*exp(3*x)*log(3) - 120*x^3*exp(5*x)*log(3)
+ 20*x^4*exp(4*x)*log(3) + 10*x^4*exp(5*x)*log(3) + x^2*exp(x)*log(3)^2))/(x^4*(exp(2*x) + exp(x))*(3*exp(x +
exp(x)) - 3*exp(2*x + 2*exp(x)) + exp(3*x + 3*exp(x)) - 1))
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sympy [A] time = 0.36, size = 26, normalized size = 1.08 \begin {gather*} \frac {10 \log {\relax (3 )}^{2} + 100 \log {\relax (3 )} + 250}{x \log {\left (- x + e^{e^{x}} + 6 \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-10*ln(3)**2-100*ln(3)-250)*exp(exp(x))+(10*x-60)*ln(3)**2+(100*x-600)*ln(3)+250*x-1500)*ln(exp(e
xp(x))-x+6)+(-20*x*ln(3)**2-200*x*ln(3)-500*x)*exp(x)*exp(exp(x))+20*x*ln(3)**2+200*x*ln(3)+500*x)/(exp(exp(x)
)*x**2-x**3+6*x**2)/ln(exp(exp(x))-x+6)**3,x)
[Out]
(10*log(3)**2 + 100*log(3) + 250)/(x*log(-x + exp(exp(x)) + 6)**2)
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