3.16.45 \(\int \frac {-20000 e^{2 e^x} x+1600 x^2+288 x^3+(100000 x+5500 x^2) \log (3)-500000 x \log ^2(3)+e^{e^x} (20000 x+1100 x^2+e^x (-10000 x^2-2500 x^3)-200000 x \log (3))}{625 e^{2 e^x}+16 x^2+1000 x \log (3)+15625 \log ^2(3)+e^{e^x} (200 x+6250 \log (3))} \, dx\)

Optimal. Leaf size=28 \[ 4 x^2 \left (-4+\frac {4+x}{e^{e^x}+\frac {4 x}{25}+5 \log (3)}\right ) \]

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Rubi [F]  time = 3.52, 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 {-20000 e^{2 e^x} x+1600 x^2+288 x^3+\left (100000 x+5500 x^2\right ) \log (3)-500000 x \log ^2(3)+e^{e^x} \left (20000 x+1100 x^2+e^x \left (-10000 x^2-2500 x^3\right )-200000 x \log (3)\right )}{625 e^{2 e^x}+16 x^2+1000 x \log (3)+15625 \log ^2(3)+e^{e^x} (200 x+6250 \log (3))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-20000*E^(2*E^x)*x + 1600*x^2 + 288*x^3 + (100000*x + 5500*x^2)*Log[3] - 500000*x*Log[3]^2 + E^E^x*(20000
*x + 1100*x^2 + E^x*(-10000*x^2 - 2500*x^3) - 200000*x*Log[3]))/(625*E^(2*E^x) + 16*x^2 + 1000*x*Log[3] + 1562
5*Log[3]^2 + E^E^x*(200*x + 6250*Log[3])),x]

[Out]

100000*Log[3]*(1 - Log[243])*Defer[Int][x/(25*E^E^x + 4*x + 125*Log[3])^2, x] + 20000*(1 - 10*Log[3])*Defer[In
t][(E^E^x*x)/(25*E^E^x + 4*x + 125*Log[3])^2, x] - 20000*Defer[Int][(E^(2*E^x)*x)/(25*E^E^x + 4*x + 125*Log[3]
)^2, x] + 100*(16 + 55*Log[3])*Defer[Int][x^2/(25*E^E^x + 4*x + 125*Log[3])^2, x] + 1100*Defer[Int][(E^E^x*x^2
)/(25*E^E^x + 4*x + 125*Log[3])^2, x] - 10000*Defer[Int][(E^(E^x + x)*x^2)/(25*E^E^x + 4*x + 125*Log[3])^2, x]
+ 288*Defer[Int][x^3/(25*E^E^x + 4*x + 125*Log[3])^2, x] - 2500*Defer[Int][(E^(E^x + x)*x^3)/(25*E^E^x + 4*x
+ 125*Log[3])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x \left (-5000 e^{2 e^x}+72 x^2-625 e^{e^x+x} x (4+x)+25 e^{e^x} (200+11 x-2000 \log (3))+25 x (16+55 \log (3))-25000 \log (3) (-1+\log (243))\right )}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx\\ &=4 \int \frac {x \left (-5000 e^{2 e^x}+72 x^2-625 e^{e^x+x} x (4+x)+25 e^{e^x} (200+11 x-2000 \log (3))+25 x (16+55 \log (3))-25000 \log (3) (-1+\log (243))\right )}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx\\ &=4 \int \left (-\frac {5000 e^{2 e^x} x}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}+\frac {72 x^3}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}-\frac {625 e^{e^x+x} x^2 (4+x)}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}+\frac {25 e^{e^x} x (200+11 x-2000 \log (3))}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}+\frac {25 x^2 (16+55 \log (3))}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}-\frac {25000 x \log (3) (-1+\log (243))}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}\right ) \, dx\\ &=100 \int \frac {e^{e^x} x (200+11 x-2000 \log (3))}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+288 \int \frac {x^3}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx-2500 \int \frac {e^{e^x+x} x^2 (4+x)}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx-20000 \int \frac {e^{2 e^x} x}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+(100 (16+55 \log (3))) \int \frac {x^2}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+(100000 \log (3) (1-\log (243))) \int \frac {x}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx\\ &=100 \int \left (\frac {11 e^{e^x} x^2}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}-\frac {200 e^{e^x} x (-1+10 \log (3))}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}\right ) \, dx+288 \int \frac {x^3}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx-2500 \int \left (\frac {4 e^{e^x+x} x^2}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}+\frac {e^{e^x+x} x^3}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2}\right ) \, dx-20000 \int \frac {e^{2 e^x} x}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+(100 (16+55 \log (3))) \int \frac {x^2}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+(100000 \log (3) (1-\log (243))) \int \frac {x}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx\\ &=288 \int \frac {x^3}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+1100 \int \frac {e^{e^x} x^2}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx-2500 \int \frac {e^{e^x+x} x^3}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx-10000 \int \frac {e^{e^x+x} x^2}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx-20000 \int \frac {e^{2 e^x} x}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+(20000 (1-10 \log (3))) \int \frac {e^{e^x} x}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+(100 (16+55 \log (3))) \int \frac {x^2}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx+(100000 \log (3) (1-\log (243))) \int \frac {x}{\left (25 e^{e^x}+4 x+125 \log (3)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.16, size = 91, normalized size = 3.25 \begin {gather*} -\frac {4 x^2 \left (-36 e^x x^2+x \left (36+400 e^{e^x+x}+25 e^x (-16+35 \log (3))\right )+100 \left (-4+125 e^x \log (3)\right ) \left (-1+e^{e^x}+\log (243)\right )\right )}{\left (25 e^{e^x}+4 x+125 \log (3)\right ) \left (-4+e^x (4 x+125 \log (3))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20000*E^(2*E^x)*x + 1600*x^2 + 288*x^3 + (100000*x + 5500*x^2)*Log[3] - 500000*x*Log[3]^2 + E^E^x*
(20000*x + 1100*x^2 + E^x*(-10000*x^2 - 2500*x^3) - 200000*x*Log[3]))/(625*E^(2*E^x) + 16*x^2 + 1000*x*Log[3]
+ 15625*Log[3]^2 + E^E^x*(200*x + 6250*Log[3])),x]

[Out]

(-4*x^2*(-36*E^x*x^2 + x*(36 + 400*E^(E^x + x) + 25*E^x*(-16 + 35*Log[3])) + 100*(-4 + 125*E^x*Log[3])*(-1 + E
^E^x + Log[243])))/((25*E^E^x + 4*x + 125*Log[3])*(-4 + E^x*(4*x + 125*Log[3])))

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fricas [A]  time = 1.13, size = 43, normalized size = 1.54 \begin {gather*} \frac {4 \, {\left (9 \, x^{3} - 100 \, x^{2} e^{\left (e^{x}\right )} - 500 \, x^{2} \log \relax (3) + 100 \, x^{2}\right )}}{4 \, x + 25 \, e^{\left (e^{x}\right )} + 125 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20000*x*exp(exp(x))^2+((-2500*x^3-10000*x^2)*exp(x)-200000*x*log(3)+1100*x^2+20000*x)*exp(exp(x))-
500000*x*log(3)^2+(5500*x^2+100000*x)*log(3)+288*x^3+1600*x^2)/(625*exp(exp(x))^2+(6250*log(3)+200*x)*exp(exp(
x))+15625*log(3)^2+1000*x*log(3)+16*x^2),x, algorithm="fricas")

[Out]

4*(9*x^3 - 100*x^2*e^(e^x) - 500*x^2*log(3) + 100*x^2)/(4*x + 25*e^(e^x) + 125*log(3))

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giac [A]  time = 0.22, size = 43, normalized size = 1.54 \begin {gather*} \frac {4 \, {\left (9 \, x^{3} - 100 \, x^{2} e^{\left (e^{x}\right )} - 500 \, x^{2} \log \relax (3) + 100 \, x^{2}\right )}}{4 \, x + 25 \, e^{\left (e^{x}\right )} + 125 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20000*x*exp(exp(x))^2+((-2500*x^3-10000*x^2)*exp(x)-200000*x*log(3)+1100*x^2+20000*x)*exp(exp(x))-
500000*x*log(3)^2+(5500*x^2+100000*x)*log(3)+288*x^3+1600*x^2)/(625*exp(exp(x))^2+(6250*log(3)+200*x)*exp(exp(
x))+15625*log(3)^2+1000*x*log(3)+16*x^2),x, algorithm="giac")

[Out]

4*(9*x^3 - 100*x^2*e^(e^x) - 500*x^2*log(3) + 100*x^2)/(4*x + 25*e^(e^x) + 125*log(3))

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maple [A]  time = 0.25, size = 30, normalized size = 1.07

method	result	size
risch	\(-16 x^{2}+\frac {100 x^{2} \left (4+x \right )}{4 x +125 \ln \relax (3)+25 \,{\mathrm e}^{{\mathrm e}^{x}}}\)	\(30\)
norman	\(\frac {\left (400-2000 \ln \relax (3)\right ) x^{2}+36 x^{3}-400 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}}{4 x +125 \ln \relax (3)+25 \,{\mathrm e}^{{\mathrm e}^{x}}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-20000*x*exp(exp(x))^2+((-2500*x^3-10000*x^2)*exp(x)-200000*x*ln(3)+1100*x^2+20000*x)*exp(exp(x))-500000*
x*ln(3)^2+(5500*x^2+100000*x)*ln(3)+288*x^3+1600*x^2)/(625*exp(exp(x))^2+(6250*ln(3)+200*x)*exp(exp(x))+15625*
ln(3)^2+1000*x*ln(3)+16*x^2),x,method=_RETURNVERBOSE)

[Out]

-16*x^2+100*x^2*(4+x)/(4*x+125*ln(3)+25*exp(exp(x)))

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maxima [A]  time = 0.55, size = 42, normalized size = 1.50 \begin {gather*} \frac {4 \, {\left (9 \, x^{3} - 100 \, x^{2} {\left (5 \, \log \relax (3) - 1\right )} - 100 \, x^{2} e^{\left (e^{x}\right )}\right )}}{4 \, x + 25 \, e^{\left (e^{x}\right )} + 125 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20000*x*exp(exp(x))^2+((-2500*x^3-10000*x^2)*exp(x)-200000*x*log(3)+1100*x^2+20000*x)*exp(exp(x))-
500000*x*log(3)^2+(5500*x^2+100000*x)*log(3)+288*x^3+1600*x^2)/(625*exp(exp(x))^2+(6250*log(3)+200*x)*exp(exp(
x))+15625*log(3)^2+1000*x*log(3)+16*x^2),x, algorithm="maxima")

[Out]

4*(9*x^3 - 100*x^2*(5*log(3) - 1) - 100*x^2*e^(e^x))/(4*x + 25*e^(e^x) + 125*log(3))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \relax (3)\,\left (5500\,x^2+100000\,x\right )+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (20000\,x-{\mathrm {e}}^x\,\left (2500\,x^3+10000\,x^2\right )-200000\,x\,\ln \relax (3)+1100\,x^2\right )-20000\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}-500000\,x\,{\ln \relax (3)}^2+1600\,x^2+288\,x^3}{625\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+1000\,x\,\ln \relax (3)+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (200\,x+6250\,\ln \relax (3)\right )+15625\,{\ln \relax (3)}^2+16\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)*(100000*x + 5500*x^2) + exp(exp(x))*(20000*x - exp(x)*(10000*x^2 + 2500*x^3) - 200000*x*log(3) + 1
100*x^2) - 20000*x*exp(2*exp(x)) - 500000*x*log(3)^2 + 1600*x^2 + 288*x^3)/(625*exp(2*exp(x)) + 1000*x*log(3)
+ exp(exp(x))*(200*x + 6250*log(3)) + 15625*log(3)^2 + 16*x^2),x)

[Out]

int((log(3)*(100000*x + 5500*x^2) + exp(exp(x))*(20000*x - exp(x)*(10000*x^2 + 2500*x^3) - 200000*x*log(3) + 1
100*x^2) - 20000*x*exp(2*exp(x)) - 500000*x*log(3)^2 + 1600*x^2 + 288*x^3)/(625*exp(2*exp(x)) + 1000*x*log(3)
+ exp(exp(x))*(200*x + 6250*log(3)) + 15625*log(3)^2 + 16*x^2), x)

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sympy [A]  time = 0.19, size = 29, normalized size = 1.04 \begin {gather*} - 16 x^{2} + \frac {100 x^{3} + 400 x^{2}}{4 x + 25 e^{e^{x}} + 125 \log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20000*x*exp(exp(x))**2+((-2500*x**3-10000*x**2)*exp(x)-200000*x*ln(3)+1100*x**2+20000*x)*exp(exp(x
))-500000*x*ln(3)**2+(5500*x**2+100000*x)*ln(3)+288*x**3+1600*x**2)/(625*exp(exp(x))**2+(6250*ln(3)+200*x)*exp
(exp(x))+15625*ln(3)**2+1000*x*ln(3)+16*x**2),x)

[Out]

-16*x**2 + (100*x**3 + 400*x**2)/(4*x + 25*exp(exp(x)) + 125*log(3))

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