3.44.80 \(\int \frac {e^{2 e^{-x^5+2 x^4 \log (\frac {45+x \log (5)}{x})-x^3 \log ^2(\frac {45+x \log (5)}{x})}-x^5+2 x^4 \log (\frac {45+x \log (5)}{x})-x^3 \log ^2(\frac {45+x \log (5)}{x})} (-180 x^3-450 x^4-10 x^5 \log (5)+(180 x^2+720 x^3+16 x^4 \log (5)) \log (\frac {45+x \log (5)}{x})+(-270 x^2-6 x^3 \log (5)) \log ^2(\frac {45+x \log (5)}{x}))}{45+x \log (5)} \, dx\)

Optimal. Leaf size=26 \[ e^{2 e^{-x^3 \left (-x+\log \left (\frac {45}{x}+\log (5)\right )\right )^2}} \]

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Rubi [F]  time = 40.43, 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 {\exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) \left (-180 x^3-450 x^4-10 x^5 \log (5)+\left (180 x^2+720 x^3+16 x^4 \log (5)\right ) \log \left (\frac {45+x \log (5)}{x}\right )+\left (-270 x^2-6 x^3 \log (5)\right ) \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )}{45+x \log (5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log
[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2)*(-180*x^3 - 450*x^4 - 10*x^5*Log[5] + (180*x^2 + 720*x^3 + 16*x^4*Log[
5])*Log[(45 + x*Log[5])/x] + (-270*x^2 - 6*x^3*Log[5])*Log[(45 + x*Log[5])/x]^2))/(45 + x*Log[5]),x]

[Out]

(-364500*Defer[Int][E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*
Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2), x])/Log[5]^3 + (8100*Defer[Int][E^(2*E^(-x^5 + 2*x^4*L
og[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*L
og[5])/x]^2)*x, x])/Log[5]^2 - (180*Defer[Int][E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*L
og[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2)*x^2, x])/Log[5] - 10*Defer[I
nt][E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[
5])/x] - x^3*Log[(45 + x*Log[5])/x]^2)*x^4, x] + (16402500*Defer[Int][E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])
/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2)/(45 +
x*Log[5]), x])/Log[5]^3 - (8100*Defer[Int][E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[
5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2)*Log[45/x + Log[5]], x])/Log[5]^
2 + (180*Defer[Int][E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*
Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2)*x*Log[45/x + Log[5]], x])/Log[5] + 16*Defer[Int][E^(2*E
^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x
^3*Log[(45 + x*Log[5])/x]^2)*x^3*Log[45/x + Log[5]], x] + (364500*Defer[Int][(E^(2*E^(-x^5 + 2*x^4*Log[(45 + x
*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^
2)*Log[45/x + Log[5]])/(45 + x*Log[5]), x])/Log[5]^2 - 6*Defer[Int][E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x
] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2)*x^2*Log
[45/x + Log[5]]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) x^2 \left (x-\log \left (\frac {45}{x}+\log (5)\right )\right ) \left (-90-225 x-5 x^2 \log (5)+135 \log \left (\frac {45}{x}+\log (5)\right )+3 x \log (5) \log \left (\frac {45}{x}+\log (5)\right )\right )}{45+x \log (5)} \, dx\\ &=2 \int \frac {\exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) x^2 \left (x-\log \left (\frac {45}{x}+\log (5)\right )\right ) \left (-90-225 x-5 x^2 \log (5)+135 \log \left (\frac {45}{x}+\log (5)\right )+3 x \log (5) \log \left (\frac {45}{x}+\log (5)\right )\right )}{45+x \log (5)} \, dx\\ &=2 \int \left (-\frac {5 \exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) x^3 \left (18+45 x+x^2 \log (5)\right )}{45+x \log (5)}+\frac {2 \exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) x^2 \left (45+180 x+4 x^2 \log (5)\right ) \log \left (\frac {45}{x}+\log (5)\right )}{45+x \log (5)}-3 \exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) x^2 \log ^2\left (\frac {45}{x}+\log (5)\right )\right ) \, dx\\ &=4 \int \frac {\exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) x^2 \left (45+180 x+4 x^2 \log (5)\right ) \log \left (\frac {45}{x}+\log (5)\right )}{45+x \log (5)} \, dx-6 \int \exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) x^2 \log ^2\left (\frac {45}{x}+\log (5)\right ) \, dx-10 \int \frac {\exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )\right ) x^3 \left (18+45 x+x^2 \log (5)\right )}{45+x \log (5)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 42, normalized size = 1.62 \begin {gather*} e^{2 e^{-x^5-x^3 \log ^2\left (\frac {45}{x}+\log (5)\right )} \left (\frac {45}{x}+\log (5)\right )^{2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2) - x^5 + 2*x^4*Log[(45 +
x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2)*(-180*x^3 - 450*x^4 - 10*x^5*Log[5] + (180*x^2 + 720*x^3 + 16*x^
4*Log[5])*Log[(45 + x*Log[5])/x] + (-270*x^2 - 6*x^3*Log[5])*Log[(45 + x*Log[5])/x]^2))/(45 + x*Log[5]),x]

[Out]

E^(2*E^(-x^5 - x^3*Log[45/x + Log[5]]^2)*(45/x + Log[5])^(2*x^4))

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fricas [A]  time = 1.05, size = 44, normalized size = 1.69 \begin {gather*} e^{\left (2 \, e^{\left (-x^{5} + 2 \, x^{4} \log \left (\frac {x \log \relax (5) + 45}{x}\right ) - x^{3} \log \left (\frac {x \log \relax (5) + 45}{x}\right )^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3*log(5)-270*x^2)*log((x*log(5)+45)/x)^2+(16*x^4*log(5)+720*x^3+180*x^2)*log((x*log(5)+45)/x)
-10*x^5*log(5)-450*x^4-180*x^3)*exp(2/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5))/(x*log(5
)+45)/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5),x, algorithm="fricas")

[Out]

e^(2*e^(-x^5 + 2*x^4*log((x*log(5) + 45)/x) - x^3*log((x*log(5) + 45)/x)^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (5 \, x^{5} \log \relax (5) + 225 \, x^{4} + 90 \, x^{3} + 3 \, {\left (x^{3} \log \relax (5) + 45 \, x^{2}\right )} \log \left (\frac {x \log \relax (5) + 45}{x}\right )^{2} - 2 \, {\left (4 \, x^{4} \log \relax (5) + 180 \, x^{3} + 45 \, x^{2}\right )} \log \left (\frac {x \log \relax (5) + 45}{x}\right )\right )} e^{\left (-x^{5} + 2 \, x^{4} \log \left (\frac {x \log \relax (5) + 45}{x}\right ) - x^{3} \log \left (\frac {x \log \relax (5) + 45}{x}\right )^{2} + 2 \, e^{\left (-x^{5} + 2 \, x^{4} \log \left (\frac {x \log \relax (5) + 45}{x}\right ) - x^{3} \log \left (\frac {x \log \relax (5) + 45}{x}\right )^{2}\right )}\right )}}{x \log \relax (5) + 45}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3*log(5)-270*x^2)*log((x*log(5)+45)/x)^2+(16*x^4*log(5)+720*x^3+180*x^2)*log((x*log(5)+45)/x)
-10*x^5*log(5)-450*x^4-180*x^3)*exp(2/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5))/(x*log(5
)+45)/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5),x, algorithm="giac")

[Out]

integrate(-2*(5*x^5*log(5) + 225*x^4 + 90*x^3 + 3*(x^3*log(5) + 45*x^2)*log((x*log(5) + 45)/x)^2 - 2*(4*x^4*lo
g(5) + 180*x^3 + 45*x^2)*log((x*log(5) + 45)/x))*e^(-x^5 + 2*x^4*log((x*log(5) + 45)/x) - x^3*log((x*log(5) +
45)/x)^2 + 2*e^(-x^5 + 2*x^4*log((x*log(5) + 45)/x) - x^3*log((x*log(5) + 45)/x)^2))/(x*log(5) + 45), x)

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maple [A]  time = 0.12, size = 45, normalized size = 1.73

method	result	size
risch	\({\mathrm e}^{2 \left (\frac {x \ln \relax (5)+45}{x}\right )^{2 x^{4}} {\mathrm e}^{-x^{3} \left (\ln \left (\frac {x \ln \relax (5)+45}{x}\right )^{2}+x^{2}\right )}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^3*ln(5)-270*x^2)*ln((x*ln(5)+45)/x)^2+(16*x^4*ln(5)+720*x^3+180*x^2)*ln((x*ln(5)+45)/x)-10*x^5*ln(5
)-450*x^4-180*x^3)*exp(2/exp(x^3*ln((x*ln(5)+45)/x)^2-2*x^4*ln((x*ln(5)+45)/x)+x^5))/(x*ln(5)+45)/exp(x^3*ln((
x*ln(5)+45)/x)^2-2*x^4*ln((x*ln(5)+45)/x)+x^5),x,method=_RETURNVERBOSE)

[Out]

exp(2/(((x*ln(5)+45)/x)^(-2*x^4))*exp(-x^3*(ln((x*ln(5)+45)/x)^2+x^2)))

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maxima [B]  time = 1.75, size = 66, normalized size = 2.54 \begin {gather*} e^{\left (2 \, e^{\left (-x^{5} + 2 \, x^{4} \log \left (x \log \relax (5) + 45\right ) - x^{3} \log \left (x \log \relax (5) + 45\right )^{2} - 2 \, x^{4} \log \relax (x) + 2 \, x^{3} \log \left (x \log \relax (5) + 45\right ) \log \relax (x) - x^{3} \log \relax (x)^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3*log(5)-270*x^2)*log((x*log(5)+45)/x)^2+(16*x^4*log(5)+720*x^3+180*x^2)*log((x*log(5)+45)/x)
-10*x^5*log(5)-450*x^4-180*x^3)*exp(2/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5))/(x*log(5
)+45)/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5),x, algorithm="maxima")

[Out]

e^(2*e^(-x^5 + 2*x^4*log(x*log(5) + 45) - x^3*log(x*log(5) + 45)^2 - 2*x^4*log(x) + 2*x^3*log(x*log(5) + 45)*l
og(x) - x^3*log(x)^2))

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mupad [B]  time = 3.84, size = 42, normalized size = 1.62 \begin {gather*} {\mathrm {e}}^{2\,{\mathrm {e}}^{-x^3\,{\ln \left (\frac {x\,\ln \relax (5)+45}{x}\right )}^2}\,{\mathrm {e}}^{-x^5}\,{\left (\ln \relax (5)+\frac {45}{x}\right )}^{2\,x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*exp(2*x^4*log((x*log(5) + 45)/x) - x^5 - x^3*log((x*log(5) + 45)/x)^2))*exp(2*x^4*log((x*log(5) +
45)/x) - x^5 - x^3*log((x*log(5) + 45)/x)^2)*(log((x*log(5) + 45)/x)^2*(6*x^3*log(5) + 270*x^2) - log((x*log(5
) + 45)/x)*(16*x^4*log(5) + 180*x^2 + 720*x^3) + 10*x^5*log(5) + 180*x^3 + 450*x^4))/(x*log(5) + 45),x)

[Out]

exp(2*exp(-x^3*log((x*log(5) + 45)/x)^2)*exp(-x^5)*(log(5) + 45/x)^(2*x^4))

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sympy [A]  time = 5.66, size = 37, normalized size = 1.42 \begin {gather*} e^{2 e^{- x^{5} + 2 x^{4} \log {\left (\frac {x \log {\relax (5 )} + 45}{x} \right )} - x^{3} \log {\left (\frac {x \log {\relax (5 )} + 45}{x} \right )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**3*ln(5)-270*x**2)*ln((x*ln(5)+45)/x)**2+(16*x**4*ln(5)+720*x**3+180*x**2)*ln((x*ln(5)+45)/x)
-10*x**5*ln(5)-450*x**4-180*x**3)*exp(2/exp(x**3*ln((x*ln(5)+45)/x)**2-2*x**4*ln((x*ln(5)+45)/x)+x**5))/(x*ln(
5)+45)/exp(x**3*ln((x*ln(5)+45)/x)**2-2*x**4*ln((x*ln(5)+45)/x)+x**5),x)

[Out]

exp(2*exp(-x**5 + 2*x**4*log((x*log(5) + 45)/x) - x**3*log((x*log(5) + 45)/x)**2))

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