3.87.33 \(\int \frac {5^{(2-e^{2 x}) x^x} (2-e^{2 x}) x^{x+(2-e^{2 x}) x^x} (-2+e^{2 x}+(-2 x+3 e^{2 x} x+(-2 x+e^{2 x} x) \log (x)) \log (5 x))}{-2 x+e^{2 x} x} \, dx\)

Optimal. Leaf size=31 \[ 5^{\left (2-e^{2 x}\right ) x^x} x^{\left (2-e^{2 x}\right ) x^x} \]

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Rubi [F]  time = 15.70, 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 {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5^((2 - E^(2*x))*x^x)*(2 - E^(2*x))*x^(x + (2 - E^(2*x))*x^x)*(-2 + E^(2*x) + (-2*x + 3*E^(2*x)*x + (-2*x
+ E^(2*x)*x)*Log[x])*Log[5*x]))/(-2*x + E^(2*x)*x),x]

[Out]

2*Defer[Int][x^(-1 + x - (-2 + E^(2*x))*x^x)/5^((-2 + E^(2*x))*x^x), x] - Defer[Int][(E^(2*x)*x^(-1 + x - (-2
+ E^(2*x))*x^x))/5^((-2 + E^(2*x))*x^x), x] + 2*Log[5*x]*Defer[Int][x^(x - (-2 + E^(2*x))*x^x)/5^((-2 + E^(2*x
))*x^x), x] + 2*Log[x]*Log[5*x]*Defer[Int][x^(x - (-2 + E^(2*x))*x^x)/5^((-2 + E^(2*x))*x^x), x] - 3*Log[5*x]*
Defer[Int][(E^(2*x)*x^(x - (-2 + E^(2*x))*x^x))/5^((-2 + E^(2*x))*x^x), x] - Log[x]*Log[5*x]*Defer[Int][(E^(2*
x)*x^(x - (-2 + E^(2*x))*x^x))/5^((-2 + E^(2*x))*x^x), x] - 2*Defer[Int][Defer[Int][x^(x - (-2 + E^(2*x))*x^x)
/5^((-2 + E^(2*x))*x^x), x]/x, x] - 2*Log[x]*Defer[Int][Defer[Int][x^(x - (-2 + E^(2*x))*x^x)/5^((-2 + E^(2*x)
)*x^x), x]/x, x] - 2*Log[5*x]*Defer[Int][Defer[Int][x^(x - (-2 + E^(2*x))*x^x)/5^((-2 + E^(2*x))*x^x), x]/x, x
] + 3*Defer[Int][Defer[Int][(E^(2*x)*x^(x - (-2 + E^(2*x))*x^x))/5^((-2 + E^(2*x))*x^x), x]/x, x] + Log[x]*Def
er[Int][Defer[Int][(E^(2*x)*x^(x - (-2 + E^(2*x))*x^x))/5^((-2 + E^(2*x))*x^x), x]/x, x] + Log[5*x]*Defer[Int]
[Defer[Int][(E^(2*x)*x^(x - (-2 + E^(2*x))*x^x))/5^((-2 + E^(2*x))*x^x), x]/x, x] + 4*Defer[Int][Defer[Int][De
fer[Int][x^(x - (-2 + E^(2*x))*x^x)/5^((-2 + E^(2*x))*x^x), x]/x, x]/x, x] - 2*Defer[Int][Defer[Int][Defer[Int
][(E^(2*x)*x^(x - (-2 + E^(2*x))*x^x))/5^((-2 + E^(2*x))*x^x), x]/x, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \left (2-e^{2 x}-x \left (-2+3 e^{2 x}+\left (-2+e^{2 x}\right ) \log (x)\right ) \log (5 x)\right ) \, dx\\ &=\int \left (2\ 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x}-5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{-1+x-\left (-2+e^{2 x}\right ) x^x}-5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \left (-2+3 e^{2 x}-2 \log (x)+e^{2 x} \log (x)\right ) \log (5 x)\right ) \, dx\\ &=2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \left (-2+3 e^{2 x}-2 \log (x)+e^{2 x} \log (x)\right ) \log (5 x) \, dx\\ &=2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int \left (-2 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} (1+\log (x)) \log (5 x)+5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} (3+\log (x)) \log (5 x)\right ) \, dx\\ &=2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx+2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} (1+\log (x)) \log (5 x) \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} (3+\log (x)) \log (5 x) \, dx\\ &=2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx+2 \int \left (5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \log (5 x)+5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \log (x) \log (5 x)\right ) \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int \left (3\ 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \log (5 x)+5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \log (x) \log (5 x)\right ) \, dx\\ &=2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx+2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \log (5 x) \, dx+2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \log (x) \log (5 x) \, dx-3 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \log (5 x) \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \log (x) \log (5 x) \, dx\\ &=2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-2 \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx-2 \int \frac {\log (x) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx-2 \int \frac {\log (5 x) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx+3 \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx+(2 \log (5 x)) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx-(3 \log (5 x)) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx-(\log (x) \log (5 x)) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx+(2 \log (x) \log (5 x)) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx+\int \frac {\log (x) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx+\int \frac {\log (5 x) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx\\ &=2 \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-2 \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx+2 \left (2 \int \frac {\int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx}{x} \, dx\right )+3 \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx+\log (x) \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx-(2 \log (x)) \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx+\log (5 x) \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx+(2 \log (5 x)) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx-(2 \log (5 x)) \int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx-(3 \log (5 x)) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx-(\log (x) \log (5 x)) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx+(2 \log (x) \log (5 x)) \int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx-\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{-1+x-\left (-2+e^{2 x}\right ) x^x} \, dx-2 \int \frac {\int \frac {\int 5^{-\left (\left (-2+e^{2 x}\right ) x^x\right )} e^{2 x} x^{x-\left (-2+e^{2 x}\right ) x^x} \, dx}{x} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 4.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5^{\left (2-e^{2 x}\right ) x^x} \left (2-e^{2 x}\right ) x^{x+\left (2-e^{2 x}\right ) x^x} \left (-2+e^{2 x}+\left (-2 x+3 e^{2 x} x+\left (-2 x+e^{2 x} x\right ) \log (x)\right ) \log (5 x)\right )}{-2 x+e^{2 x} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(5^((2 - E^(2*x))*x^x)*(2 - E^(2*x))*x^(x + (2 - E^(2*x))*x^x)*(-2 + E^(2*x) + (-2*x + 3*E^(2*x)*x +
(-2*x + E^(2*x)*x)*Log[x])*Log[5*x]))/(-2*x + E^(2*x)*x),x]

[Out]

Integrate[(5^((2 - E^(2*x))*x^x)*(2 - E^(2*x))*x^(x + (2 - E^(2*x))*x^x)*(-2 + E^(2*x) + (-2*x + 3*E^(2*x)*x +
(-2*x + E^(2*x)*x)*Log[x])*Log[5*x]))/(-2*x + E^(2*x)*x), x]

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fricas [A]  time = 0.67, size = 22, normalized size = 0.71 \begin {gather*} e^{\left ({\left (\log \relax (5) + \log \relax (x)\right )} e^{\left (x \log \relax (x) + \log \left (-e^{\left (2 \, x\right )} + 2\right )\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x*exp(x)^2-2*x)*log(x)+3*x*exp(x)^2-2*x)*log(5*x)+exp(x)^2-2)*exp(log(-exp(x)^2+2)+x*log(x))*exp(
log(5*x)*exp(log(-exp(x)^2+2)+x*log(x)))/(x*exp(x)^2-2*x),x, algorithm="fricas")

[Out]

e^((log(5) + log(x))*e^(x*log(x) + log(-e^(2*x) + 2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x*exp(x)^2-2*x)*log(x)+3*x*exp(x)^2-2*x)*log(5*x)+exp(x)^2-2)*exp(log(-exp(x)^2+2)+x*log(x))*exp(
log(5*x)*exp(log(-exp(x)^2+2)+x*log(x)))/(x*exp(x)^2-2*x),x, algorithm="giac")

[Out]

integrate(((3*x*e^(2*x) + (x*e^(2*x) - 2*x)*log(x) - 2*x)*log(5*x) + e^(2*x) - 2)*(5*x)^e^(x*log(x) + log(-e^(
2*x) + 2))*e^(x*log(x) + log(-e^(2*x) + 2))/(x*e^(2*x) - 2*x), x)

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maple [A]  time = 0.10, size = 18, normalized size = 0.58

method	result	size
risch	\({\mathrm e}^{-\left (\ln \relax (5)+\ln \relax (x )\right ) \left (-2+{\mathrm e}^{2 x}\right ) x^{x}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x*exp(x)^2-2*x)*ln(x)+3*x*exp(x)^2-2*x)*ln(5*x)+exp(x)^2-2)*exp(ln(-exp(x)^2+2)+x*ln(x))*exp(ln(5*x)*ex
p(ln(-exp(x)^2+2)+x*ln(x)))/(x*exp(x)^2-2*x),x,method=_RETURNVERBOSE)

[Out]

exp(-(ln(5)+ln(x))*(-2+exp(2*x))*x^x)

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maxima [B]  time = 0.55, size = 42, normalized size = 1.35 \begin {gather*} e^{\left (2 \, x^{x} \log \relax (5) - e^{\left (x \log \relax (x) + 2 \, x\right )} \log \relax (5) + 2 \, x^{x} \log \relax (x) - e^{\left (x \log \relax (x) + 2 \, x\right )} \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x*exp(x)^2-2*x)*log(x)+3*x*exp(x)^2-2*x)*log(5*x)+exp(x)^2-2)*exp(log(-exp(x)^2+2)+x*log(x))*exp(
log(5*x)*exp(log(-exp(x)^2+2)+x*log(x)))/(x*exp(x)^2-2*x),x, algorithm="maxima")

[Out]

e^(2*x^x*log(5) - e^(x*log(x) + 2*x)*log(5) + 2*x^x*log(x) - e^(x*log(x) + 2*x)*log(x))

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mupad [B]  time = 5.46, size = 19, normalized size = 0.61 \begin {gather*} {\left (5\,x\right )}^{2\,x^x-x^x\,{\mathrm {e}}^{2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(2 - exp(2*x)) + x*log(x))*exp(log(5*x)*exp(log(2 - exp(2*x)) + x*log(x)))*(log(5*x)*(2*x - 3*x*ex
p(2*x) + log(x)*(2*x - x*exp(2*x))) - exp(2*x) + 2))/(2*x - x*exp(2*x)),x)

[Out]

(5*x)^(2*x^x - x^x*exp(2*x))

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sympy [A]  time = 22.03, size = 20, normalized size = 0.65 \begin {gather*} e^{\left (2 - e^{2 x}\right ) \left (\log {\relax (x )} + \log {\relax (5 )}\right ) e^{x \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x*exp(x)**2-2*x)*ln(x)+3*x*exp(x)**2-2*x)*ln(5*x)+exp(x)**2-2)*exp(ln(-exp(x)**2+2)+x*ln(x))*exp(
ln(5*x)*exp(ln(-exp(x)**2+2)+x*ln(x)))/(x*exp(x)**2-2*x),x)

[Out]

exp((2 - exp(2*x))*(log(x) + log(5))*exp(x*log(x)))

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