3.57.44 \(\int \frac {e^{-2 x+e^{-2 x} (4 e^{2 x}+x)} (e^{2 x}+9 x-18 x^2+(x-2 x^2) \log (x))}{x} \, dx\)

Optimal. Leaf size=16 \[ e^{4+e^{-2 x} x} (9+\log (x)) \]

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Rubi [F]  time = 1.26, 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 {e^{-2 x+e^{-2 x} \left (4 e^{2 x}+x\right )} \left (e^{2 x}+9 x-18 x^2+\left (x-2 x^2\right ) \log (x)\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-2*x + (4*E^(2*x) + x)/E^(2*x))*(E^(2*x) + 9*x - 18*x^2 + (x - 2*x^2)*Log[x]))/x,x]

[Out]

9*Defer[Int][E^(4 + (-2 + E^(-2*x))*x), x] + Log[x]*Defer[Int][E^(4 + (-2 + E^(-2*x))*x), x] + Defer[Int][E^(4
+ x/E^(2*x))/x, x] - 18*Defer[Int][E^(4 + (-2 + E^(-2*x))*x)*x, x] - 2*Log[x]*Defer[Int][E^(4 + (-2 + E^(-2*x
))*x)*x, x] - Defer[Int][Defer[Int][E^(4 + (-2 + E^(-2*x))*x), x]/x, x] + 2*Defer[Int][Defer[Int][E^(4 + (-2 +
E^(-2*x))*x)*x, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{e^{-2 x} \left (4 e^{2 x}+x\right )}}{x}-e^{-2 x+e^{-2 x} \left (4 e^{2 x}+x\right )} (-1+2 x) (9+\log (x))\right ) \, dx\\ &=\int \frac {e^{e^{-2 x} \left (4 e^{2 x}+x\right )}}{x} \, dx-\int e^{-2 x+e^{-2 x} \left (4 e^{2 x}+x\right )} (-1+2 x) (9+\log (x)) \, dx\\ &=\int \frac {e^{4+e^{-2 x} x}}{x} \, dx-\int \left (9 e^{-2 x+e^{-2 x} \left (4 e^{2 x}+x\right )} (-1+2 x)+e^{-2 x+e^{-2 x} \left (4 e^{2 x}+x\right )} (-1+2 x) \log (x)\right ) \, dx\\ &=-\left (9 \int e^{-2 x+e^{-2 x} \left (4 e^{2 x}+x\right )} (-1+2 x) \, dx\right )+\int \frac {e^{4+e^{-2 x} x}}{x} \, dx-\int e^{-2 x+e^{-2 x} \left (4 e^{2 x}+x\right )} (-1+2 x) \log (x) \, dx\\ &=-\left (9 \int e^{4+\left (-2+e^{-2 x}\right ) x} (-1+2 x) \, dx\right )+\log (x) \int e^{4+\left (-2+e^{-2 x}\right ) x} \, dx-(2 \log (x)) \int e^{4+\left (-2+e^{-2 x}\right ) x} x \, dx+\int \frac {e^{4+e^{-2 x} x}}{x} \, dx+\int \frac {-\int e^{4+\left (-2+e^{-2 x}\right ) x} \, dx+2 \int e^{4+\left (-2+e^{-2 x}\right ) x} x \, dx}{x} \, dx\\ &=-\left (9 \int \left (-e^{4+\left (-2+e^{-2 x}\right ) x}+2 e^{4+\left (-2+e^{-2 x}\right ) x} x\right ) \, dx\right )+\log (x) \int e^{4+\left (-2+e^{-2 x}\right ) x} \, dx-(2 \log (x)) \int e^{4+\left (-2+e^{-2 x}\right ) x} x \, dx+\int \frac {e^{4+e^{-2 x} x}}{x} \, dx+\int \left (-\frac {\int e^{4+\left (-2+e^{-2 x}\right ) x} \, dx}{x}+\frac {2 \int e^{4+\left (-2+e^{-2 x}\right ) x} x \, dx}{x}\right ) \, dx\\ &=2 \int \frac {\int e^{4+\left (-2+e^{-2 x}\right ) x} x \, dx}{x} \, dx+9 \int e^{4+\left (-2+e^{-2 x}\right ) x} \, dx-18 \int e^{4+\left (-2+e^{-2 x}\right ) x} x \, dx+\log (x) \int e^{4+\left (-2+e^{-2 x}\right ) x} \, dx-(2 \log (x)) \int e^{4+\left (-2+e^{-2 x}\right ) x} x \, dx+\int \frac {e^{4+e^{-2 x} x}}{x} \, dx-\int \frac {\int e^{4+\left (-2+e^{-2 x}\right ) x} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.53, size = 16, normalized size = 1.00 \begin {gather*} e^{4+e^{-2 x} x} (9+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-2*x + (4*E^(2*x) + x)/E^(2*x))*(E^(2*x) + 9*x - 18*x^2 + (x - 2*x^2)*Log[x]))/x,x]

[Out]

E^(4 + x/E^(2*x))*(9 + Log[x])

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fricas [B]  time = 0.51, size = 35, normalized size = 2.19 \begin {gather*} {\left (e^{\left (2 \, x\right )} \log \relax (x) + 9 \, e^{\left (2 \, x\right )}\right )} e^{\left (-{\left (2 \, {\left (x - 2\right )} e^{\left (2 \, x\right )} - x\right )} e^{\left (-2 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+x)*log(x)+exp(x)^2-18*x^2+9*x)*exp((4*exp(x)^2+x)/exp(x)^2)/x/exp(x)^2,x, algorithm="fricas
")

[Out]

(e^(2*x)*log(x) + 9*e^(2*x))*e^(-(2*(x - 2)*e^(2*x) - x)*e^(-2*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (18 \, x^{2} + {\left (2 \, x^{2} - x\right )} \log \relax (x) - 9 \, x - e^{\left (2 \, x\right )}\right )} e^{\left ({\left (x + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )} - 2 \, x\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+x)*log(x)+exp(x)^2-18*x^2+9*x)*exp((4*exp(x)^2+x)/exp(x)^2)/x/exp(x)^2,x, algorithm="giac")

[Out]

integrate(-(18*x^2 + (2*x^2 - x)*log(x) - 9*x - e^(2*x))*e^((x + 4*e^(2*x))*e^(-2*x) - 2*x)/x, x)

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maple [A]  time = 0.07, size = 20, normalized size = 1.25

method	result	size
risch	\(\left (9+\ln \relax (x )\right ) {\mathrm e}^{\left (4 \,{\mathrm e}^{2 x}+x \right ) {\mathrm e}^{-2 x}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2+x)*ln(x)+exp(x)^2-18*x^2+9*x)*exp((4*exp(x)^2+x)/exp(x)^2)/x/exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

(9+ln(x))*exp((4*exp(2*x)+x)*exp(-2*x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (18 \, x^{2} + {\left (2 \, x^{2} - x\right )} \log \relax (x) - 9 \, x - e^{\left (2 \, x\right )}\right )} e^{\left ({\left (x + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )} - 2 \, x\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+x)*log(x)+exp(x)^2-18*x^2+9*x)*exp((4*exp(x)^2+x)/exp(x)^2)/x/exp(x)^2,x, algorithm="maxima
")

[Out]

-integrate((18*x^2 + (2*x^2 - x)*log(x) - 9*x - e^(2*x))*e^((x + 4*e^(2*x))*e^(-2*x) - 2*x)/x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}\,\left (x+4\,{\mathrm {e}}^{2\,x}\right )}\,\left (9\,x+{\mathrm {e}}^{2\,x}+\ln \relax (x)\,\left (x-2\,x^2\right )-18\,x^2\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*x)*exp(exp(-2*x)*(x + 4*exp(2*x)))*(9*x + exp(2*x) + log(x)*(x - 2*x^2) - 18*x^2))/x,x)

[Out]

int((exp(-2*x)*exp(exp(-2*x)*(x + 4*exp(2*x)))*(9*x + exp(2*x) + log(x)*(x - 2*x^2) - 18*x^2))/x, x)

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sympy [A]  time = 0.43, size = 19, normalized size = 1.19 \begin {gather*} \left (\log {\relax (x )} + 9\right ) e^{\left (x + 4 e^{2 x}\right ) e^{- 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2+x)*ln(x)+exp(x)**2-18*x**2+9*x)*exp((4*exp(x)**2+x)/exp(x)**2)/x/exp(x)**2,x)

[Out]

(log(x) + 9)*exp((x + 4*exp(2*x))*exp(-2*x))

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