3.58.94 \(\int \frac {1+e+x^x (e (6+4 x)+e (2+4 x) \log (x))}{e} \, dx\)

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

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Rubi [F]  time = 0.14, 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 {1+e+x^x (e (6+4 x)+e (2+4 x) \log (x))}{e} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + E + x^x*(E*(6 + 4*x) + E*(2 + 4*x)*Log[x]))/E,x]

[Out]

((1 + E)*x)/E + 2*x^x + 4*Defer[Int][x^x, x] + 4*Defer[Int][x^(1 + x), x] + 4*Log[x]*Defer[Int][x^(1 + x), x]
- 4*Defer[Int][Defer[Int][x^(1 + x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (1+e+x^x (e (6+4 x)+e (2+4 x) \log (x))\right ) \, dx}{e}\\ &=\frac {(1+e) x}{e}+\frac {\int x^x (e (6+4 x)+e (2+4 x) \log (x)) \, dx}{e}\\ &=\frac {(1+e) x}{e}+\frac {\int 2 e x^x (3+2 x+\log (x)+2 x \log (x)) \, dx}{e}\\ &=\frac {(1+e) x}{e}+2 \int x^x (3+2 x+\log (x)+2 x \log (x)) \, dx\\ &=\frac {(1+e) x}{e}+2 \int \left (3 x^x+2 x^{1+x}+x^x \log (x)+2 x^{1+x} \log (x)\right ) \, dx\\ &=\frac {(1+e) x}{e}+2 \int x^x \log (x) \, dx+4 \int x^{1+x} \, dx+4 \int x^{1+x} \log (x) \, dx+6 \int x^x \, dx\\ &=\frac {(1+e) x}{e}+2 x^x-2 \int x^x \, dx+4 \int x^{1+x} \, dx-4 \int \frac {\int x^{1+x} \, dx}{x} \, dx+6 \int x^x \, dx+(4 \log (x)) \int x^{1+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 17, normalized size = 1.06 \begin {gather*} x+\frac {x}{e}+2 x^x (1+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E + x^x*(E*(6 + 4*x) + E*(2 + 4*x)*Log[x]))/E,x]

[Out]

x + x/E + 2*x^x*(1 + 2*x)

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fricas [A]  time = 0.61, size = 21, normalized size = 1.31 \begin {gather*} {\left (2 \, {\left (2 \, x + 1\right )} x^{x} e + x e + x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+2)*exp(1)*log(x)+(4*x+6)*exp(1))*exp(x*log(x))+1+exp(1))/exp(1),x, algorithm="fricas")

[Out]

(2*(2*x + 1)*x^x*e + x*e + x)*e^(-1)

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giac [A]  time = 0.23, size = 29, normalized size = 1.81 \begin {gather*} 2 \, {\left (2 \, x x^{x} e + x^{x} e\right )} e^{\left (-1\right )} + {\left (x e + x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+2)*exp(1)*log(x)+(4*x+6)*exp(1))*exp(x*log(x))+1+exp(1))/exp(1),x, algorithm="giac")

[Out]

2*(2*x*x^x*e + x^x*e)*e^(-1) + (x*e + x)*e^(-1)

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maple [A]  time = 0.10, size = 27, normalized size = 1.69

method	result	size
norman	\(\left (1+{\mathrm e}\right ) {\mathrm e}^{-1} x +4 x \,{\mathrm e}^{x \ln \relax (x )}+2 \,{\mathrm e}^{x \ln \relax (x )}\)	\(27\)
risch	\({\mathrm e}^{-1} x \,{\mathrm e}+{\mathrm e}^{-1} x +{\mathrm e}^{-1} \left (4 x \,{\mathrm e}+2 \,{\mathrm e}\right ) x^{x}\)	\(28\)
default	\({\mathrm e}^{-1} \left (x +4 \,{\mathrm e} \,{\mathrm e}^{x \ln \relax (x )} x +2 \,{\mathrm e} \,{\mathrm e}^{x \ln \relax (x )}+x \,{\mathrm e}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x+2)*exp(1)*ln(x)+(4*x+6)*exp(1))*exp(x*ln(x))+1+exp(1))/exp(1),x,method=_RETURNVERBOSE)

[Out]

(1+exp(1))/exp(1)*x+4*x*exp(x*ln(x))+2*exp(x*ln(x))

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maxima [A]  time = 0.36, size = 22, normalized size = 1.38 \begin {gather*} {\left (2 \, {\left (2 \, x e + e\right )} x^{x} + x e + x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+2)*exp(1)*log(x)+(4*x+6)*exp(1))*exp(x*log(x))+1+exp(1))/exp(1),x, algorithm="maxima")

[Out]

(2*(2*x*e + e)*x^x + x*e + x)*e^(-1)

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mupad [B]  time = 4.21, size = 17, normalized size = 1.06 \begin {gather*} x+4\,x\,x^x+x\,{\mathrm {e}}^{-1}+2\,x^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-1)*(exp(1) + exp(x*log(x))*(exp(1)*(4*x + 6) + exp(1)*log(x)*(4*x + 2)) + 1),x)

[Out]

x + 4*x*x^x + x*exp(-1) + 2*x^x

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sympy [A]  time = 0.30, size = 20, normalized size = 1.25 \begin {gather*} \frac {x \left (1 + e\right )}{e} + \left (4 x + 2\right ) e^{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+2)*exp(1)*ln(x)+(4*x+6)*exp(1))*exp(x*ln(x))+1+exp(1))/exp(1),x)

[Out]

x*(1 + E)*exp(-1) + (4*x + 2)*exp(x*log(x))

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