3.92.47 \(\int \frac {1+e^{e^x+x} (5 x-e^{2 x} x+x^2+e^{9+x} (-2 x-e^x x)+e^x (2 x+x^2))}{x} \, dx\)

Optimal. Leaf size=26 \[ e^{e^x+x} \left (4-e^x-e^{9+x}+x\right )+\log (x) \]

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Rubi [F]  time = 0.26, 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 {1+e^{e^x+x} \left (5 x-e^{2 x} x+x^2+e^{9+x} \left (-2 x-e^x x\right )+e^x \left (2 x+x^2\right )\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + E^(E^x + x)*(5*x - E^(2*x)*x + x^2 + E^(9 + x)*(-2*x - E^x*x) + E^x*(2*x + x^2)))/x,x]

[Out]

5*E^E^x - 2*E^E^x*(1 - E^9) + 2*E^(E^x + x)*(1 - E^9) - 2*E^E^x*(1 + E^9) + 2*E^(E^x + x)*(1 + E^9) - E^(E^x +
2*x)*(1 + E^9) + Log[x] + Defer[Int][E^(E^x + x)*x, x] + Defer[Int][E^(E^x + 2*x)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^{e^x+3 x} \left (1+e^9\right )-e^{e^x+2 x} \left (-2+2 e^9-x\right )+\frac {1}{x}+e^{e^x+x} (5+x)\right ) \, dx\\ &=\log (x)+\left (-1-e^9\right ) \int e^{e^x+3 x} \, dx-\int e^{e^x+2 x} \left (-2+2 e^9-x\right ) \, dx+\int e^{e^x+x} (5+x) \, dx\\ &=\log (x)+\left (-1-e^9\right ) \operatorname {Subst}\left (\int e^x x^2 \, dx,x,e^x\right )+\int \left (5 e^{e^x+x}+e^{e^x+x} x\right ) \, dx-\int \left (-2 e^{e^x+2 x} \left (1-e^9\right )-e^{e^x+2 x} x\right ) \, dx\\ &=-e^{e^x+2 x} \left (1+e^9\right )+\log (x)+5 \int e^{e^x+x} \, dx+\left (2 \left (1-e^9\right )\right ) \int e^{e^x+2 x} \, dx+\left (2 \left (1+e^9\right )\right ) \operatorname {Subst}\left (\int e^x x \, dx,x,e^x\right )+\int e^{e^x+x} x \, dx+\int e^{e^x+2 x} x \, dx\\ &=2 e^{e^x+x} \left (1+e^9\right )-e^{e^x+2 x} \left (1+e^9\right )+\log (x)+5 \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )+\left (2 \left (1-e^9\right )\right ) \operatorname {Subst}\left (\int e^x x \, dx,x,e^x\right )-\left (2 \left (1+e^9\right )\right ) \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )+\int e^{e^x+x} x \, dx+\int e^{e^x+2 x} x \, dx\\ &=5 e^{e^x}+2 e^{e^x+x} \left (1-e^9\right )-2 e^{e^x} \left (1+e^9\right )+2 e^{e^x+x} \left (1+e^9\right )-e^{e^x+2 x} \left (1+e^9\right )+\log (x)-\left (2 \left (1-e^9\right )\right ) \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )+\int e^{e^x+x} x \, dx+\int e^{e^x+2 x} x \, dx\\ &=5 e^{e^x}-2 e^{e^x} \left (1-e^9\right )+2 e^{e^x+x} \left (1-e^9\right )-2 e^{e^x} \left (1+e^9\right )+2 e^{e^x+x} \left (1+e^9\right )-e^{e^x+2 x} \left (1+e^9\right )+\log (x)+\int e^{e^x+x} x \, dx+\int e^{e^x+2 x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.83, size = 30, normalized size = 1.15 \begin {gather*} e^{e^x} \left (e^{2 x} \left (-1-e^9\right )+e^x (4+x)\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^(E^x + x)*(5*x - E^(2*x)*x + x^2 + E^(9 + x)*(-2*x - E^x*x) + E^x*(2*x + x^2)))/x,x]

[Out]

E^E^x*(E^(2*x)*(-1 - E^9) + E^x*(4 + x)) + Log[x]

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fricas [A]  time = 0.55, size = 40, normalized size = 1.54 \begin {gather*} {\left ({\left ({\left (x + 4\right )} e^{9} - {\left (e^{9} + 1\right )} e^{\left (x + 9\right )}\right )} e^{\left ({\left (x e^{9} + e^{\left (x + 9\right )}\right )} e^{\left (-9\right )}\right )} + e^{9} \log \relax (x)\right )} e^{\left (-9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x-2*x)*exp(x+9)-x*exp(x)^2+(x^2+2*x)*exp(x)+x^2+5*x)*exp(exp(x)+x)+1)/x,x, algorithm="fri
cas")

[Out]

(((x + 4)*e^9 - (e^9 + 1)*e^(x + 9))*e^((x*e^9 + e^(x + 9))*e^(-9)) + e^9*log(x))*e^(-9)

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giac [B]  time = 0.14, size = 50, normalized size = 1.92 \begin {gather*} {\left (x e^{\left (4 \, x + e^{x}\right )} + e^{\left (3 \, x\right )} \log \relax (x) - e^{\left (5 \, x + e^{x} + 9\right )} - e^{\left (5 \, x + e^{x}\right )} + 4 \, e^{\left (4 \, x + e^{x}\right )}\right )} e^{\left (-3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x-2*x)*exp(x+9)-x*exp(x)^2+(x^2+2*x)*exp(x)+x^2+5*x)*exp(exp(x)+x)+1)/x,x, algorithm="gia
c")

[Out]

(x*e^(4*x + e^x) + e^(3*x)*log(x) - e^(5*x + e^x + 9) - e^(5*x + e^x) + 4*e^(4*x + e^x))*e^(-3*x)

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maple [A]  time = 0.06, size = 23, normalized size = 0.88

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{x}+x} \left (x -{\mathrm e}^{x}-{\mathrm e}^{x +9}+4\right )+\ln \relax (x )\)	\(23\)
norman	\({\mathrm e}^{{\mathrm e}^{x}+x} x +\left (-{\mathrm e}^{9}-1\right ) {\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}+x}+4 \,{\mathrm e}^{{\mathrm e}^{x}+x}+\ln \relax (x )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-exp(x)*x-2*x)*exp(x+9)-x*exp(x)^2+(x^2+2*x)*exp(x)+x^2+5*x)*exp(exp(x)+x)+1)/x,x,method=_RETURNVERBOSE
)

[Out]

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

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maxima [A]  time = 0.41, size = 31, normalized size = 1.19 \begin {gather*} -{\left ({\left (e^{9} + 1\right )} e^{\left (2 \, x\right )} - {\left (x + 4\right )} e^{x} + 5\right )} e^{\left (e^{x}\right )} + 5 \, e^{\left (e^{x}\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x-2*x)*exp(x+9)-x*exp(x)^2+(x^2+2*x)*exp(x)+x^2+5*x)*exp(exp(x)+x)+1)/x,x, algorithm="max
ima")

[Out]

-((e^9 + 1)*e^(2*x) - (x + 4)*e^x + 5)*e^(e^x) + 5*e^(e^x) + log(x)

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mupad [B]  time = 5.35, size = 30, normalized size = 1.15 \begin {gather*} 4\,{\mathrm {e}}^{x+{\mathrm {e}}^x}+\ln \relax (x)-{\mathrm {e}}^{2\,x+{\mathrm {e}}^x}\,\left ({\mathrm {e}}^9+1\right )+x\,{\mathrm {e}}^{x+{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + exp(x))*(5*x - x*exp(2*x) + exp(x)*(2*x + x^2) - exp(x + 9)*(2*x + x*exp(x)) + x^2) + 1)/x,x)

[Out]

4*exp(x + exp(x)) + log(x) - exp(2*x + exp(x))*(exp(9) + 1) + x*exp(x + exp(x))

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sympy [A]  time = 0.24, size = 22, normalized size = 0.85 \begin {gather*} \left (x - e^{9} e^{x} - e^{x} + 4\right ) e^{x + e^{x}} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)*x-2*x)*exp(x+9)-x*exp(x)**2+(x**2+2*x)*exp(x)+x**2+5*x)*exp(exp(x)+x)+1)/x,x)

[Out]

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

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