3.70.71 \(\int \frac {4+e^4 x+e^{4-x+e^{-x} x} (-1+x) x}{x} \, dx\)

Optimal. Leaf size=25 \[ -3+e^4 \left (-e^{e^{-x} x}+x\right )+2 \log \left (x^2\right ) \]

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Rubi [F]  time = 0.22, 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 {4+e^4 x+e^{4-x+e^{-x} x} (-1+x) x}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 23, normalized size = 0.92 \begin {gather*} -e^{4+e^{-x} x}+e^4 x+4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + E^4*x + E^(4 - x + x/E^x)*(-1 + x)*x)/x,x]

[Out]

-E^(4 + x/E^x) + E^4*x + 4*Log[x]

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fricas [A]  time = 0.67, size = 43, normalized size = 1.72 \begin {gather*} {\left ({\left (x e^{4} + 4 \, \log \relax (x)\right )} e^{\left (-x + \log \relax (x)\right )} - e^{\left (-x + e^{\left (-x + \log \relax (x)\right )} + \log \relax (x) + 4\right )}\right )} e^{\left (x - \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.29, size = 34, normalized size = 1.36 \begin {gather*} {\left (x e^{\left (-x + 4\right )} + 4 \, e^{\left (-x\right )} \log \relax (x) - e^{\left (x e^{\left (-x\right )} - x + 4\right )}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 21, normalized size = 0.84

method	result	size
risch	\(x \,{\mathrm e}^{4}+4 \ln \relax (x )-{\mathrm e}^{4+x \,{\mathrm e}^{-x}}\)	\(21\)
default	\(x \,{\mathrm e}^{4}+4 \ln \relax (x )-{\mathrm e}^{4} {\mathrm e}^{{\mathrm e}^{\ln \relax (x )-x}}\)	\(22\)
norman	\(x \,{\mathrm e}^{4}+4 \ln \relax (x )-{\mathrm e}^{4} {\mathrm e}^{{\mathrm e}^{\ln \relax (x )-x}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 20, normalized size = 0.80 \begin {gather*} x e^{4} - e^{\left (x e^{\left (-x\right )} + 4\right )} + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^4 - e^(x*e^(-x) + 4) + 4*log(x)

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mupad [B]  time = 4.15, size = 20, normalized size = 0.80 \begin {gather*} 4\,\ln \relax (x)+x\,{\mathrm {e}}^4-{\mathrm {e}}^4\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.18, size = 19, normalized size = 0.76 \begin {gather*} x e^{4} - e^{4} e^{x e^{- x}} + 4 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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