3.7 \(\int \frac{e^{x+\frac{1}{\log (x)}} (-1+(1+x) \log ^2(x))}{\log ^2(x)} \, dx\)

Optimal. Leaf size=10 \[ x e^{x+\frac{1}{\log (x)}} \]

[Out]

E^(x + Log[x]^(-1))*x

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Rubi [F]  time = 0.187568, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{x+\frac{1}{\log (x)}} \left (-1+(1+x) \log ^2(x)\right )}{\log ^2(x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{e^{x+\frac{1}{\log (x)}} \left (-1+(1+x) \log ^2(x)\right )}{\log ^2(x)} \, dx &=\int \left (e^{x+\frac{1}{\log (x)}}+e^{x+\frac{1}{\log (x)}} x-\frac{e^{x+\frac{1}{\log (x)}}}{\log ^2(x)}\right ) \, dx\\ &=\int e^{x+\frac{1}{\log (x)}} \, dx+\int e^{x+\frac{1}{\log (x)}} x \, dx-\int \frac{e^{x+\frac{1}{\log (x)}}}{\log ^2(x)} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0841447, size = 10, normalized size = 1. \[ x e^{x+\frac{1}{\log (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(x + Log[x]^(-1))*x

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Maple [A]  time = 0.018, size = 10, normalized size = 1. \begin{align*}{{\rm e}^{x+ \left ( \ln \left ( x \right ) \right ) ^{-1}}}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14151, size = 12, normalized size = 1.2 \begin{align*} x e^{\left (x + \frac{1}{\log \left (x\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(x + 1/log(x))

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Fricas [A]  time = 1.5703, size = 39, normalized size = 3.9 \begin{align*} x e^{\left (\frac{x \log \left (x\right ) + 1}{\log \left (x\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^((x*log(x) + 1)/log(x))

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Sympy [A]  time = 1.83324, size = 8, normalized size = 0.8 \begin{align*} x e^{x + \frac{1}{\log{\left (x \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(x + 1/log(x))

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Giac [A]  time = 1.10192, size = 19, normalized size = 1.9 \begin{align*} x e^{\left (\frac{x \log \left (x\right ) + 1}{\log \left (x\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^((x*log(x) + 1)/log(x))