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

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

[Out]

E^(1 + Log[x]^(-1))*x

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Rubi [A]  time = 0.026896, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2288} \[ x e^{\frac{1}{\log (x)}+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(1 + Log[x]^(-1))*x

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

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

Mathematica [A]  time = 0.0060003, size = 10, normalized size = 1. \[ x e^{\frac{1}{\log (x)}+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(1 + Log[x]^(-1))*x

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Maple [A]  time = 0.015, size = 10, normalized size = 1. \begin{align*}{{\rm e}^{1+ \left ( \ln \left ( x \right ) \right ) ^{-1}}}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14036, size = 12, normalized size = 1.2 \begin{align*} x e^{\left (\frac{1}{\log \left (x\right )} + 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(1/log(x) + 1)

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Fricas [A]  time = 1.51604, size = 36, normalized size = 3.6 \begin{align*} x e^{\left (\frac{\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(1+1/log(x))*(-1+log(x)^2)/log(x)^2,x, algorithm="fricas")

[Out]

x*e^((log(x) + 1)/log(x))

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Sympy [A]  time = 1.29315, size = 8, normalized size = 0.8 \begin{align*} x e^{1 + \frac{1}{\log{\left (x \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(1 + 1/log(x))

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Giac [A]  time = 1.09356, size = 12, normalized size = 1.2 \begin{align*} x e^{\left (\frac{1}{\log \left (x\right )} + 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(1/log(x) + 1)