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

Optimal. Leaf size=15 \[ 3+e^{2 x+x^{\frac {1}{x}}} x \]

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Rubi [B]  time = 0.21, antiderivative size = 54, normalized size of antiderivative = 3.60, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2288} \begin {gather*} \frac {e^{x^{\frac {1}{x}}+2 x} \left (x^{\frac {1}{x}} (1-\log (x))+2 x^2\right )}{x \left (x^{\frac {1}{x}} \left (\frac {1}{x^2}-\frac {\log (x)}{x^2}\right )+2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.15, size = 13, normalized size = 0.87 \begin {gather*} e^{2 x+x^{\frac {1}{x}}} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*x + x^x^(-1))*x

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fricas [A]  time = 0.76, size = 12, normalized size = 0.80 \begin {gather*} x e^{\left (2 \, x + x^{\left (\frac {1}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(2*x + x^(1/x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{2} - x^{\left (\frac {1}{x}\right )} {\left (\log \relax (x) - 1\right )} + x\right )} e^{\left (2 \, x + x^{\left (\frac {1}{x}\right )}\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^2 - x^(1/x)*(log(x) - 1) + x)*e^(2*x + x^(1/x))/x, x)

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maple [A]  time = 0.06, size = 13, normalized size = 0.87

method	result	size
risch	\({\mathrm e}^{x^{\frac {1}{x}}+2 x} x\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x^(1/x)+2*x)*x

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maxima [A]  time = 0.43, size = 12, normalized size = 0.80 \begin {gather*} x e^{\left (2 \, x + x^{\left (\frac {1}{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(2*x + x^(1/x))

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mupad [B]  time = 4.33, size = 12, normalized size = 0.80 \begin {gather*} x\,{\mathrm {e}}^{x^{1/x}}\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(x^(1/x))*exp(2*x)

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sympy [A]  time = 11.99, size = 12, normalized size = 0.80 \begin {gather*} x e^{2 x + e^{\frac {\log {\relax (x )}}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(2*x + exp(log(x)/x))

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