3.61.41 \(\int \frac {-4+x^{.\frac {1}{2}/x} (-1+\log (x))+2 \log (\frac {x^2}{5})}{2 x^2} \, dx\)

Optimal. Leaf size=26 \[ 1-x^{\left .\frac {1}{2}\right /x}-\frac {\log \left (\frac {x^2}{5}\right )}{x} \]

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Rubi [F]  time = 0.10, 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+x^{\left .\frac {1}{2}\right /x} (-1+\log (x))+2 \log \left (\frac {x^2}{5}\right )}{2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 + x^(1/(2*x))*(-1 + Log[x]) + 2*Log[x^2/5])/(2*x^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-4+x^{\left .\frac {1}{2}\right /x} (-1+\log (x))+2 \log \left (\frac {x^2}{5}\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \left (x^{-2+\frac {1}{2 x}} (-1+\log (x))+\frac {2 \left (-2+\log \left (\frac {x^2}{5}\right )\right )}{x^2}\right ) \, dx\\ &=\frac {1}{2} \int x^{-2+\frac {1}{2 x}} (-1+\log (x)) \, dx+\int \frac {-2+\log \left (\frac {x^2}{5}\right )}{x^2} \, dx\\ &=-\frac {\log \left (\frac {x^2}{5}\right )}{x}+\frac {1}{2} \int \left (-x^{-2+\frac {1}{2 x}}+x^{-2+\frac {1}{2 x}} \log (x)\right ) \, dx\\ &=-\frac {\log \left (\frac {x^2}{5}\right )}{x}-\frac {1}{2} \int x^{-2+\frac {1}{2 x}} \, dx+\frac {1}{2} \int x^{-2+\frac {1}{2 x}} \log (x) \, dx\\ &=-\frac {\log \left (\frac {x^2}{5}\right )}{x}-\frac {1}{2} \int x^{-2+\frac {1}{2 x}} \, dx-\frac {1}{2} \int \frac {\int x^{-2+\frac {1}{2 x}} \, dx}{x} \, dx+\frac {1}{2} \log (x) \int x^{-2+\frac {1}{2 x}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 25, normalized size = 0.96 \begin {gather*} -\frac {x^{1+\frac {1}{2 x}}+\log \left (\frac {x^2}{5}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + x^(1/(2*x))*(-1 + Log[x]) + 2*Log[x^2/5])/(2*x^2),x]

[Out]

-((x^(1 + 1/(2*x)) + Log[x^2/5])/x)

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fricas [A]  time = 0.85, size = 23, normalized size = 0.88 \begin {gather*} -\frac {x x^{\frac {1}{2 \, x}} - \log \relax (5) + 2 \, \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((log(x)-1)*exp(1/2*log(x)/x)+2*log(1/5*x^2)-4)/x^2,x, algorithm="fricas")

[Out]

-(x*x^(1/2/x) - log(5) + 2*log(x))/x

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giac [A]  time = 0.22, size = 23, normalized size = 0.88 \begin {gather*} -x^{\frac {1}{2 \, x}} + \frac {\log \relax (5)}{x} - \frac {2 \, \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((log(x)-1)*exp(1/2*log(x)/x)+2*log(1/5*x^2)-4)/x^2,x, algorithm="giac")

[Out]

-x^(1/2/x) + log(5)/x - 2*log(x)/x

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maple [A]  time = 0.08, size = 27, normalized size = 1.04

method	result	size
default	\(-{\mathrm e}^{\frac {\ln \relax (x )}{2 x}}-\frac {\ln \left (x^{2}\right )}{x}+\frac {\ln \relax (5)}{x}\)	\(27\)
risch	\(-\frac {2 \ln \relax (x )}{x}+\frac {i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 \ln \relax (5)}{2 x}-x^{\frac {1}{2 x}}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((ln(x)-1)*exp(1/2*ln(x)/x)+2*ln(1/5*x^2)-4)/x^2,x,method=_RETURNVERBOSE)

[Out]

-exp(1/2*ln(x)/x)-ln(x^2)/x+ln(5)/x

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maxima [A]  time = 0.49, size = 30, normalized size = 1.15 \begin {gather*} -\frac {x x^{\frac {1}{2 \, x}} - \log \relax (5) + 2 \, \log \relax (x) + 2}{x} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((log(x)-1)*exp(1/2*log(x)/x)+2*log(1/5*x^2)-4)/x^2,x, algorithm="maxima")

[Out]

-(x*x^(1/2/x) - log(5) + 2*log(x) + 2)/x + 2/x

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mupad [B]  time = 4.38, size = 25, normalized size = 0.96 \begin {gather*} -{\mathrm {e}}^{\frac {\ln \relax (x)}{2\,x}}-\frac {\ln \left (x^2\right )-\ln \relax (5)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- exp(log(x)/(2*x)) - (log(x^2) - log(5))/x

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sympy [A]  time = 0.32, size = 19, normalized size = 0.73 \begin {gather*} - e^{\frac {\log {\relax (x )}}{2 x}} - \frac {2 \log {\relax (x )}}{x} + \frac {\log {\relax (5 )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((ln(x)-1)*exp(1/2*ln(x)/x)+2*ln(1/5*x**2)-4)/x**2,x)

[Out]

-exp(log(x)/(2*x)) - 2*log(x)/x + log(5)/x

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