3.11 \(\int (\frac {1}{x}+\frac {1+\frac {1}{x}}{(x+\log (x))^{3/2}}) \, dx\)

Optimal. Leaf size=13 \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]

[Out]

ln(x)-2/(x+ln(x))^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6686} \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1) + (1 + x^(-1))/(x + Log[x])^(3/2),x]

[Out]

Log[x] - 2/Sqrt[x + Log[x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \left (\frac {1}{x}+\frac {1+\frac {1}{x}}{(x+\log (x))^{3/2}}\right ) \, dx &=\log (x)+\int \frac {1+\frac {1}{x}}{(x+\log (x))^{3/2}} \, dx\\ &=\log (x)-\frac {2}{\sqrt {x+\log (x)}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 13, normalized size = 1.00 \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1) + (1 + x^(-1))/(x + Log[x])^(3/2),x]

[Out]

Log[x] - 2/Sqrt[x + Log[x]]

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fricas [B]  time = 0.41, size = 24, normalized size = 1.85 \[ \frac {x \log \relax (x) + \log \relax (x)^{2} - 2 \, \sqrt {x + \log \relax (x)}}{x + \log \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x+(1+1/x)/(x+log(x))^(3/2),x, algorithm="fricas")

[Out]

(x*log(x) + log(x)^2 - 2*sqrt(x + log(x)))/(x + log(x))

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giac [A]  time = 0.91, size = 12, normalized size = 0.92 \[ -\frac {2}{\sqrt {x + \log \relax (x)}} + \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x+(1+1/x)/(x+log(x))^(3/2),x, algorithm="giac")

[Out]

-2/sqrt(x + log(x)) + log(abs(x))

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maple [A]  time = 0.01, size = 12, normalized size = 0.92 \[ \ln \relax (x )-\frac {2}{\sqrt {x +\ln \relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x+(1/x+1)/(x+ln(x))^(3/2),x)

[Out]

ln(x)-2/(x+ln(x))^(1/2)

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maxima [A]  time = 0.41, size = 11, normalized size = 0.85 \[ -\frac {2}{\sqrt {x + \log \relax (x)}} + \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x+(1+1/x)/(x+log(x))^(3/2),x, algorithm="maxima")

[Out]

-2/sqrt(x + log(x)) + log(x)

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mupad [B]  time = 0.19, size = 11, normalized size = 0.85 \[ \ln \relax (x)-\frac {2}{\sqrt {x+\ln \relax (x)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/x + 1)/(x + log(x))^(3/2) + 1/x,x)

[Out]

log(x) - 2/(x + log(x))^(1/2)

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sympy [A]  time = 1.49, size = 12, normalized size = 0.92 \[ \log {\relax (x )} - \frac {2}{\sqrt {x + \log {\relax (x )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x+(1+1/x)/(x+ln(x))**(3/2),x)

[Out]

log(x) - 2/sqrt(x + log(x))

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