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]

Log[x] - 2/Sqrt[x + Log[x]]

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Rubi [A]  time = 0.0294411, antiderivative size = 13, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0227658, size = 13, normalized size = 1. \[ \log (x)-\frac{2}{\sqrt{x+\log (x)}} \]

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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Maple [A]  time = 0.004, size = 12, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -2\,{\frac{1}{\sqrt{x+\ln \left ( x \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.948737, size = 15, normalized size = 1.15 \begin{align*} -\frac{2}{\sqrt{x + \log \left (x\right )}} + \log \left (x\right ) \end{align*}

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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Fricas [B]  time = 2.65604, size = 77, normalized size = 5.92 \begin{align*} \frac{x \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \sqrt{x + \log \left (x\right )}}{x + \log \left (x\right )} \end{align*}

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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Sympy [A]  time = 1.62125, size = 12, normalized size = 0.92 \begin{align*} \log{\left (x \right )} - \frac{2}{\sqrt{x + \log{\left (x \right )}}} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\frac{1}{x} + 1}{{\left (x + \log \left (x\right )\right )}^{\frac{3}{2}}} + \frac{1}{x}\,{d x} \end{align*}

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]

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