3.1.11 \(\int (\frac {1}{x}+\frac {1+\frac {1}{x}}{(x+\log (x))^{3/2}}) \, dx\) [11]

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 = 0.02, 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 = {6818} \begin {gather*} \log (x)-\frac {2}{\sqrt {x+\log (x)}} \end {gather*}

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 6818

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 \begin {gather*} \log (x)-\frac {2}{\sqrt {x+\log (x)}} \end {gather*}

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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Mathics [A]
time = 1.86, size = 11, normalized size = 0.85 \begin {gather*} \text {Log}\left [x\right ]-\frac {2}{\sqrt {x+\text {Log}\left [x\right ]}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - 2 / Sqrt[x + Log[x]]

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Maple [A]
time = 0.01, size = 12, normalized size = 0.92

method	result	size
default	\(\ln \left (x \right )-\frac {2}{\sqrt {x +\ln \left (x \right )}}\)	\(12\)
risch	\(\ln \left (x \right )-\frac {2}{\sqrt {x +\ln \left (x \right )}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 11, normalized size = 0.85 \begin {gather*} -\frac {2}{\sqrt {x + \log \left (x\right )}} + \log \left (x\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).
time = 0.33, size = 24, normalized size = 1.85 \begin {gather*} \frac {x \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \sqrt {x + \log \left (x\right )}}{x + \log \left (x\right )} \end {gather*}

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 = 0.39, size = 12, normalized size = 0.92 \begin {gather*} \log {\left (x \right )} - \frac {2}{\sqrt {x + \log {\left (x \right )}}} \end {gather*}

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 [A]
time = 0.00, size = 16, normalized size = 1.23 \begin {gather*} \ln \left |x\right |-\frac {2}{\sqrt {x+\ln x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/sqrt(x + log(x)) + log(abs(x))

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Mupad [B]
time = 0.19, size = 11, normalized size = 0.85 \begin {gather*} \ln \left (x\right )-\frac {2}{\sqrt {x+\ln \left (x\right )}} \end {gather*}

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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