3.2.30 \(\int \frac {-1+\log ^2(x)}{x \log ^2(x)} \, dx\) [130]

Optimal. Leaf size=7 \[ \frac {1}{\log (x)}+\log (x) \]

[Out]

1/ln(x)+ln(x)

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Rubi [A]
time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14} \begin {gather*} \log (x)+\frac {1}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + Log[x]^2)/(x*Log[x]^2),x]

[Out]

Log[x]^(-1) + Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {-1+x^2}{x^2} \, dx,x,\log (x)\right )\\ &=\text {Subst}\left (\int \left (1-\frac {1}{x^2}\right ) \, dx,x,\log (x)\right )\\ &=\frac {1}{\log (x)}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} \frac {1}{\log (x)}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]^(-1) + Log[x]

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Maple [A]
time = 0.02, size = 8, normalized size = 1.14

method	result	size
derivativedivides	\(\frac {1}{\ln \left (x \right )}+\ln \left (x \right )\)	\(8\)
default	\(\frac {1}{\ln \left (x \right )}+\ln \left (x \right )\)	\(8\)
risch	\(\frac {1}{\ln \left (x \right )}+\ln \left (x \right )\)	\(8\)
parts	\(\frac {1}{\ln \left (x \right )}+\ln \left (x \right )\)	\(8\)
norman	\(\frac {1+\ln \left (x \right )^{2}}{\ln \left (x \right )}\)	\(12\)
parallelrisch	\(\frac {1+\ln \left (x \right )^{2}}{\ln \left (x \right )}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/ln(x)+ln(x)

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Maxima [A]
time = 0.39, size = 7, normalized size = 1.00 \begin {gather*} \frac {1}{\log \left (x\right )} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/log(x) + log(x)

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Fricas [A]
time = 0.58, size = 11, normalized size = 1.57 \begin {gather*} \frac {\log \left (x\right )^{2} + 1}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(x)^2 + 1)/log(x)

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Sympy [A]
time = 0.02, size = 7, normalized size = 1.00 \begin {gather*} \log {\left (x \right )} + \frac {1}{\log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + 1/log(x)

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Giac [A]
time = 0.50, size = 7, normalized size = 1.00 \begin {gather*} \frac {1}{\log \left (x\right )} + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/log(x) + log(x)

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Mupad [B]
time = 0.11, size = 7, normalized size = 1.00 \begin {gather*} \ln \left (x\right )+\frac {1}{\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2 - 1)/(x*log(x)^2),x)

[Out]

log(x) + 1/log(x)

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Chatgpt [F] Failed to verify
time = 1.00, size = 9, normalized size = 1.29 \begin {gather*} \ln \left (x \right )-\frac {1}{\ln \left (x \right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((ln(x)^2-1)/x/ln(x)^2,x)

[Out]

ln(x)-1/ln(x)

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