3.67.70 \(\int \frac {-1+\log (x^2)}{x-2 x^2+x \log (x^2)} \, dx\)

Optimal. Leaf size=20 \[ -\frac {5}{3}+\log (x)-\log \left (-1+2 x-\log \left (x^2\right )\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 15, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6742, 6684} \begin {gather*} \log (x)-\log \left (\log \left (x^2\right )-2 x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - Log[1 - 2*x + Log[x^2]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 15, normalized size = 0.75 \begin {gather*} \log (x)-\log \left (1-2 x+\log \left (x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - Log[1 - 2*x + Log[x^2]]

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fricas [A]  time = 0.53, size = 19, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, \log \left (x^{2}\right ) - \log \left (-2 \, x + \log \left (x^{2}\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.23, size = 15, normalized size = 0.75 \begin {gather*} \log \relax (x) - \log \left (-2 \, x + \log \left (x^{2}\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(-2*x + log(x^2) + 1)

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maple [A]  time = 0.02, size = 16, normalized size = 0.80

method	result	size
risch	\(\ln \relax (x )-\ln \left (\ln \left (x^{2}\right )-2 x +1\right )\)	\(16\)
norman	\(\ln \relax (x )-\ln \left (2 x -1-\ln \left (x^{2}\right )\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.40, size = 13, normalized size = 0.65 \begin {gather*} \log \relax (x) - \log \left (-x + \log \relax (x) + \frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(-x + log(x) + 1/2)

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mupad [B]  time = 4.21, size = 21, normalized size = 1.05 \begin {gather*} \frac {\ln \left (x^2\right )}{2}-\ln \left (2\,x-\ln \left (x^2\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2)/2 - log(2*x - log(x^2) - 1)

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sympy [A]  time = 0.12, size = 14, normalized size = 0.70 \begin {gather*} \log {\relax (x )} - \log {\left (- 2 x + \log {\left (x^{2} \right )} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(-2*x + log(x**2) + 1)

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