3.59.72 \(\int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx\) [5872]

Optimal. Leaf size=11 \[ -4+e^2+\log (-4+x+\log (x)) \]

[Out]

-4+ln(x+ln(x)-4)+exp(2)

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Rubi [F]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x)/(-4*x + x^2 + x*Log[x]),x]

[Out]

Defer[Int][(-4 + x + Log[x])^(-1), x] + Defer[Int][1/(x*(-4 + x + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{-4+x+\log (x)}+\frac {1}{x (-4+x+\log (x))}\right ) \, dx\\ &=\int \frac {1}{-4+x+\log (x)} \, dx+\int \frac {1}{x (-4+x+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 10, normalized size = 0.91 \begin {gather*} \log (4-x-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/(-4*x + x^2 + x*Log[x]),x]

[Out]

Log[4 - x - Log[x]]

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Maple [A]
time = 1.78, size = 7, normalized size = 0.64

method	result	size
default	\(\ln \left (x +\ln \left (x \right )-4\right )\)	\(7\)
norman	\(\ln \left (x +\ln \left (x \right )-4\right )\)	\(7\)
risch	\(\ln \left (x +\ln \left (x \right )-4\right )\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)/(x*ln(x)+x^2-4*x),x,method=_RETURNVERBOSE)

[Out]

ln(x+ln(x)-4)

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Maxima [A]
time = 0.30, size = 6, normalized size = 0.55 \begin {gather*} \log \left (x + \log \left (x\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x*log(x)+x^2-4*x),x, algorithm="maxima")

[Out]

log(x + log(x) - 4)

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Fricas [A]
time = 0.33, size = 6, normalized size = 0.55 \begin {gather*} \log \left (x + \log \left (x\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x*log(x)+x^2-4*x),x, algorithm="fricas")

[Out]

log(x + log(x) - 4)

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Sympy [A]
time = 0.04, size = 7, normalized size = 0.64 \begin {gather*} \log {\left (x + \log {\left (x \right )} - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x*ln(x)+x**2-4*x),x)

[Out]

log(x + log(x) - 4)

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Giac [A]
time = 0.40, size = 6, normalized size = 0.55 \begin {gather*} \log \left (x + \log \left (x\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x*log(x)+x^2-4*x),x, algorithm="giac")

[Out]

log(x + log(x) - 4)

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Mupad [B]
time = 4.06, size = 6, normalized size = 0.55 \begin {gather*} \ln \left (x+\ln \left (x\right )-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(x*log(x) - 4*x + x^2),x)

[Out]

log(x + log(x) - 4)

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