3.86.40 \(\int \frac {1+2 x^2}{-3 x+x^3+x \log (x)} \, dx\) [8540]

Optimal. Leaf size=12 \[ \log \left (3-x^2-\log (x)\right ) \]

[Out]

ln(3-ln(x)-x^2)

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Rubi [F]
time = 0.09, 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+2 x^2}{-3 x+x^3+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 2*x^2)/(-3*x + x^3 + x*Log[x]),x]

[Out]

Defer[Int][1/(x*(-3 + x^2 + Log[x])), x] + 2*Defer[Int][x/(-3 + x^2 + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{x \left (-3+x^2+\log (x)\right )}+\frac {2 x}{-3+x^2+\log (x)}\right ) \, dx\\ &=2 \int \frac {x}{-3+x^2+\log (x)} \, dx+\int \frac {1}{x \left (-3+x^2+\log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 8, normalized size = 0.67 \begin {gather*} \log \left (-3+x^2+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^2)/(-3*x + x^3 + x*Log[x]),x]

[Out]

Log[-3 + x^2 + Log[x]]

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Maple [A]
time = 2.09, size = 9, normalized size = 0.75

method	result	size
default	\(\ln \left (x^{2}+\ln \left (x \right )-3\right )\)	\(9\)
norman	\(\ln \left (x^{2}+\ln \left (x \right )-3\right )\)	\(9\)
risch	\(\ln \left (x^{2}+\ln \left (x \right )-3\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x^2+ln(x)-3)

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Maxima [A]
time = 0.30, size = 8, normalized size = 0.67 \begin {gather*} \log \left (x^{2} + \log \left (x\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(x*log(x)+x^3-3*x),x, algorithm="maxima")

[Out]

log(x^2 + log(x) - 3)

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Fricas [A]
time = 0.40, size = 8, normalized size = 0.67 \begin {gather*} \log \left (x^{2} + \log \left (x\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(x*log(x)+x^3-3*x),x, algorithm="fricas")

[Out]

log(x^2 + log(x) - 3)

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Sympy [A]
time = 0.05, size = 8, normalized size = 0.67 \begin {gather*} \log {\left (x^{2} + \log {\left (x \right )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+1)/(x*ln(x)+x**3-3*x),x)

[Out]

log(x**2 + log(x) - 3)

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Giac [A]
time = 0.41, size = 8, normalized size = 0.67 \begin {gather*} \log \left (x^{2} + \log \left (x\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(x*log(x)+x^3-3*x),x, algorithm="giac")

[Out]

log(x^2 + log(x) - 3)

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Mupad [B]
time = 5.21, size = 8, normalized size = 0.67 \begin {gather*} \ln \left (\ln \left (x\right )+x^2-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 + 1)/(x*log(x) - 3*x + x^3),x)

[Out]

log(log(x) + x^2 - 3)

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