3.4.14 \(\int \frac {-1+2 x+3 \log (x)}{x^2+2 x^4+x \log (x)} \, dx\) [314]

Optimal. Leaf size=17 \[ 3 \log (x)-\log \left (x+2 x^3+\log (x)\right ) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6874, 6816} \begin {gather*} 3 \log (x)-\log \left (2 x^3+x+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x + 3*Log[x])/(x^2 + 2*x^4 + x*Log[x]),x]

[Out]

3*Log[x] - Log[x + 2*x^3 + Log[x]]

Rule 6816

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

Rule 6874

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 {3}{x}+\frac {-1-x-6 x^3}{x \left (x+2 x^3+\log (x)\right )}\right ) \, dx\\ &=3 \log (x)+\int \frac {-1-x-6 x^3}{x \left (x+2 x^3+\log (x)\right )} \, dx\\ &=3 \log (x)-\log \left (x+2 x^3+\log (x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 17, normalized size = 1.00 \begin {gather*} 3 \log (x)-\log \left (x+2 x^3+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x + 3*Log[x])/(x^2 + 2*x^4 + x*Log[x]),x]

[Out]

3*Log[x] - Log[x + 2*x^3 + Log[x]]

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Maple [A]
time = 0.13, size = 18, normalized size = 1.06

method	result	size
default	\(3 \ln \left (x \right )-\ln \left (x +2 x^{3}+\ln \left (x \right )\right )\)	\(18\)
norman	\(3 \ln \left (x \right )-\ln \left (x +2 x^{3}+\ln \left (x \right )\right )\)	\(18\)
risch	\(3 \ln \left (x \right )-\ln \left (x +2 x^{3}+\ln \left (x \right )\right )\)	\(18\)
parallelrisch	\(-\ln \left (x^{3}+\frac {x}{2}+\frac {\ln \left (x \right )}{2}\right )+3 \ln \left (x \right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*ln(x)-1+2*x)/(x*ln(x)+x^2+2*x^4),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.36, size = 17, normalized size = 1.00 \begin {gather*} -\log \left (2 \, x^{3} + x + \log \left (x\right )\right ) + 3 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)-1+2*x)/(x*log(x)+x^2+2*x^4),x, algorithm="maxima")

[Out]

-log(2*x^3 + x + log(x)) + 3*log(x)

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Fricas [A]
time = 0.59, size = 17, normalized size = 1.00 \begin {gather*} -\log \left (2 \, x^{3} + x + \log \left (x\right )\right ) + 3 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)-1+2*x)/(x*log(x)+x^2+2*x^4),x, algorithm="fricas")

[Out]

-log(2*x^3 + x + log(x)) + 3*log(x)

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Sympy [A]
time = 0.06, size = 15, normalized size = 0.88 \begin {gather*} 3 \log {\left (x \right )} - \log {\left (2 x^{3} + x + \log {\left (x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*ln(x)-1+2*x)/(x*ln(x)+x**2+2*x**4),x)

[Out]

3*log(x) - log(2*x**3 + x + log(x))

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Giac [A]
time = 0.44, size = 21, normalized size = 1.24 \begin {gather*} -\log \left (-2 \, x^{3} - x - \log \left (x\right )\right ) + 3 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*log(x)-1+2*x)/(x*log(x)+x^2+2*x^4),x, algorithm="giac")

[Out]

-log(-2*x^3 - x - log(x)) + 3*log(x)

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Mupad [B]
time = 0.21, size = 17, normalized size = 1.00 \begin {gather*} 3\,\ln \left (x\right )-\ln \left (x+\ln \left (x\right )+2\,x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 3*log(x) - 1)/(x*log(x) + x^2 + 2*x^4),x)

[Out]

3*log(x) - log(x + log(x) + 2*x^3)

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Chatgpt [F] Failed to verify
time = 1.00, size = 17, normalized size = 1.00 \begin {gather*} 3 \ln \left (x \right )+\arctan \left (x^{2}\right )-\ln \left (x^{2}+1\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((3*ln(x)-1+2*x)/(x*ln(x)+x^2+2*x^4),x)

[Out]

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

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