3.22.83 \(\int \frac {-7+20 x+e^x (16+4 x)+4 \log (x)}{-2 x+4 e^x x+4 x^2+x \log (x)} \, dx\)

Optimal. Leaf size=24 \[ \log \left (x^4 \left (e^x+x\right ) \left (4+\frac {-2+\log (x)}{e^x+x}\right )\right ) \]

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Rubi [F]  time = 0.79, 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 {-7+20 x+e^x (16+4 x)+4 \log (x)}{-2 x+4 e^x x+4 x^2+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-7 + 20*x + E^x*(16 + 4*x) + 4*Log[x])/(-2*x + 4*E^x*x + 4*x^2 + x*Log[x]),x]

[Out]

x + 4*Log[x] + 6*Defer[Int][(-2 + 4*E^x + 4*x + Log[x])^(-1), x] + Defer[Int][1/(x*(-2 + 4*E^x + 4*x + Log[x])
), x] - 4*Defer[Int][x/(-2 + 4*E^x + 4*x + Log[x]), x] - Defer[Int][Log[x]/(-2 + 4*E^x + 4*x + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4+x}{x}-\frac {-1-6 x+4 x^2+x \log (x)}{x \left (-2+4 e^x+4 x+\log (x)\right )}\right ) \, dx\\ &=\int \frac {4+x}{x} \, dx-\int \frac {-1-6 x+4 x^2+x \log (x)}{x \left (-2+4 e^x+4 x+\log (x)\right )} \, dx\\ &=\int \left (1+\frac {4}{x}\right ) \, dx-\int \left (-\frac {6}{-2+4 e^x+4 x+\log (x)}-\frac {1}{x \left (-2+4 e^x+4 x+\log (x)\right )}+\frac {4 x}{-2+4 e^x+4 x+\log (x)}+\frac {\log (x)}{-2+4 e^x+4 x+\log (x)}\right ) \, dx\\ &=x+4 \log (x)-4 \int \frac {x}{-2+4 e^x+4 x+\log (x)} \, dx+6 \int \frac {1}{-2+4 e^x+4 x+\log (x)} \, dx+\int \frac {1}{x \left (-2+4 e^x+4 x+\log (x)\right )} \, dx-\int \frac {\log (x)}{-2+4 e^x+4 x+\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 20, normalized size = 0.83 \begin {gather*} 4 \log (x)+\log \left (2-4 e^x-4 x-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7 + 20*x + E^x*(16 + 4*x) + 4*Log[x])/(-2*x + 4*E^x*x + 4*x^2 + x*Log[x]),x]

[Out]

4*Log[x] + Log[2 - 4*E^x - 4*x - Log[x]]

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fricas [A]  time = 0.85, size = 17, normalized size = 0.71 \begin {gather*} \log \left (4 \, x + 4 \, e^{x} + \log \relax (x) - 2\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)+(4*x+16)*exp(x)+20*x-7)/(x*log(x)+4*exp(x)*x+4*x^2-2*x),x, algorithm="fricas")

[Out]

log(4*x + 4*e^x + log(x) - 2) + 4*log(x)

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giac [A]  time = 0.21, size = 17, normalized size = 0.71 \begin {gather*} \log \left (4 \, x + 4 \, e^{x} + \log \relax (x) - 2\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)+(4*x+16)*exp(x)+20*x-7)/(x*log(x)+4*exp(x)*x+4*x^2-2*x),x, algorithm="giac")

[Out]

log(4*x + 4*e^x + log(x) - 2) + 4*log(x)

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maple [A]  time = 0.03, size = 18, normalized size = 0.75

method	result	size
norman	\(4 \ln \relax (x )+\ln \left (\ln \relax (x )+4 \,{\mathrm e}^{x}+4 x -2\right )\)	\(18\)
risch	\(4 \ln \relax (x )+\ln \left (\ln \relax (x )+4 \,{\mathrm e}^{x}+4 x -2\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*ln(x)+(4*x+16)*exp(x)+20*x-7)/(x*ln(x)+4*exp(x)*x+4*x^2-2*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.48, size = 15, normalized size = 0.62 \begin {gather*} \log \left (x + e^{x} + \frac {1}{4} \, \log \relax (x) - \frac {1}{2}\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)+(4*x+16)*exp(x)+20*x-7)/(x*log(x)+4*exp(x)*x+4*x^2-2*x),x, algorithm="maxima")

[Out]

log(x + e^x + 1/4*log(x) - 1/2) + 4*log(x)

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mupad [B]  time = 1.33, size = 17, normalized size = 0.71 \begin {gather*} \ln \left (4\,x+4\,{\mathrm {e}}^x+\ln \relax (x)-2\right )+4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x + 4*log(x) + exp(x)*(4*x + 16) - 7)/(4*x*exp(x) - 2*x + x*log(x) + 4*x^2),x)

[Out]

log(4*x + 4*exp(x) + log(x) - 2) + 4*log(x)

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sympy [A]  time = 0.33, size = 19, normalized size = 0.79 \begin {gather*} 4 \log {\relax (x )} + \log {\left (x + e^{x} + \frac {\log {\relax (x )}}{4} - \frac {1}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*ln(x)+(4*x+16)*exp(x)+20*x-7)/(x*ln(x)+4*exp(x)*x+4*x**2-2*x),x)

[Out]

4*log(x) + log(x + exp(x) + log(x)/4 - 1/2)

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