3.6.31 \(\int \frac {-86+28 x-2 x^2+e^x (-56+22 x-2 x^2)+(-46+10 x) \log (x)}{92-40 x+4 x^2+e^x (23-10 x+x^2)+(23 x-10 x^2+x^3) \log (x)} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\frac {16 \left (2-(-5+x)^2\right )}{\left (4+e^x+x \log (x)\right )^2}\right ) \]

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Rubi [F]  time = 0.93, 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 {-86+28 x-2 x^2+e^x \left (-56+22 x-2 x^2\right )+(-46+10 x) \log (x)}{92-40 x+4 x^2+e^x \left (23-10 x+x^2\right )+\left (23 x-10 x^2+x^3\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-86 + 28*x - 2*x^2 + E^x*(-56 + 22*x - 2*x^2) + (-46 + 10*x)*Log[x])/(92 - 40*x + 4*x^2 + E^x*(23 - 10*x
+ x^2) + (23*x - 10*x^2 + x^3)*Log[x]),x]

[Out]

-2*x + Log[23 - 10*x + x^2] + 6*Defer[Int][(4 + E^x + x*Log[x])^(-1), x] - 2*Defer[Int][Log[x]/(4 + E^x + x*Lo
g[x]), x] + 2*Defer[Int][(x*Log[x])/(4 + E^x + x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-86+28 x-2 x^2+e^x \left (-56+22 x-2 x^2\right )+(-46+10 x) \log (x)}{\left (23-10 x+x^2\right ) \left (4+e^x+x \log (x)\right )} \, dx\\ &=\int \left (-\frac {2 \left (28-11 x+x^2\right )}{23-10 x+x^2}+\frac {2 (3-\log (x)+x \log (x))}{4+e^x+x \log (x)}\right ) \, dx\\ &=-\left (2 \int \frac {28-11 x+x^2}{23-10 x+x^2} \, dx\right )+2 \int \frac {3-\log (x)+x \log (x)}{4+e^x+x \log (x)} \, dx\\ &=-\left (2 \int \left (1+\frac {5-x}{23-10 x+x^2}\right ) \, dx\right )+2 \int \left (\frac {3}{4+e^x+x \log (x)}-\frac {\log (x)}{4+e^x+x \log (x)}+\frac {x \log (x)}{4+e^x+x \log (x)}\right ) \, dx\\ &=-2 x-2 \int \frac {5-x}{23-10 x+x^2} \, dx-2 \int \frac {\log (x)}{4+e^x+x \log (x)} \, dx+2 \int \frac {x \log (x)}{4+e^x+x \log (x)} \, dx+6 \int \frac {1}{4+e^x+x \log (x)} \, dx\\ &=-2 x+\log \left (23-10 x+x^2\right )-2 \int \frac {\log (x)}{4+e^x+x \log (x)} \, dx+2 \int \frac {x \log (x)}{4+e^x+x \log (x)} \, dx+6 \int \frac {1}{4+e^x+x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 26, normalized size = 1.13 \begin {gather*} -2 \left (-\frac {1}{2} \log \left (23-10 x+x^2\right )+\log \left (4+e^x+x \log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-86 + 28*x - 2*x^2 + E^x*(-56 + 22*x - 2*x^2) + (-46 + 10*x)*Log[x])/(92 - 40*x + 4*x^2 + E^x*(23 -
10*x + x^2) + (23*x - 10*x^2 + x^3)*Log[x]),x]

[Out]

-2*(-1/2*Log[23 - 10*x + x^2] + Log[4 + E^x + x*Log[x]])

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fricas [A]  time = 0.65, size = 29, normalized size = 1.26 \begin {gather*} \log \left (x^{2} - 10 \, x + 23\right ) - 2 \, \log \relax (x) - 2 \, \log \left (\frac {x \log \relax (x) + e^{x} + 4}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-46)*log(x)+(-2*x^2+22*x-56)*exp(x)-2*x^2+28*x-86)/((x^3-10*x^2+23*x)*log(x)+(x^2-10*x+23)*exp
(x)+4*x^2-40*x+92),x, algorithm="fricas")

[Out]

log(x^2 - 10*x + 23) - 2*log(x) - 2*log((x*log(x) + e^x + 4)/x)

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giac [A]  time = 0.74, size = 21, normalized size = 0.91 \begin {gather*} \log \left (x^{2} - 10 \, x + 23\right ) - 2 \, \log \left (x \log \relax (x) + e^{x} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-46)*log(x)+(-2*x^2+22*x-56)*exp(x)-2*x^2+28*x-86)/((x^3-10*x^2+23*x)*log(x)+(x^2-10*x+23)*exp
(x)+4*x^2-40*x+92),x, algorithm="giac")

[Out]

log(x^2 - 10*x + 23) - 2*log(x*log(x) + e^x + 4)

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maple [A]  time = 0.09, size = 22, normalized size = 0.96

method	result	size
norman	\(-2 \ln \left ({\mathrm e}^{x}+x \ln \relax (x )+4\right )+\ln \left (x^{2}-10 x +23\right )\)	\(22\)
risch	\(-2 \ln \relax (x )+\ln \left (x^{2}-10 x +23\right )-2 \ln \left (\ln \relax (x )+\frac {{\mathrm e}^{x}+4}{x}\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x-46)*ln(x)+(-2*x^2+22*x-56)*exp(x)-2*x^2+28*x-86)/((x^3-10*x^2+23*x)*ln(x)+(x^2-10*x+23)*exp(x)+4*x^
2-40*x+92),x,method=_RETURNVERBOSE)

[Out]

-2*ln(exp(x)+x*ln(x)+4)+ln(x^2-10*x+23)

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maxima [A]  time = 0.90, size = 21, normalized size = 0.91 \begin {gather*} \log \left (x^{2} - 10 \, x + 23\right ) - 2 \, \log \left (x \log \relax (x) + e^{x} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-46)*log(x)+(-2*x^2+22*x-56)*exp(x)-2*x^2+28*x-86)/((x^3-10*x^2+23*x)*log(x)+(x^2-10*x+23)*exp
(x)+4*x^2-40*x+92),x, algorithm="maxima")

[Out]

log(x^2 - 10*x + 23) - 2*log(x*log(x) + e^x + 4)

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mupad [B]  time = 0.81, size = 29, normalized size = 1.26 \begin {gather*} \ln \left (x^2-10\,x+23\right )-2\,\ln \left (\frac {{\mathrm {e}}^x+x\,\ln \relax (x)+4}{x}\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(2*x^2 - 22*x + 56) - log(x)*(10*x - 46) - 28*x + 2*x^2 + 86)/(exp(x)*(x^2 - 10*x + 23) - 40*x +
log(x)*(23*x - 10*x^2 + x^3) + 4*x^2 + 92),x)

[Out]

log(x^2 - 10*x + 23) - 2*log((exp(x) + x*log(x) + 4)/x) - 2*log(x)

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sympy [A]  time = 0.40, size = 22, normalized size = 0.96 \begin {gather*} \log {\left (x^{2} - 10 x + 23 \right )} - 2 \log {\left (x \log {\relax (x )} + e^{x} + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-46)*ln(x)+(-2*x**2+22*x-56)*exp(x)-2*x**2+28*x-86)/((x**3-10*x**2+23*x)*ln(x)+(x**2-10*x+23)*
exp(x)+4*x**2-40*x+92),x)

[Out]

log(x**2 - 10*x + 23) - 2*log(x*log(x) + exp(x) + 4)

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