3.34.36 \(\int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 (24-10 x+x^2)+(160+120 x-72 x^2+8 x^3) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 (20-9 x+x^2)+(160 x-72 x^2+8 x^3) \log (4-x)} \, dx\)

Optimal. Leaf size=26 \[ x+\log \left (\frac {e^2+x+8 x (4+x+\log (4-x))}{-5+x}\right ) \]

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Rubi [F]  time = 1.17, 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 {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{660 x-137 x^2-39 x^3+8 x^4+e^2 \left (20-9 x+x^2\right )+\left (160 x-72 x^2+8 x^3\right ) \log (4-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(660 + 775*x - 241*x^2 - 31*x^3 + 8*x^4 + E^2*(24 - 10*x + x^2) + (160 + 120*x - 72*x^2 + 8*x^3)*Log[4 - x
])/(660*x - 137*x^2 - 39*x^3 + 8*x^4 + E^2*(20 - 9*x + x^2) + (160*x - 72*x^2 + 8*x^3)*Log[4 - x]),x]

[Out]

x - Log[5 - x] + Log[x] + 8*Defer[Int][(E^2 + 33*x + 8*x^2 + 8*x*Log[4 - x])^(-1), x] + 32*Defer[Int][1/((-4 +
x)*(E^2 + 33*x + 8*x^2 + 8*x*Log[4 - x])), x] - E^2*Defer[Int][1/(x*(E^2 + 33*x + 8*x^2 + 8*x*Log[4 - x])), x
] + 8*Defer[Int][x/(E^2 + 33*x + 8*x^2 + 8*x*Log[4 - x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {660+775 x-241 x^2-31 x^3+8 x^4+e^2 \left (24-10 x+x^2\right )+\left (160+120 x-72 x^2+8 x^3\right ) \log (4-x)}{\left (20-9 x+x^2\right ) \left (e^2+33 x+8 x^2+8 x \log (4-x)\right )} \, dx\\ &=\int \left (\frac {-5-5 x+x^2}{(-5+x) x}+\frac {4 e^2-e^2 x-24 x^2+8 x^3}{(-4+x) x \left (e^2+33 x+8 x^2+8 x \log (4-x)\right )}\right ) \, dx\\ &=\int \frac {-5-5 x+x^2}{(-5+x) x} \, dx+\int \frac {4 e^2-e^2 x-24 x^2+8 x^3}{(-4+x) x \left (e^2+33 x+8 x^2+8 x \log (4-x)\right )} \, dx\\ &=\int \left (1+\frac {1}{5-x}+\frac {1}{x}\right ) \, dx+\int \left (\frac {8}{e^2+33 x+8 x^2+8 x \log (4-x)}+\frac {32}{(-4+x) \left (e^2+33 x+8 x^2+8 x \log (4-x)\right )}-\frac {e^2}{x \left (e^2+33 x+8 x^2+8 x \log (4-x)\right )}+\frac {8 x}{e^2+33 x+8 x^2+8 x \log (4-x)}\right ) \, dx\\ &=x-\log (5-x)+\log (x)+8 \int \frac {1}{e^2+33 x+8 x^2+8 x \log (4-x)} \, dx+8 \int \frac {x}{e^2+33 x+8 x^2+8 x \log (4-x)} \, dx+32 \int \frac {1}{(-4+x) \left (e^2+33 x+8 x^2+8 x \log (4-x)\right )} \, dx-e^2 \int \frac {1}{x \left (e^2+33 x+8 x^2+8 x \log (4-x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 32, normalized size = 1.23 \begin {gather*} x-\log (5-x)+\log \left (e^2+33 x+8 x^2+8 x \log (4-x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(660 + 775*x - 241*x^2 - 31*x^3 + 8*x^4 + E^2*(24 - 10*x + x^2) + (160 + 120*x - 72*x^2 + 8*x^3)*Log
[4 - x])/(660*x - 137*x^2 - 39*x^3 + 8*x^4 + E^2*(20 - 9*x + x^2) + (160*x - 72*x^2 + 8*x^3)*Log[4 - x]),x]

[Out]

x - Log[5 - x] + Log[E^2 + 33*x + 8*x^2 + 8*x*Log[4 - x]]

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fricas [A]  time = 0.65, size = 35, normalized size = 1.35 \begin {gather*} x - \log \left (x - 5\right ) + \log \relax (x) + \log \left (\frac {8 \, x^{2} + 8 \, x \log \left (-x + 4\right ) + 33 \, x + e^{2}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-72*x^2+120*x+160)*log(-x+4)+(x^2-10*x+24)*exp(2)+8*x^4-31*x^3-241*x^2+775*x+660)/((8*x^3-72*
x^2+160*x)*log(-x+4)+(x^2-9*x+20)*exp(2)+8*x^4-39*x^3-137*x^2+660*x),x, algorithm="fricas")

[Out]

x - log(x - 5) + log(x) + log((8*x^2 + 8*x*log(-x + 4) + 33*x + e^2)/x)

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giac [A]  time = 0.54, size = 29, normalized size = 1.12 \begin {gather*} x + \log \left (8 \, x^{2} + 8 \, x \log \left (-x + 4\right ) + 33 \, x + e^{2}\right ) - \log \left (x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-72*x^2+120*x+160)*log(-x+4)+(x^2-10*x+24)*exp(2)+8*x^4-31*x^3-241*x^2+775*x+660)/((8*x^3-72*
x^2+160*x)*log(-x+4)+(x^2-9*x+20)*exp(2)+8*x^4-39*x^3-137*x^2+660*x),x, algorithm="giac")

[Out]

x + log(8*x^2 + 8*x*log(-x + 4) + 33*x + e^2) - log(x - 5)

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maple [A]  time = 0.12, size = 30, normalized size = 1.15

method	result	size
norman	\(x -\ln \left (x -5\right )+\ln \left (8 x^{2}+8 \ln \left (-x +4\right ) x +{\mathrm e}^{2}+33 x \right )\)	\(30\)
risch	\(x -\ln \left (x -5\right )+\ln \relax (x )+\ln \left (\ln \left (-x +4\right )+\frac {8 x^{2}+{\mathrm e}^{2}+33 x}{8 x}\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^3-72*x^2+120*x+160)*ln(-x+4)+(x^2-10*x+24)*exp(2)+8*x^4-31*x^3-241*x^2+775*x+660)/((8*x^3-72*x^2+160
*x)*ln(-x+4)+(x^2-9*x+20)*exp(2)+8*x^4-39*x^3-137*x^2+660*x),x,method=_RETURNVERBOSE)

[Out]

x-ln(x-5)+ln(8*x^2+8*ln(-x+4)*x+exp(2)+33*x)

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maxima [A]  time = 0.55, size = 36, normalized size = 1.38 \begin {gather*} x - \log \left (x - 5\right ) + \log \relax (x) + \log \left (\frac {8 \, x^{2} + 8 \, x \log \left (-x + 4\right ) + 33 \, x + e^{2}}{8 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-72*x^2+120*x+160)*log(-x+4)+(x^2-10*x+24)*exp(2)+8*x^4-31*x^3-241*x^2+775*x+660)/((8*x^3-72*
x^2+160*x)*log(-x+4)+(x^2-9*x+20)*exp(2)+8*x^4-39*x^3-137*x^2+660*x),x, algorithm="maxima")

[Out]

x - log(x - 5) + log(x) + log(1/8*(8*x^2 + 8*x*log(-x + 4) + 33*x + e^2)/x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {775\,x+\ln \left (4-x\right )\,\left (8\,x^3-72\,x^2+120\,x+160\right )+{\mathrm {e}}^2\,\left (x^2-10\,x+24\right )-241\,x^2-31\,x^3+8\,x^4+660}{660\,x+\ln \left (4-x\right )\,\left (8\,x^3-72\,x^2+160\,x\right )+{\mathrm {e}}^2\,\left (x^2-9\,x+20\right )-137\,x^2-39\,x^3+8\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((775*x + log(4 - x)*(120*x - 72*x^2 + 8*x^3 + 160) + exp(2)*(x^2 - 10*x + 24) - 241*x^2 - 31*x^3 + 8*x^4 +
660)/(660*x + log(4 - x)*(160*x - 72*x^2 + 8*x^3) + exp(2)*(x^2 - 9*x + 20) - 137*x^2 - 39*x^3 + 8*x^4),x)

[Out]

int((775*x + log(4 - x)*(120*x - 72*x^2 + 8*x^3 + 160) + exp(2)*(x^2 - 10*x + 24) - 241*x^2 - 31*x^3 + 8*x^4 +
660)/(660*x + log(4 - x)*(160*x - 72*x^2 + 8*x^3) + exp(2)*(x^2 - 9*x + 20) - 137*x^2 - 39*x^3 + 8*x^4), x)

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sympy [A]  time = 0.43, size = 31, normalized size = 1.19 \begin {gather*} x + \log {\relax (x )} - \log {\left (x - 5 \right )} + \log {\left (\log {\left (4 - x \right )} + \frac {8 x^{2} + 33 x + e^{2}}{8 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**3-72*x**2+120*x+160)*ln(-x+4)+(x**2-10*x+24)*exp(2)+8*x**4-31*x**3-241*x**2+775*x+660)/((8*x*
*3-72*x**2+160*x)*ln(-x+4)+(x**2-9*x+20)*exp(2)+8*x**4-39*x**3-137*x**2+660*x),x)

[Out]

x + log(x) - log(x - 5) + log(log(4 - x) + (8*x**2 + 33*x + exp(2))/(8*x))

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