3.96.75 \(\int \frac {-2 x-x^2-2 \log (\frac {5}{4})}{4 x+4 x^2+x^3+(4+4 x+x^2) \log (\frac {5}{4})+(-4 x^2-2 x^3+(-4 x-2 x^2) \log (\frac {5}{4})) \log (x+\log (\frac {5}{4}))+(x^3+x^2 \log (\frac {5}{4})) \log ^2(x+\log (\frac {5}{4}))} \, dx\)

Optimal. Leaf size=18 \[ \frac {x}{-2-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )} \]

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Rubi [F]  time = 0.42, 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 {-2 x-x^2-2 \log \left (\frac {5}{4}\right )}{4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log \left (\frac {5}{4}\right )+\left (-4 x^2-2 x^3+\left (-4 x-2 x^2\right ) \log \left (\frac {5}{4}\right )\right ) \log \left (x+\log \left (\frac {5}{4}\right )\right )+\left (x^3+x^2 \log \left (\frac {5}{4}\right )\right ) \log ^2\left (x+\log \left (\frac {5}{4}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x - x^2 - 2*Log[5/4])/(4*x + 4*x^2 + x^3 + (4 + 4*x + x^2)*Log[5/4] + (-4*x^2 - 2*x^3 + (-4*x - 2*x^2)
*Log[5/4])*Log[x + Log[5/4]] + (x^3 + x^2*Log[5/4])*Log[x + Log[5/4]]^2),x]

[Out]

-2*(1 + Log[2/Sqrt[5]])*Defer[Int][(-2 - x + x*Log[x + Log[5/4]])^(-2), x] - Defer[Int][x/(-2 - x + x*Log[x +
Log[5/4]])^2, x] - Log[5/4]^2*Defer[Int][1/((x + Log[5/4])*(-2 - x + x*Log[x + Log[5/4]])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x-x^2-\log \left (\frac {25}{16}\right )}{\left (x+\log \left (\frac {5}{4}\right )\right ) \left (2+x-x \log \left (x+\log \left (\frac {5}{4}\right )\right )\right )^2} \, dx\\ &=\int \left (-\frac {x}{\left (-2-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )\right )^2}-\frac {\log ^2\left (\frac {5}{4}\right )}{\left (x+\log \left (\frac {5}{4}\right )\right ) \left (-2-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )\right )^2}-\frac {2 \left (1+\log \left (\frac {2}{\sqrt {5}}\right )\right )}{\left (-2-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )\right )^2}\right ) \, dx\\ &=-\left (\log ^2\left (\frac {5}{4}\right ) \int \frac {1}{\left (x+\log \left (\frac {5}{4}\right )\right ) \left (-2-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )\right )^2} \, dx\right )-\left (2 \left (1+\log \left (\frac {2}{\sqrt {5}}\right )\right )\right ) \int \frac {1}{\left (-2-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )\right )^2} \, dx-\int \frac {x}{\left (-2-x+x \log \left (x+\log \left (\frac {5}{4}\right )\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.66, size = 18, normalized size = 1.00 \begin {gather*} -\frac {x}{2+x-x \log \left (x+\log \left (\frac {5}{4}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x - x^2 - 2*Log[5/4])/(4*x + 4*x^2 + x^3 + (4 + 4*x + x^2)*Log[5/4] + (-4*x^2 - 2*x^3 + (-4*x -
2*x^2)*Log[5/4])*Log[x + Log[5/4]] + (x^3 + x^2*Log[5/4])*Log[x + Log[5/4]]^2),x]

[Out]

-(x/(2 + x - x*Log[x + Log[5/4]]))

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fricas [A]  time = 0.86, size = 16, normalized size = 0.89 \begin {gather*} \frac {x}{x \log \left (x + \log \left (\frac {5}{4}\right )\right ) - x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(5/4)-x^2-2*x)/((x^2*log(5/4)+x^3)*log(log(5/4)+x)^2+((-2*x^2-4*x)*log(5/4)-2*x^3-4*x^2)*log(
log(5/4)+x)+(x^2+4*x+4)*log(5/4)+x^3+4*x^2+4*x),x, algorithm="fricas")

[Out]

x/(x*log(x + log(5/4)) - x - 2)

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giac [B]  time = 0.40, size = 247, normalized size = 13.72 \begin {gather*} \frac {x^{4} + x^{3} \log \left (\frac {5}{4}\right ) + 2 \, x^{3} + 2 \, x^{2} \log \relax (5) - 4 \, x^{2} \log \relax (2) + 2 \, x^{2} \log \left (\frac {5}{4}\right ) + 2 \, x \log \relax (5) \log \left (\frac {5}{4}\right ) - 4 \, x \log \relax (2) \log \left (\frac {5}{4}\right )}{x^{4} \log \left (x + \log \left (\frac {5}{4}\right )\right ) + x^{3} \log \relax (5) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - 2 \, x^{3} \log \relax (2) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - x^{4} - x^{3} \log \relax (5) + 2 \, x^{3} \log \relax (2) + 2 \, x^{3} \log \left (x + \log \left (\frac {5}{4}\right )\right ) + 2 \, x^{2} \log \relax (5) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - 4 \, x^{2} \log \relax (2) \log \left (x + \log \left (\frac {5}{4}\right )\right ) + 2 \, x^{2} \log \left (\frac {5}{4}\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) + 2 \, x \log \relax (5) \log \left (\frac {5}{4}\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - 4 \, x \log \relax (2) \log \left (\frac {5}{4}\right ) \log \left (x + \log \left (\frac {5}{4}\right )\right ) - 4 \, x^{3} - 4 \, x^{2} \log \relax (5) + 8 \, x^{2} \log \relax (2) - 2 \, x^{2} \log \left (\frac {5}{4}\right ) - 2 \, x \log \relax (5) \log \left (\frac {5}{4}\right ) + 4 \, x \log \relax (2) \log \left (\frac {5}{4}\right ) - 4 \, x^{2} - 4 \, x \log \relax (5) + 8 \, x \log \relax (2) - 4 \, x \log \left (\frac {5}{4}\right ) - 4 \, \log \relax (5) \log \left (\frac {5}{4}\right ) + 8 \, \log \relax (2) \log \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(5/4)-x^2-2*x)/((x^2*log(5/4)+x^3)*log(log(5/4)+x)^2+((-2*x^2-4*x)*log(5/4)-2*x^3-4*x^2)*log(
log(5/4)+x)+(x^2+4*x+4)*log(5/4)+x^3+4*x^2+4*x),x, algorithm="giac")

[Out]

(x^4 + x^3*log(5/4) + 2*x^3 + 2*x^2*log(5) - 4*x^2*log(2) + 2*x^2*log(5/4) + 2*x*log(5)*log(5/4) - 4*x*log(2)*
log(5/4))/(x^4*log(x + log(5/4)) + x^3*log(5)*log(x + log(5/4)) - 2*x^3*log(2)*log(x + log(5/4)) - x^4 - x^3*l
og(5) + 2*x^3*log(2) + 2*x^3*log(x + log(5/4)) + 2*x^2*log(5)*log(x + log(5/4)) - 4*x^2*log(2)*log(x + log(5/4
)) + 2*x^2*log(5/4)*log(x + log(5/4)) + 2*x*log(5)*log(5/4)*log(x + log(5/4)) - 4*x*log(2)*log(5/4)*log(x + lo
g(5/4)) - 4*x^3 - 4*x^2*log(5) + 8*x^2*log(2) - 2*x^2*log(5/4) - 2*x*log(5)*log(5/4) + 4*x*log(2)*log(5/4) - 4
*x^2 - 4*x*log(5) + 8*x*log(2) - 4*x*log(5/4) - 4*log(5)*log(5/4) + 8*log(2)*log(5/4))

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maple [A]  time = 0.20, size = 17, normalized size = 0.94

method	result	size
norman	\(\frac {x}{\ln \left (\ln \left (\frac {5}{4}\right )+x \right ) x -2-x}\)	\(17\)
risch	\(\frac {x}{x \ln \left (\ln \relax (5)-2 \ln \relax (2)+x \right )-x -2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(5/4)-x^2-2*x)/((x^2*ln(5/4)+x^3)*ln(ln(5/4)+x)^2+((-2*x^2-4*x)*ln(5/4)-2*x^3-4*x^2)*ln(ln(5/4)+x)+(
x^2+4*x+4)*ln(5/4)+x^3+4*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

x/(ln(ln(5/4)+x)*x-2-x)

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maxima [A]  time = 0.49, size = 20, normalized size = 1.11 \begin {gather*} \frac {x}{x \log \left (x + \log \relax (5) - 2 \, \log \relax (2)\right ) - x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(5/4)-x^2-2*x)/((x^2*log(5/4)+x^3)*log(log(5/4)+x)^2+((-2*x^2-4*x)*log(5/4)-2*x^3-4*x^2)*log(
log(5/4)+x)+(x^2+4*x+4)*log(5/4)+x^3+4*x^2+4*x),x, algorithm="maxima")

[Out]

x/(x*log(x + log(5) - 2*log(2)) - x - 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {x^2+2\,x+2\,\ln \left (\frac {5}{4}\right )}{4\,x-\ln \left (x+\ln \left (\frac {5}{4}\right )\right )\,\left (\ln \left (\frac {5}{4}\right )\,\left (2\,x^2+4\,x\right )+4\,x^2+2\,x^3\right )+{\ln \left (x+\ln \left (\frac {5}{4}\right )\right )}^2\,\left (x^3+\ln \left (\frac {5}{4}\right )\,x^2\right )+4\,x^2+x^3+\ln \left (\frac {5}{4}\right )\,\left (x^2+4\,x+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 2*log(5/4) + x^2)/(4*x - log(x + log(5/4))*(log(5/4)*(4*x + 2*x^2) + 4*x^2 + 2*x^3) + log(x + log(
5/4))^2*(x^2*log(5/4) + x^3) + 4*x^2 + x^3 + log(5/4)*(4*x + x^2 + 4)),x)

[Out]

int(-(2*x + 2*log(5/4) + x^2)/(4*x - log(x + log(5/4))*(log(5/4)*(4*x + 2*x^2) + 4*x^2 + 2*x^3) + log(x + log(
5/4))^2*(x^2*log(5/4) + x^3) + 4*x^2 + x^3 + log(5/4)*(4*x + x^2 + 4)), x)

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sympy [A]  time = 0.21, size = 14, normalized size = 0.78 \begin {gather*} \frac {x}{x \log {\left (x + \log {\left (\frac {5}{4} \right )} \right )} - x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(5/4)-x**2-2*x)/((x**2*ln(5/4)+x**3)*ln(ln(5/4)+x)**2+((-2*x**2-4*x)*ln(5/4)-2*x**3-4*x**2)*ln
(ln(5/4)+x)+(x**2+4*x+4)*ln(5/4)+x**3+4*x**2+4*x),x)

[Out]

x/(x*log(x + log(5/4)) - x - 2)

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