3.24.48 \(\int \frac {-8-18 x+40 x^2-8 x^4+(50+18 x-20 x^2+2 x^4+8 \log (4 x)) \log (25+9 x-10 x^2+x^4+4 \log (4 x)) \log (\log (25+9 x-10 x^2+x^4+4 \log (4 x)))}{(25+9 x-10 x^2+x^4+4 \log (4 x)) \log (25+9 x-10 x^2+x^4+4 \log (4 x)) \log ^2(\log (25+9 x-10 x^2+x^4+4 \log (4 x)))} \, dx\)

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

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Rubi [F]  time = 2.25, 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 {-8-18 x+40 x^2-8 x^4+\left (50+18 x-20 x^2+2 x^4+8 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8 - 18*x + 40*x^2 - 8*x^4 + (50 + 18*x - 20*x^2 + 2*x^4 + 8*Log[4*x])*Log[25 + 9*x - 10*x^2 + x^4 + 4*Lo
g[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]])/((25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x])*Log[25 + 9*x -
10*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]]^2),x]

[Out]

-8*Defer[Int][1/((25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x])*Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 +
9*x - 10*x^2 + x^4 + 4*Log[4*x]]]^2), x] - 18*Defer[Int][x/((25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x])*Log[25 + 9
*x - 10*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]]^2), x] + 40*Defer[Int][x^2/((25
+ 9*x - 10*x^2 + x^4 + 4*Log[4*x])*Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4
+ 4*Log[4*x]]]^2), x] - 8*Defer[Int][x^4/((25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x])*Log[25 + 9*x - 10*x^2 + x^4 +
4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]]^2), x] + 2*Defer[Int][Log[Log[25 + 9*x - 10*x^2 +
x^4 + 4*Log[4*x]]]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (4+9 x-20 x^2+4 x^4\right )}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}+\frac {2}{\log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {4+9 x-20 x^2+4 x^4}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx\right )+2 \int \frac {1}{\log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx\\ &=-\left (2 \int \left (\frac {4}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}+\frac {9 x}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}-\frac {20 x^2}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}+\frac {4 x^4}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )}\right ) \, dx\right )+2 \int \frac {1}{\log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx\\ &=2 \int \frac {1}{\log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx-8 \int \frac {1}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx-8 \int \frac {x^4}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx-18 \int \frac {x}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx+40 \int \frac {x^2}{\left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right ) \log ^2\left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 x}{\log \left (\log \left (25+9 x-10 x^2+x^4+4 \log (4 x)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 - 18*x + 40*x^2 - 8*x^4 + (50 + 18*x - 20*x^2 + 2*x^4 + 8*Log[4*x])*Log[25 + 9*x - 10*x^2 + x^4
+ 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]])/((25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x])*Log[25 +
9*x - 10*x^2 + x^4 + 4*Log[4*x]]*Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]]^2),x]

[Out]

(2*x)/Log[Log[25 + 9*x - 10*x^2 + x^4 + 4*Log[4*x]]]

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fricas [A]  time = 0.50, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 \, x}{\log \left (\log \left (x^{4} - 10 \, x^{2} + 9 \, x + 4 \, \log \left (4 \, x\right ) + 25\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(4*x)+2*x^4-20*x^2+18*x+50)*log(4*log(4*x)+x^4-10*x^2+9*x+25)*log(log(4*log(4*x)+x^4-10*x^2+9
*x+25))-8*x^4+40*x^2-18*x-8)/(4*log(4*x)+x^4-10*x^2+9*x+25)/log(4*log(4*x)+x^4-10*x^2+9*x+25)/log(log(4*log(4*
x)+x^4-10*x^2+9*x+25))^2,x, algorithm="fricas")

[Out]

2*x/log(log(x^4 - 10*x^2 + 9*x + 4*log(4*x) + 25))

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giac [A]  time = 1.13, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 \, x}{\log \left (\log \left (x^{4} - 10 \, x^{2} + 9 \, x + 4 \, \log \left (4 \, x\right ) + 25\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(4*x)+2*x^4-20*x^2+18*x+50)*log(4*log(4*x)+x^4-10*x^2+9*x+25)*log(log(4*log(4*x)+x^4-10*x^2+9
*x+25))-8*x^4+40*x^2-18*x-8)/(4*log(4*x)+x^4-10*x^2+9*x+25)/log(4*log(4*x)+x^4-10*x^2+9*x+25)/log(log(4*log(4*
x)+x^4-10*x^2+9*x+25))^2,x, algorithm="giac")

[Out]

2*x/log(log(x^4 - 10*x^2 + 9*x + 4*log(4*x) + 25))

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maple [A]  time = 0.05, size = 27, normalized size = 1.04

method	result	size
risch	\(\frac {2 x}{\ln \left (\ln \left (4 \ln \left (4 x \right )+x^{4}-10 x^{2}+9 x +25\right )\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*ln(4*x)+2*x^4-20*x^2+18*x+50)*ln(4*ln(4*x)+x^4-10*x^2+9*x+25)*ln(ln(4*ln(4*x)+x^4-10*x^2+9*x+25))-8*x^
4+40*x^2-18*x-8)/(4*ln(4*x)+x^4-10*x^2+9*x+25)/ln(4*ln(4*x)+x^4-10*x^2+9*x+25)/ln(ln(4*ln(4*x)+x^4-10*x^2+9*x+
25))^2,x,method=_RETURNVERBOSE)

[Out]

2*x/ln(ln(4*ln(4*x)+x^4-10*x^2+9*x+25))

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maxima [A]  time = 0.68, size = 28, normalized size = 1.08 \begin {gather*} \frac {2 \, x}{\log \left (\log \left (x^{4} - 10 \, x^{2} + 9 \, x + 8 \, \log \relax (2) + 4 \, \log \relax (x) + 25\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(4*x)+2*x^4-20*x^2+18*x+50)*log(4*log(4*x)+x^4-10*x^2+9*x+25)*log(log(4*log(4*x)+x^4-10*x^2+9
*x+25))-8*x^4+40*x^2-18*x-8)/(4*log(4*x)+x^4-10*x^2+9*x+25)/log(4*log(4*x)+x^4-10*x^2+9*x+25)/log(log(4*log(4*
x)+x^4-10*x^2+9*x+25))^2,x, algorithm="maxima")

[Out]

2*x/log(log(x^4 - 10*x^2 + 9*x + 8*log(2) + 4*log(x) + 25))

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mupad [B]  time = 2.41, size = 26, normalized size = 1.00 \begin {gather*} \frac {2\,x}{\ln \left (\ln \left (9\,x+4\,\ln \left (4\,x\right )-10\,x^2+x^4+25\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*x - 40*x^2 + 8*x^4 - log(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25)*log(log(9*x + 4*log(4*x) - 10*x^2 + x^
4 + 25))*(18*x + 8*log(4*x) - 20*x^2 + 2*x^4 + 50) + 8)/(log(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25)*log(log(9*x
+ 4*log(4*x) - 10*x^2 + x^4 + 25))^2*(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25)),x)

[Out]

(2*x)/log(log(9*x + 4*log(4*x) - 10*x^2 + x^4 + 25))

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sympy [A]  time = 6.91, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 x}{\log {\left (\log {\left (x^{4} - 10 x^{2} + 9 x + 4 \log {\left (4 x \right )} + 25 \right )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*ln(4*x)+2*x**4-20*x**2+18*x+50)*ln(4*ln(4*x)+x**4-10*x**2+9*x+25)*ln(ln(4*ln(4*x)+x**4-10*x**2+9
*x+25))-8*x**4+40*x**2-18*x-8)/(4*ln(4*x)+x**4-10*x**2+9*x+25)/ln(4*ln(4*x)+x**4-10*x**2+9*x+25)/ln(ln(4*ln(4*
x)+x**4-10*x**2+9*x+25))**2,x)

[Out]

2*x/log(log(x**4 - 10*x**2 + 9*x + 4*log(4*x) + 25))

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