3.74.50 \(\int \frac {((-320-16 x-1280 x^2-64 x^3) \log (\frac {20+x}{4})+(1280 x^2+64 x^3) \log (x^2) \log (\frac {20+x}{4})) \log (\frac {16+64 x^2}{\log (x^2)})+(-4 x-16 x^3) \log (x^2) \log ^2(\frac {16+64 x^2}{\log (x^2)})}{(20 x+x^2+80 x^3+4 x^4) \log (x^2) \log ^2(\frac {20+x}{4})} \, dx\)

Optimal. Leaf size=30 \[ \frac {4 \log ^2\left (\frac {4 \left (4+16 x^2\right )}{\log \left (x^2\right )}\right )}{\log \left (5+\frac {x}{4}\right )} \]

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Rubi [F]  time = 8.64, 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 {\left (\left (-320-16 x-1280 x^2-64 x^3\right ) \log \left (\frac {20+x}{4}\right )+\left (1280 x^2+64 x^3\right ) \log \left (x^2\right ) \log \left (\frac {20+x}{4}\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )+\left (-4 x-16 x^3\right ) \log \left (x^2\right ) \log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (20 x+x^2+80 x^3+4 x^4\right ) \log \left (x^2\right ) \log ^2\left (\frac {20+x}{4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(((-320 - 16*x - 1280*x^2 - 64*x^3)*Log[(20 + x)/4] + (1280*x^2 + 64*x^3)*Log[x^2]*Log[(20 + x)/4])*Log[(1
6 + 64*x^2)/Log[x^2]] + (-4*x - 16*x^3)*Log[x^2]*Log[(16 + 64*x^2)/Log[x^2]]^2)/((20*x + x^2 + 80*x^3 + 4*x^4)
*Log[x^2]*Log[(20 + x)/4]^2),x]

[Out]

-16*Defer[Int][Log[(16 + 64*x^2)/Log[x^2]]/((I - 2*x)*Log[5 + x/4]), x] + 16*Defer[Int][Log[(16 + 64*x^2)/Log[
x^2]]/((I + 2*x)*Log[5 + x/4]), x] - 16*Defer[Int][Log[(16 + 64*x^2)/Log[x^2]]/(x*Log[5 + x/4]*Log[x^2]), x] -
4*Defer[Int][Log[(16 + 64*x^2)/Log[x^2]]^2/((20 + x)*Log[5 + x/4]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (\left (-320-16 x-1280 x^2-64 x^3\right ) \log \left (\frac {20+x}{4}\right )+\left (1280 x^2+64 x^3\right ) \log \left (x^2\right ) \log \left (\frac {20+x}{4}\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )+\left (-4 x-16 x^3\right ) \log \left (x^2\right ) \log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \left (20+x+80 x^2+4 x^3\right ) \log ^2\left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx\\ &=\int \left (\frac {16 \left (-1-4 x^2+4 x^2 \log \left (x^2\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}-\frac {4 \log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )}\right ) \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )+16 \int \frac {\left (-1-4 x^2+4 x^2 \log \left (x^2\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )+16 \int \left (\frac {\left (-1-4 x^2+4 x^2 \log \left (x^2\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}-\frac {4 x \left (-1-4 x^2+4 x^2 \log \left (x^2\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}\right ) \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )+16 \int \frac {\left (-1-4 x^2+4 x^2 \log \left (x^2\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx-64 \int \frac {x \left (-1-4 x^2+4 x^2 \log \left (x^2\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )+16 \int \left (\frac {4 x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\log \left (5+\frac {x}{4}\right )}-\frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}-\frac {4 x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}\right ) \, dx-64 \int \left (\frac {4 x^3 \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right )}-\frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}-\frac {4 x^3 \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}\right ) \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )-16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx+64 \int \frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\log \left (5+\frac {x}{4}\right )} \, dx-64 \int \frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx+64 \int \frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx-256 \int \frac {x^3 \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right )} \, dx+256 \int \frac {x^3 \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )-16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx+64 \int \frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\log \left (5+\frac {x}{4}\right )} \, dx-64 \int \frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx+64 \int \left (-\frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 (i-2 x) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}+\frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 (i+2 x) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}\right ) \, dx-256 \int \left (\frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 \log \left (5+\frac {x}{4}\right )}-\frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 \left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right )}\right ) \, dx+256 \int \left (\frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}-\frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 \left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}\right ) \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )-16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(i-2 x) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx-16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx+16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(i+2 x) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx+64 \int \frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right )} \, dx-64 \int \frac {x \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (1+4 x^2\right ) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )-16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(i-2 x) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx-16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx+16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(i+2 x) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx+64 \int \left (-\frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 (i-2 x) \log \left (5+\frac {x}{4}\right )}+\frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 (i+2 x) \log \left (5+\frac {x}{4}\right )}\right ) \, dx-64 \int \left (-\frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 (i-2 x) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}+\frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{4 (i+2 x) \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )}\right ) \, dx\\ &=-\left (4 \int \frac {\log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(20+x) \log ^2\left (5+\frac {x}{4}\right )} \, dx\right )-16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(i-2 x) \log \left (5+\frac {x}{4}\right )} \, dx+16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{(i+2 x) \log \left (5+\frac {x}{4}\right )} \, dx-16 \int \frac {\log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{x \log \left (5+\frac {x}{4}\right ) \log \left (x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (-320-16 x-1280 x^2-64 x^3\right ) \log \left (\frac {20+x}{4}\right )+\left (1280 x^2+64 x^3\right ) \log \left (x^2\right ) \log \left (\frac {20+x}{4}\right )\right ) \log \left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )+\left (-4 x-16 x^3\right ) \log \left (x^2\right ) \log ^2\left (\frac {16+64 x^2}{\log \left (x^2\right )}\right )}{\left (20 x+x^2+80 x^3+4 x^4\right ) \log \left (x^2\right ) \log ^2\left (\frac {20+x}{4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(((-320 - 16*x - 1280*x^2 - 64*x^3)*Log[(20 + x)/4] + (1280*x^2 + 64*x^3)*Log[x^2]*Log[(20 + x)/4])*
Log[(16 + 64*x^2)/Log[x^2]] + (-4*x - 16*x^3)*Log[x^2]*Log[(16 + 64*x^2)/Log[x^2]]^2)/((20*x + x^2 + 80*x^3 +
4*x^4)*Log[x^2]*Log[(20 + x)/4]^2),x]

[Out]

Integrate[(((-320 - 16*x - 1280*x^2 - 64*x^3)*Log[(20 + x)/4] + (1280*x^2 + 64*x^3)*Log[x^2]*Log[(20 + x)/4])*
Log[(16 + 64*x^2)/Log[x^2]] + (-4*x - 16*x^3)*Log[x^2]*Log[(16 + 64*x^2)/Log[x^2]]^2)/((20*x + x^2 + 80*x^3 +
4*x^4)*Log[x^2]*Log[(20 + x)/4]^2), x]

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fricas [A]  time = 0.68, size = 28, normalized size = 0.93 \begin {gather*} \frac {4 \, \log \left (\frac {16 \, {\left (4 \, x^{2} + 1\right )}}{\log \left (x^{2}\right )}\right )^{2}}{\log \left (\frac {1}{4} \, x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3-4*x)*log(x^2)*log((64*x^2+16)/log(x^2))^2+((64*x^3+1280*x^2)*log(5+1/4*x)*log(x^2)+(-64*x^
3-1280*x^2-16*x-320)*log(5+1/4*x))*log((64*x^2+16)/log(x^2)))/(4*x^4+80*x^3+x^2+20*x)/log(5+1/4*x)^2/log(x^2),
x, algorithm="fricas")

[Out]

4*log(16*(4*x^2 + 1)/log(x^2))^2/log(1/4*x + 5)

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giac [B]  time = 0.83, size = 137, normalized size = 4.57 \begin {gather*} -8 \, {\left (\frac {4 \, \log \relax (2)}{2 \, \log \relax (2) - \log \left (x + 20\right )} - \frac {\log \left (\log \left (x^{2}\right )\right )}{2 \, \log \relax (2) - \log \left (x + 20\right )}\right )} \log \left (4 \, x^{2} + 1\right ) - \frac {64 \, \log \relax (2)^{2}}{2 \, \log \relax (2) - \log \left (x + 20\right )} - \frac {4 \, \log \left (4 \, x^{2} + 1\right )^{2}}{2 \, \log \relax (2) - \log \left (x + 20\right )} + \frac {32 \, \log \relax (2) \log \left (\log \left (x^{2}\right )\right )}{2 \, \log \relax (2) - \log \left (x + 20\right )} - \frac {4 \, \log \left (\log \left (x^{2}\right )\right )^{2}}{2 \, \log \relax (2) - \log \left (x + 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3-4*x)*log(x^2)*log((64*x^2+16)/log(x^2))^2+((64*x^3+1280*x^2)*log(5+1/4*x)*log(x^2)+(-64*x^
3-1280*x^2-16*x-320)*log(5+1/4*x))*log((64*x^2+16)/log(x^2)))/(4*x^4+80*x^3+x^2+20*x)/log(5+1/4*x)^2/log(x^2),
x, algorithm="giac")

[Out]

-8*(4*log(2)/(2*log(2) - log(x + 20)) - log(log(x^2))/(2*log(2) - log(x + 20)))*log(4*x^2 + 1) - 64*log(2)^2/(
2*log(2) - log(x + 20)) - 4*log(4*x^2 + 1)^2/(2*log(2) - log(x + 20)) + 32*log(2)*log(log(x^2))/(2*log(2) - lo
g(x + 20)) - 4*log(log(x^2))^2/(2*log(2) - log(x + 20))

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maple [C]  time = 1.00, size = 7928, normalized size = 264.27

method	result	size
risch	\(\text {Expression too large to display}\)	\(7928\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^3-4*x)*ln(x^2)*ln((64*x^2+16)/ln(x^2))^2+((64*x^3+1280*x^2)*ln(5+1/4*x)*ln(x^2)+(-64*x^3-1280*x^2-
16*x-320)*ln(5+1/4*x))*ln((64*x^2+16)/ln(x^2)))/(4*x^4+80*x^3+x^2+20*x)/ln(5+1/4*x)^2/ln(x^2),x,method=_RETURN
VERBOSE)

[Out]

result too large to display

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maxima [B]  time = 0.56, size = 64, normalized size = 2.13 \begin {gather*} -\frac {4 \, {\left (9 \, \log \relax (2)^{2} + 2 \, {\left (3 \, \log \relax (2) - \log \left (\log \relax (x)\right )\right )} \log \left (4 \, x^{2} + 1\right ) + \log \left (4 \, x^{2} + 1\right )^{2} - 6 \, \log \relax (2) \log \left (\log \relax (x)\right ) + \log \left (\log \relax (x)\right )^{2}\right )}}{2 \, \log \relax (2) - \log \left (x + 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^3-4*x)*log(x^2)*log((64*x^2+16)/log(x^2))^2+((64*x^3+1280*x^2)*log(5+1/4*x)*log(x^2)+(-64*x^
3-1280*x^2-16*x-320)*log(5+1/4*x))*log((64*x^2+16)/log(x^2)))/(4*x^4+80*x^3+x^2+20*x)/log(5+1/4*x)^2/log(x^2),
x, algorithm="maxima")

[Out]

-4*(9*log(2)^2 + 2*(3*log(2) - log(log(x)))*log(4*x^2 + 1) + log(4*x^2 + 1)^2 - 6*log(2)*log(log(x)) + log(log
(x))^2)/(2*log(2) - log(x + 20))

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mupad [B]  time = 5.21, size = 28, normalized size = 0.93 \begin {gather*} \frac {4\,{\ln \left (\frac {16\,\left (4\,x^2+1\right )}{\ln \left (x^2\right )}\right )}^2}{\ln \left (\frac {x}{4}+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((64*x^2 + 16)/log(x^2))*(log(x/4 + 5)*(16*x + 1280*x^2 + 64*x^3 + 320) - log(x^2)*log(x/4 + 5)*(1280
*x^2 + 64*x^3)) + log((64*x^2 + 16)/log(x^2))^2*log(x^2)*(4*x + 16*x^3))/(log(x^2)*log(x/4 + 5)^2*(20*x + x^2
+ 80*x^3 + 4*x^4)),x)

[Out]

(4*log((16*(4*x^2 + 1))/log(x^2))^2)/log(x/4 + 5)

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sympy [A]  time = 0.84, size = 22, normalized size = 0.73 \begin {gather*} \frac {4 \log {\left (\frac {64 x^{2} + 16}{\log {\left (x^{2} \right )}} \right )}^{2}}{\log {\left (\frac {x}{4} + 5 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**3-4*x)*ln(x**2)*ln((64*x**2+16)/ln(x**2))**2+((64*x**3+1280*x**2)*ln(5+1/4*x)*ln(x**2)+(-64
*x**3-1280*x**2-16*x-320)*ln(5+1/4*x))*ln((64*x**2+16)/ln(x**2)))/(4*x**4+80*x**3+x**2+20*x)/ln(5+1/4*x)**2/ln
(x**2),x)

[Out]

4*log((64*x**2 + 16)/log(x**2))**2/log(x/4 + 5)

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