3.59.21 \(\int \frac {(-4-2 x) \log ^2(8 x+4 x^2)+(4+4 x) \log (x) \log (8 x+4 x^2) \log (\log (x))+((2 x+x^2) \log (x)+(2 x+x^2) \log ^2(x)) \log ^3(\log (x))}{(2 x+x^2) \log (x) \log ^3(\log (x))} \, dx\)

Optimal. Leaf size=26 \[ 4+\log \left (3 e^{\frac {\log ^2(4 x (2+x))}{\log ^2(\log (x))}}\right )+x \log (x) \]

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Rubi [F]  time = 1.42, 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 {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-4 - 2*x)*Log[8*x + 4*x^2]^2 + (4 + 4*x)*Log[x]*Log[8*x + 4*x^2]*Log[Log[x]] + ((2*x + x^2)*Log[x] + (2*
x + x^2)*Log[x]^2)*Log[Log[x]]^3)/((2*x + x^2)*Log[x]*Log[Log[x]]^3),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{x (2+x) \log (x) \log ^3(\log (x))} \, dx\\ &=\int \left (1+\log (x)-\frac {2 \log ^2(4 x (2+x))}{x \log (x) \log ^3(\log (x))}+\frac {4 (1+x) \log (4 x (2+x))}{x (2+x) \log ^2(\log (x))}\right ) \, dx\\ &=x-2 \int \frac {\log ^2(4 x (2+x))}{x \log (x) \log ^3(\log (x))} \, dx+4 \int \frac {(1+x) \log (4 x (2+x))}{x (2+x) \log ^2(\log (x))} \, dx+\int \log (x) \, dx\\ &=x \log (x)-2 \int \frac {\log ^2(4 x (2+x))}{x \log (x) \log ^3(\log (x))} \, dx+4 \int \left (\frac {\log (4 x (2+x))}{2 x \log ^2(\log (x))}+\frac {\log (4 x (2+x))}{2 (2+x) \log ^2(\log (x))}\right ) \, dx\\ &=x \log (x)-2 \int \frac {\log ^2(4 x (2+x))}{x \log (x) \log ^3(\log (x))} \, dx+2 \int \frac {\log (4 x (2+x))}{x \log ^2(\log (x))} \, dx+2 \int \frac {\log (4 x (2+x))}{(2+x) \log ^2(\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 20, normalized size = 0.77 \begin {gather*} x \log (x)+\frac {\log ^2(4 x (2+x))}{\log ^2(\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4 - 2*x)*Log[8*x + 4*x^2]^2 + (4 + 4*x)*Log[x]*Log[8*x + 4*x^2]*Log[Log[x]] + ((2*x + x^2)*Log[x]
+ (2*x + x^2)*Log[x]^2)*Log[Log[x]]^3)/((2*x + x^2)*Log[x]*Log[Log[x]]^3),x]

[Out]

x*Log[x] + Log[4*x*(2 + x)]^2/Log[Log[x]]^2

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fricas [A]  time = 0.81, size = 28, normalized size = 1.08 \begin {gather*} \frac {x \log \relax (x) \log \left (\log \relax (x)\right )^{2} + \log \left (4 \, x^{2} + 8 \, x\right )^{2}}{\log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+2*x)*log(x)^2+(x^2+2*x)*log(x))*log(log(x))^3+(4*x+4)*log(x)*log(4*x^2+8*x)*log(log(x))+(-2*x
-4)*log(4*x^2+8*x)^2)/(x^2+2*x)/log(x)/log(log(x))^3,x, algorithm="fricas")

[Out]

(x*log(x)*log(log(x))^2 + log(4*x^2 + 8*x)^2)/log(log(x))^2

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giac [A]  time = 0.30, size = 44, normalized size = 1.69 \begin {gather*} x \log \relax (x) + \frac {\log \left (4 \, x + 8\right )^{2}}{\log \left (\log \relax (x)\right )^{2}} + \frac {2 \, \log \left (4 \, x + 8\right ) \log \relax (x)}{\log \left (\log \relax (x)\right )^{2}} + \frac {\log \relax (x)^{2}}{\log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+2*x)*log(x)^2+(x^2+2*x)*log(x))*log(log(x))^3+(4*x+4)*log(x)*log(4*x^2+8*x)*log(log(x))+(-2*x
-4)*log(4*x^2+8*x)^2)/(x^2+2*x)/log(x)/log(log(x))^3,x, algorithm="giac")

[Out]

x*log(x) + log(4*x + 8)^2/log(log(x))^2 + 2*log(4*x + 8)*log(x)/log(log(x))^2 + log(x)^2/log(log(x))^2

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maple [C]  time = 0.42, size = 541, normalized size = 20.81

method	result	size
risch	\(x \ln \relax (x )+\frac {16 \ln \relax (2)^{2}+4 \ln \relax (x )^{2}+16 \ln \relax (2) \ln \relax (x )-8 i \ln \relax (2) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{3}-4 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{3}-4 i \ln \left (2+x \right ) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )-\pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (2+x \right )\right )+2 \pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{3} \mathrm {csgn}\left (i \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i x \right )+4 \ln \left (2+x \right )^{2}+8 \ln \relax (x ) \ln \left (2+x \right )-\pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{4} \mathrm {csgn}\left (i \left (2+x \right )\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{5} \mathrm {csgn}\left (i x \right )+2 \pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{5} \mathrm {csgn}\left (i \left (2+x \right )\right )-\pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{4} \mathrm {csgn}\left (i x \right )^{2}+16 \ln \relax (2) \ln \left (2+x \right )-\pi ^{2} \mathrm {csgn}\left (i x \left (2+x \right )\right )^{6}-4 i \ln \left (2+x \right ) \pi \,\mathrm {csgn}\left (i x \left (2+x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (2+x \right )\right )-4 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \left (2+x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (2+x \right )\right )-8 i \ln \relax (2) \pi \,\mathrm {csgn}\left (i x \left (2+x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (2+x \right )\right )+4 i \ln \left (2+x \right ) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i x \right )+4 i \ln \left (2+x \right ) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )+8 i \ln \relax (2) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i x \right )+8 i \ln \relax (2) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )+4 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i x \right )+4 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )}{4 \ln \left (\ln \relax (x )\right )^{2}}\)	\(541\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2+2*x)*ln(x)^2+(x^2+2*x)*ln(x))*ln(ln(x))^3+(4*x+4)*ln(x)*ln(4*x^2+8*x)*ln(ln(x))+(-2*x-4)*ln(4*x^2+8
*x)^2)/(x^2+2*x)/ln(x)/ln(ln(x))^3,x,method=_RETURNVERBOSE)

[Out]

x*ln(x)+1/4*(16*ln(2)^2+4*ln(x)^2+16*ln(2)*ln(x)-8*I*ln(2)*Pi*csgn(I*x*(2+x))^3-4*I*ln(x)*Pi*csgn(I*x*(2+x))^3
-4*I*ln(2+x)*Pi*csgn(I*x*(2+x))^3+2*Pi^2*csgn(I*x*(2+x))^3*csgn(I*x)^2*csgn(I*(2+x))+2*Pi^2*csgn(I*x*(2+x))^3*
csgn(I*(2+x))^2*csgn(I*x)-Pi^2*csgn(I*x*(2+x))^2*csgn(I*x)^2*csgn(I*(2+x))^2+4*ln(2+x)^2+4*I*ln(2+x)*Pi*csgn(I
*x*(2+x))^2*csgn(I*x)+4*I*ln(2+x)*Pi*csgn(I*x*(2+x))^2*csgn(I*(2+x))+8*ln(x)*ln(2+x)-Pi^2*csgn(I*x*(2+x))^4*cs
gn(I*(2+x))^2-Pi^2*csgn(I*x*(2+x))^6+16*ln(2)*ln(2+x)-8*I*ln(2)*Pi*csgn(I*x*(2+x))*csgn(I*x)*csgn(I*(2+x))-4*I
*ln(2+x)*Pi*csgn(I*x*(2+x))*csgn(I*x)*csgn(I*(2+x))-4*I*ln(x)*Pi*csgn(I*x*(2+x))*csgn(I*x)*csgn(I*(2+x))+2*Pi^
2*csgn(I*x*(2+x))^5*csgn(I*x)+2*Pi^2*csgn(I*x*(2+x))^5*csgn(I*(2+x))-Pi^2*csgn(I*x*(2+x))^4*csgn(I*x)^2-4*Pi^2
*csgn(I*x*(2+x))^4*csgn(I*x)*csgn(I*(2+x))+8*I*ln(2)*Pi*csgn(I*x*(2+x))^2*csgn(I*x)+8*I*ln(2)*Pi*csgn(I*x*(2+x
))^2*csgn(I*(2+x))+4*I*ln(x)*Pi*csgn(I*x*(2+x))^2*csgn(I*x)+4*I*ln(x)*Pi*csgn(I*x*(2+x))^2*csgn(I*(2+x)))/ln(l
n(x))^2

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maxima [B]  time = 0.51, size = 51, normalized size = 1.96 \begin {gather*} \frac {x \log \relax (x) \log \left (\log \relax (x)\right )^{2} + 4 \, \log \relax (2)^{2} + 2 \, {\left (2 \, \log \relax (2) + \log \relax (x)\right )} \log \left (x + 2\right ) + \log \left (x + 2\right )^{2} + 4 \, \log \relax (2) \log \relax (x) + \log \relax (x)^{2}}{\log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+2*x)*log(x)^2+(x^2+2*x)*log(x))*log(log(x))^3+(4*x+4)*log(x)*log(4*x^2+8*x)*log(log(x))+(-2*x
-4)*log(4*x^2+8*x)^2)/(x^2+2*x)/log(x)/log(log(x))^3,x, algorithm="maxima")

[Out]

(x*log(x)*log(log(x))^2 + 4*log(2)^2 + 2*(2*log(2) + log(x))*log(x + 2) + log(x + 2)^2 + 4*log(2)*log(x) + log
(x)^2)/log(log(x))^2

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mupad [B]  time = 4.64, size = 190, normalized size = 7.31 \begin {gather*} 2\,\ln \left (4\,x^2+8\,x\right )\,\ln \relax (x)-\frac {16\,{\ln \relax (x)}^2}{x^2+4\,x+4}+4\,{\ln \relax (x)}^2+\frac {{\ln \left (4\,x^2+8\,x\right )}^2}{{\ln \left (\ln \relax (x)\right )}^2}+x\,\ln \relax (x)-\frac {2\,\ln \left (4\,x^2+8\,x\right )\,\ln \relax (x)}{x+2}-\frac {16\,x\,{\ln \relax (x)}^2}{x^2+4\,x+4}-\frac {4\,\ln \left (4\,x^2+8\,x\right )\,\ln \relax (x)}{x^2+4\,x+4}-\frac {4\,x^2\,{\ln \relax (x)}^2}{x^2+4\,x+4}-\frac {6\,x\,\ln \left (4\,x^2+8\,x\right )\,\ln \relax (x)}{x^2+4\,x+4}-\frac {2\,x^2\,\ln \left (4\,x^2+8\,x\right )\,\ln \relax (x)}{x^2+4\,x+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x))^3*(log(x)*(2*x + x^2) + log(x)^2*(2*x + x^2)) - log(8*x + 4*x^2)^2*(2*x + 4) + log(log(x))*lo
g(8*x + 4*x^2)*log(x)*(4*x + 4))/(log(log(x))^3*log(x)*(2*x + x^2)),x)

[Out]

2*log(8*x + 4*x^2)*log(x) - (16*log(x)^2)/(4*x + x^2 + 4) + 4*log(x)^2 + log(8*x + 4*x^2)^2/log(log(x))^2 + x*
log(x) - (2*log(8*x + 4*x^2)*log(x))/(x + 2) - (16*x*log(x)^2)/(4*x + x^2 + 4) - (4*log(8*x + 4*x^2)*log(x))/(
4*x + x^2 + 4) - (4*x^2*log(x)^2)/(4*x + x^2 + 4) - (6*x*log(8*x + 4*x^2)*log(x))/(4*x + x^2 + 4) - (2*x^2*log
(8*x + 4*x^2)*log(x))/(4*x + x^2 + 4)

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sympy [A]  time = 0.34, size = 22, normalized size = 0.85 \begin {gather*} x \log {\relax (x )} + \frac {\log {\left (4 x^{2} + 8 x \right )}^{2}}{\log {\left (\log {\relax (x )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2+2*x)*ln(x)**2+(x**2+2*x)*ln(x))*ln(ln(x))**3+(4*x+4)*ln(x)*ln(4*x**2+8*x)*ln(ln(x))+(-2*x-4)
*ln(4*x**2+8*x)**2)/(x**2+2*x)/ln(x)/ln(ln(x))**3,x)

[Out]

x*log(x) + log(4*x**2 + 8*x)**2/log(log(x))**2

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