3.21.100 \(\int \frac {4+8 x+(8 x+4 x^2) \log (x)+(-2+(-2-2 x) \log (x)) \log (\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})}{5 x^4+(5 x^4+5 x^5) \log (x)+(-10 x^3+(-10 x^3-10 x^4) \log (x)) \log (\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})+(5 x^2+(5 x^2+5 x^3) \log (x)) \log ^2(\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})} \, dx\)

Optimal. Leaf size=31 \[ 5-\frac {2}{5 x \left (x-\log \left (\frac {x^2}{(x+x (1+x) \log (x))^2}\right )\right )} \]

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Rubi [F]  time = 4.41, 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+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + 8*x + (8*x + 4*x^2)*Log[x] + (-2 + (-2 - 2*x)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] + (1 + 2*x + x^2)*Log
[x]^2)^(-1)])/(5*x^4 + (5*x^4 + 5*x^5)*Log[x] + (-10*x^3 + (-10*x^3 - 10*x^4)*Log[x])*Log[(1 + (2 + 2*x)*Log[x
] + (1 + 2*x + x^2)*Log[x]^2)^(-1)] + (5*x^2 + (5*x^2 + 5*x^3)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] + (1 + 2*x +
x^2)*Log[x]^2)^(-1)]^2),x]

[Out]

(4*Defer[Int][1/(x^2*(1 + Log[x] + x*Log[x])*(x - Log[(1 + (1 + x)*Log[x])^(-2)])^2), x])/5 + (6*Defer[Int][1/
(x*(1 + Log[x] + x*Log[x])*(x - Log[(1 + (1 + x)*Log[x])^(-2)])^2), x])/5 + (2*Defer[Int][Log[x]/((1 + Log[x]
+ x*Log[x])*(x - Log[(1 + (1 + x)*Log[x])^(-2)])^2), x])/5 + (6*Defer[Int][Log[x]/(x*(1 + Log[x] + x*Log[x])*(
x - Log[(1 + (1 + x)*Log[x])^(-2)])^2), x])/5 + (2*Defer[Int][1/(x^2*(1 + Log[x] + x*Log[x])*(x - Log[(1 + (1
+ x)*Log[x])^(-2)])), x])/5 + (2*Defer[Int][Log[x]/(x^2*(1 + Log[x] + x*Log[x])*(x - Log[(1 + (1 + x)*Log[x])^
(-2)])), x])/5 + (2*Defer[Int][Log[x]/(x*(1 + Log[x] + x*Log[x])*(x - Log[(1 + (1 + x)*Log[x])^(-2)])), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+8 x-2 \log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )+\log (x) \left (4 x (2+x)-2 (1+x) \log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )\right )}{5 x^2 (1+(1+x) \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {4+8 x-2 \log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )+\log (x) \left (4 x (2+x)-2 (1+x) \log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )\right )}{x^2 (1+(1+x) \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {4}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}+\frac {8}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}+\frac {4 \log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}+\frac {8 \log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {2 \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {2 \log (x) \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {2 \log (x) \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}\right ) \, dx\\ &=-\left (\frac {2}{5} \int \frac {\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\right )-\frac {2}{5} \int \frac {\log (x) \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx-\frac {2}{5} \int \frac {\log (x) \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {4}{5} \int \frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {4}{5} \int \frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ &=-\left (\frac {2}{5} \int \left (\frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )}\right ) \, dx\right )-\frac {2}{5} \int \left (\frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {\log (x)}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )}\right ) \, dx-\frac {2}{5} \int \left (\frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )}\right ) \, dx+\frac {4}{5} \int \frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {4}{5} \int \frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ &=-\left (\frac {2}{5} \int \frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\right )-\frac {2}{5} \int \frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx-\frac {2}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {2}{5} \int \frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )} \, dx+\frac {2}{5} \int \frac {\log (x)}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )} \, dx+\frac {2}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )} \, dx+\frac {4}{5} \int \frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {4}{5} \int \frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.06, size = 24, normalized size = 0.77 \begin {gather*} \frac {2}{5 x \left (-x+\log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 8*x + (8*x + 4*x^2)*Log[x] + (-2 + (-2 - 2*x)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] + (1 + 2*x + x^
2)*Log[x]^2)^(-1)])/(5*x^4 + (5*x^4 + 5*x^5)*Log[x] + (-10*x^3 + (-10*x^3 - 10*x^4)*Log[x])*Log[(1 + (2 + 2*x)
*Log[x] + (1 + 2*x + x^2)*Log[x]^2)^(-1)] + (5*x^2 + (5*x^2 + 5*x^3)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] + (1 +
2*x + x^2)*Log[x]^2)^(-1)]^2),x]

[Out]

2/(5*x*(-x + Log[(1 + Log[x] + x*Log[x])^(-2)]))

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fricas [A]  time = 0.74, size = 36, normalized size = 1.16 \begin {gather*} -\frac {2}{5 \, {\left (x^{2} - x \log \left (\frac {1}{{\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} + 2 \, {\left (x + 1\right )} \log \relax (x) + 1}\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-2)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x
^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(
1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, algorithm="fricas")

[Out]

-2/5/(x^2 - x*log(1/((x^2 + 2*x + 1)*log(x)^2 + 2*(x + 1)*log(x) + 1)))

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giac [A]  time = 4.08, size = 41, normalized size = 1.32 \begin {gather*} -\frac {2}{5 \, {\left (x^{2} + x \log \left (x^{2} \log \relax (x)^{2} + 2 \, x \log \relax (x)^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-2)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x
^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(
1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, algorithm="giac")

[Out]

-2/5/(x^2 + x*log(x^2*log(x)^2 + 2*x*log(x)^2 + 2*x*log(x) + log(x)^2 + 2*log(x) + 1))

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maple [C]  time = 0.19, size = 105, normalized size = 3.39

method	result	size
risch	\(-\frac {4 i}{5 x \left (\pi \mathrm {csgn}\left (i \left (x \ln \relax (x )+\ln \relax (x )+1\right )\right )^{2} \mathrm {csgn}\left (i \left (x \ln \relax (x )+\ln \relax (x )+1\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (x \ln \relax (x )+\ln \relax (x )+1\right )\right ) \mathrm {csgn}\left (i \left (x \ln \relax (x )+\ln \relax (x )+1\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (x \ln \relax (x )+\ln \relax (x )+1\right )^{2}\right )^{3}+2 i x +4 i \ln \left (x \ln \relax (x )+\ln \relax (x )+1\right )\right )}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x-2)*ln(x)-2)*ln(1/((x^2+2*x+1)*ln(x)^2+(2*x+2)*ln(x)+1))+(4*x^2+8*x)*ln(x)+8*x+4)/(((5*x^3+5*x^2)*l
n(x)+5*x^2)*ln(1/((x^2+2*x+1)*ln(x)^2+(2*x+2)*ln(x)+1))^2+((-10*x^4-10*x^3)*ln(x)-10*x^3)*ln(1/((x^2+2*x+1)*ln
(x)^2+(2*x+2)*ln(x)+1))+(5*x^5+5*x^4)*ln(x)+5*x^4),x,method=_RETURNVERBOSE)

[Out]

-4/5*I/x/(Pi*csgn(I*(x*ln(x)+ln(x)+1))^2*csgn(I*(x*ln(x)+ln(x)+1)^2)-2*Pi*csgn(I*(x*ln(x)+ln(x)+1))*csgn(I*(x*
ln(x)+ln(x)+1)^2)^2+Pi*csgn(I*(x*ln(x)+ln(x)+1)^2)^3+2*I*x+4*I*ln(x*ln(x)+ln(x)+1))

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maxima [A]  time = 0.85, size = 20, normalized size = 0.65 \begin {gather*} -\frac {2}{5 \, {\left (x^{2} + 2 \, x \log \left ({\left (x + 1\right )} \log \relax (x) + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-2)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x
^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(
1/((x^2+2*x+1)*log(x)^2+(2*x+2)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, algorithm="maxima")

[Out]

-2/5/(x^2 + 2*x*log((x + 1)*log(x) + 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {8\,x-\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \relax (x)}^2+\left (2\,x+2\right )\,\ln \relax (x)+1}\right )\,\left (\ln \relax (x)\,\left (2\,x+2\right )+2\right )+\ln \relax (x)\,\left (4\,x^2+8\,x\right )+4}{\ln \relax (x)\,\left (5\,x^5+5\,x^4\right )-\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \relax (x)}^2+\left (2\,x+2\right )\,\ln \relax (x)+1}\right )\,\left (\ln \relax (x)\,\left (10\,x^4+10\,x^3\right )+10\,x^3\right )+{\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \relax (x)}^2+\left (2\,x+2\right )\,\ln \relax (x)+1}\right )}^2\,\left (\ln \relax (x)\,\left (5\,x^3+5\,x^2\right )+5\,x^2\right )+5\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(2*x + 2) + 2) + log(x)*(8*x + 4*x
^2) + 4)/(log(x)*(5*x^4 + 5*x^5) - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(10*x^3 +
10*x^4) + 10*x^3) + log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))^2*(log(x)*(5*x^2 + 5*x^3) + 5*x^2
) + 5*x^4),x)

[Out]

int((8*x - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(2*x + 2) + 2) + log(x)*(8*x + 4*x
^2) + 4)/(log(x)*(5*x^4 + 5*x^5) - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(10*x^3 +
10*x^4) + 10*x^3) + log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))^2*(log(x)*(5*x^2 + 5*x^3) + 5*x^2
) + 5*x^4), x)

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sympy [A]  time = 0.43, size = 36, normalized size = 1.16 \begin {gather*} \frac {2}{- 5 x^{2} + 5 x \log {\left (\frac {1}{\left (2 x + 2\right ) \log {\relax (x )} + \left (x^{2} + 2 x + 1\right ) \log {\relax (x )}^{2} + 1} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-2)*ln(x)-2)*ln(1/((x**2+2*x+1)*ln(x)**2+(2*x+2)*ln(x)+1))+(4*x**2+8*x)*ln(x)+8*x+4)/(((5*x**
3+5*x**2)*ln(x)+5*x**2)*ln(1/((x**2+2*x+1)*ln(x)**2+(2*x+2)*ln(x)+1))**2+((-10*x**4-10*x**3)*ln(x)-10*x**3)*ln
(1/((x**2+2*x+1)*ln(x)**2+(2*x+2)*ln(x)+1))+(5*x**5+5*x**4)*ln(x)+5*x**4),x)

[Out]

2/(-5*x**2 + 5*x*log(1/((2*x + 2)*log(x) + (x**2 + 2*x + 1)*log(x)**2 + 1)))

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